Particle Physics Planet


January 24, 2018

The n-Category Cafe

FreeTikZ

Guest post by Chris Heunen

I don’t have to tell you, dear <semantics>n<annotation encoding="application/x-tex">n</annotation></semantics>-Category Café reader, that string diagrams are extremely useful. They speed up computations massively, reveal what makes a proof tick without an impenetrable forest of details, and suggest properties that you might not even have thought about in algebraic notation. They also make your paper friendlier to read.

However, string diagrams are also a pain to typeset. First, there is the entrance fee of learning a modern LaTeX drawing package, like TikZ. Then, there is the learning period of setting up macros tailored to string diagrams and internalizing them in your muscle memory. But even after all that investment, it still takes a lot of time. And unlike that glorious moment when you realise that you have cycled about twice the circumference of the Earth in your life, this time is mostly wasted. I estimate I’ve wasted over 2000 string diagrams’ worth of time by now.

Wouldn’t it be great if you could simply draw your diagram by hand, and have it magically converted into TikZ? Now you can!

Meet FreeTikZ. It uses the magic of SVG, HTML, and Javascript, to turn

into the following LaTeX code:

\begin{tikzpicture}
  \node[dot] (d0) at (4.25, 5.5) {};
  \node[dot] (d1) at (6.25, 3.8) {};
  \node[morphism] (m0) at (7.5, 5.3) {};
  \draw (d0.center) to[out=-90, in=180] (d1.center);
  \draw (d1.center) to[out=60, in=-90] (m0.south west);
  \draw (m0.south east) to[out=-60, in=90] (8.25, 2);
  \draw (m0.north) to[out=90, in=-90] (7.5, 8.8);
  \draw (d1.center) to[out=-90, in=90] (6.25, 1.8);
\end{tikzpicture}

which turns into the following pdf:

Neat, eh?

Some things to notice. To help LaTeX make sense of things like \node[dot] and \node[morphism], you will need a small latex package. This defines the shapes of the nodes. You see that FreeTikZ is geared towards morphisms that can be rotated four ways, and dots for Frobenius algebras. If you prefer different shapes, such as rectangles with a marked corner instead of morphisms, or ribbons instead of wires, or cobordism-style pictures instead of dots, this is the place to change that.

You can only draw or erase in FreeTikZ, as if you were working on paper. There is no way to move dots around once they are drawn, or to change the colour of things, or to write labels anywhere. The idea is that you use a touchscreen, or stylus, or mouse, to draw your diagram. After that, you can tweak the LaTeX code by hand, moving morphisms around, changing angles, adding labels, or whatever you like. So you still need to know a bit of TikZ, but this is much faster than typesetting your diagram from scratch. (If you really wanted, you could load the LaTeX code into TikZiT, and change the diagram with your mouse before going to your actual LaTex file and using your keyboard.)

Finally, you can download the whole source, so you can also use FreeTikZ locally. Handy for when you don’t have an internet connection, en route to your conference, desparately making slides.

How does it work?

To let this post not completely be a blatant commercial, let’s briefly explain how FreeTikZ works. Up to some webpage bookkeeping and generating LaTeX code, the heart of the algorithm solves the following problem. You’re given a list of strokes, each stroke being a list of vectors in <semantics> 2<annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics>. For each stroke, you’re asked whether it is a dot, a morphism (and if so in which orientation), or something else (that is, a wire, and if so what it connects).

How does it do that? First we compute some numerical properties of the stroke. Thinking of the stroke as spanning a polygon, we can calculate some basic properties:

From those, we can find some more advanced properties.

  • Compactness: This has nothing to do with topological compactness. Instead it measures “how close together” the points of the stroke are. For example, a circle will be very compact, but a straight line will not.
  • Eccentricity is another fundamental characteristic of polygons. It measures the shortest distance between a point and all others, and takes the maximum of that over all points.

The two most useful properties for us are the following.

  • Rectangularity is the ratio between the area of the polygon itself and the area of the smallest rectangle that contains it. So a perfect rectangle will have rectangularity 1, whereas a straight line will have rectangularity 0.
  • Circularity is the ratio between the area of the polygon itself and the area of the smallest circle that contains it. So a perfect circle will have circularity 1, whereas a straight line will have circularity 0.

We then make a simple decision. If circularity is high enough and rectangularity low enough, then our stroke is probably a dot. If rectangularity is high enough and circularity is low enough, it is probably a morphism. (In that case, we can find out its orientation by checking whether the centroid is left/right/above/below the point furthest from the centroid.) Otherwise, we leave the stroke as a wire.

Can it be improved?

This simple decision procedure is a terrible hack, but it seems to work well enough. It would be better to train a support vector machine to make the decision for us based on all these numerical characteristics. Indeed, this is what Ruiting Cao did for her MSc thesis project, on which FreeTikZ is based. Similarly, there are other areas for improvement, but the current heuristics seem to work well enough.

  • Right now dots and morphisms need to be drawn in a single stroke. If you take your stylus/finger/mouse off the screen when drawing a wire, it will end up as two wires. Segmenting and joining strokes cleverly could improve the recognition.

  • If you would like a graphical language with more shapes than just dots and morphisms, you could tweak the characteristic properties, or ideally train and use a state vector machine.

  • Wires are simplified in a somewhat buggy way. You’re given a stroke with many many points. But we would like the LaTeX code not to have hundreds of points in it. At the moment wires are simplified as follows. Annotate each point with its incoming and outgoing angles. Then throw away all points that keep the same angle, up to a certain threshold. So if your stroke has angles 90, 90, 90, 90, 89, 87, 80, 60, 45, 40, 0, you would delete all points except the ones with angles 90, 60, 45, 0 (if the threshold is 30 degrees). But, for example, especially at the start of a stroke when the user just put the pen to the screen, the angle could be off, but that is the one that will be used. There are probably better heuristics.

Luckily, all you need to know is a little JavaScript to make FreeTikZ fit all your needs! What would you like to implement?

by leinster (Tom.Leinster@gmx.com) at January 24, 2018 01:38 AM

January 23, 2018

Christian P. Robert - xi'an's og

Le Monde puzzle [#1037]

A purely geometric Le Monde mathematical puzzle this (or two independent ones, rather):

Find whether or not there are inscribed and circumscribed circles to a convex polygon with 2018 sides of lengths ranging 1,2,…,2018.

In the first (or rather second) case, the circle of radius R that is tangential to the polygon and going through all nodes (assuming such a circle exists) is such that a side L and its corresponding inner angle θ satisfy

L²=R²2(1-cos(θ))

leading to the following R code

R=3.2e5
step=1e2
anglz=sum(acos(1-(1:2018)^2/(2*R^2)))
while (abs(anglz-2*pi)>1e-4){
R=R-step+2*step*(anglz>2*pi)*(R>step)
anglz=sum(acos(1-(1:2018)^2/(2*R^2))) 
step=step/1.01}

and the result is

> R=324221
> sum(acos(1-(1:2018)^2/(2*R^2)))-2*pi
[1] 9.754153e-05

(which is very close to the solution of Le Monde when replacing sin(α) by α!). The authors of the quoted paper do not seem to consider the existence an issue.

In the second case, there is a theorem that states that if the equations

x¹+x²=L¹,…,x²⁰¹⁸+x¹=L²⁰¹⁸

have a solution on R⁺ then there exists a circle such that the polygon is tangential to this circle. Quite interestingly, if the number n of sides is even there are an infinitude of tangential polygons if any.  Now, and rather obviously, the matrix A associated with the above system of linear equations is singular with a kernel induced by the vector (1,-1,…,1,-1). Hence the collection of the sides must satisfy

L¹-L²…+L²⁰¹⁷-L²⁰¹⁸ =0

which puts a constraint on the sequence of sides, namely to divide them into two groups with equal sum, 2018×2019/4, which is not an integer. Hence, a conclusion of impossibility! [Thanks to my office neighbours François and Julien for discussing the puzzle with me.]

by xi'an at January 23, 2018 11:18 PM

The n-Category Cafe

Statebox: A Universal Language of Distributed Systems

We’re getting a lot of great posts here this week, but I also want to point out this, by one grad students:

A brief teaser follows, in case you’re wondering what this is about.

The Azimuth Project’s research on networks and applied category theory has taken an interesting new turn. I always meant for it to do something useful, but I’m too theoretical to pull that off myself. Luckily there are plenty of other people with similar visions whose feet are a bit more firmly on the ground.

I first met the young Dutch hacktivist Jelle Herold at a meeting on network theory that I helped run in Torinio. I saw him again at a Simons Institute meeting on compositionality in computer science. He was already talking about his new startup.

Now it’s here. It’s called Statebox. Among other things, it’s an ambitious attempt to combine categories, open games, dependent types, Petri nets, string diagrams, and blockchains into a universal language for distributed systems.

Herold is inviting academics to help. I want to. But I couldn’t go to the Croatian island of Zlarin at the drop of a hat during classes. Luckily, my grad student Christian Williams is fascinated by the idea of using category theory and blockchain technology to do something good for the world: that’s why he came to work with me! So, I sent him to the first Statebox summit as my deputy. Now he has reported back. A snippet:

Zlarin is a lovely place, but we haven’t gotten to the best part — the people. All who attended are brilliant, creative, and spirited. Everyone’s eyes had a unique spark to light. I don’t think I’ve ever met such a fascinating group in my life. The crew: Jelle, Anton, Emi Gheorghe, Fabrizio Genovese, Daniel van Dijk, Neil Ghani, Viktor Winschel, Philipp Zahn, Pawel Sobocinski, Jules Hedges, Andrew Polonsky, Robin Piedeleu, Alex Norta, Anthony di Franco, Florian Glatz, Fredrik Nordvall Forsberg. These innovators have provocative and complementary ideas in category theory, computer science, open game theory, functional programming, and the blockchain industry; and they came to share an important goal. These are people who work earnestly to better humanity, motivated by progress, not profit. Talking with them gave me hope, that there are enough intelligent, open-minded, and caring people to fix this mess of modern society. In our short time together, we connected — now, almost all continue to contribute and grow the endeavor.

Ah, the starry-eyed idealism of youth! I’m feeling a bit beaten down by the events of the last year, so it’s nice (though somewhat unnerving) to see someone who is not. Read the whole article for more details about this endeavor.

by john (baez@math.ucr.edu) at January 23, 2018 08:26 PM

ZapperZ - Physics and Physicists

Putting Science Back Into Popular Culture
Clifford Johnson of USC has an interesting article on ways to introduce science (or physics in particular), back into things that the public usually gravitate to. In particular, he asks the question on how we can put legitimate science into popular culture so that the public will get to see it more regularly.

Science, though, gets portrayed as opposite to art, intuition and mystery, as though knowing in detail how that flower works somehow undermines its beauty. As a practicing physicist, I disagree. Science can enhance our appreciation of the world around us. It should be part of our general culture, accessible to all. Those “special talents” required in order to engage with and even contribute to science are present in all of us.

So how do we bring about a change? I think using the tools of the general culture to integrate science with everything else in our lives can be a big part of the solution.

Read the rest of the article on how to inject science into popular entertainment, etc.

Zz.

by ZapperZ (noreply@blogger.com) at January 23, 2018 04:02 PM

Peter Coles - In the Dark

R.I.P. Hugh Masekela (1939-2018)

I woke up this morning to the very sad news that South African jazz trumpeter Hugh Masekela had lost the long and courageous battle he had been fighting against cancer and has passed away at the age of 78. Hugh Masakela did a huge amount to establish a uniquely South African jazz tradition and much of his music was a response to the struggle against apartheid. Although some “serious” jazz fans have criticised him for `selling out’ in his forays into pop – for which he simplified his playing style considerably – this approach definitely succeeded in bringing many new people to his music. His was exactly the same approach as Louis Armstrong, actually, and I for one don’t begrudge either his commercial success.

I was fortunate to hear Hugh Masekela live many years ago at Ronnie Scott’s Club. He had a wonderful stage presence, and played a typically eclectic mix of music and it was a wonderful night that I’ll remember for the rest of my life.

Here’s a clip of him playing and singing that gives an idea of what the man and his music were like and just how much he will be missed.

R.I.P. Hugh Masekela (1939-2018).

by telescoper at January 23, 2018 01:50 PM

January 22, 2018

Christian P. Robert - xi'an's og

algorithm for predicting when kids are in danger [guest post]

[Last week, I read this article in The New York Times about child abuse prediction software and approached Kristian Lum, of HRDAG, for her opinion on the approach, possibly for a guest post which she kindly and quickly provided!]

A week or so ago, an article about the use of statistical models to predict child abuse was published in the New York Times. The article recounts a heart-breaking story of two young boys who died in a fire due to parental neglect. Despite the fact that social services had received “numerous calls” to report the family, human screeners had not regarded the reports as meeting the criteria to warrant a full investigation. Offered as a solution to imperfect and potentially biased human screeners is the use of computer models that compile data from a variety of sources (jails, alcohol and drug treatment centers, etc.) to output a predicted risk score. The implication here is that had the human screeners had access to such technology, the software might issued a warning that the case was high risk and, based on this warning, the screener might have sent out investigators to intervene, thus saving the children.

These types of models bring up all sorts of interesting questions regarding fairness, equity, transparency, and accountability (which, by the way, are an exciting area of statistical research that I hope some readers here will take up!). For example, most risk assessment models that I have seen are just logistic regressions of [characteristics] on [indicator of undesirable outcome]. In this case, the outcome is likely an indicator of whether child abuse had been determined to take place in the home or not. This raises the issue of whether past determinations of abuse– which make up  the training data that is used to make the risk assessment tool–  are objective, or whether they encode systemic bias against certain groups that will be passed through the tool to result in systematically biased predictions. To quote the article, “All of the data on which the algorithm is based is biased. Black children are, relatively speaking, over-surveilled in our systems, and white children are under-surveilled.” And one need not look further than the same news outlet to find cases in which there have been egregiously unfair determinations of abuse, which disproportionately impact poor and minority communities.  Child abuse isn’t my immediate area of expertise, and so I can’t responsibly comment on whether these types of cases are prevalent enough that the bias they introduce will swamp the utility of the tool.

At the end of the day, we obviously want to prevent all instances of child abuse, and this tool seems to get a lot of things right in terms of transparency and responsible use. And according to the original article, it (at least on the surface) seems to be effective at more efficiently allocating scarce resources to investigate reports of child abuse. As these types of models become used more and more for a wider variety of prediction types, we need to be cognizant that (to quote my brilliant colleague, Josh Norkin) we don’t “lose sight of the fact that because this system is so broken all we are doing is finding new ways to sort our country’s poorest citizens. What we should be finding are new ways to lift people out of poverty.”

by xi'an at January 22, 2018 11:18 PM

The n-Category Cafe

A Categorical Semantics for Causal Structure

Guest post by Joseph Moeller and Dmitry Vagner

We begin the Applied Category Theory Seminar by discussing the paper A categorical semantics for causal structure by Aleks Kissinger and Sander Uijlen.

Compact closed categories have been used in categorical quantum mechanics to give a structure for talking about quantum processes. However, they prove insufficient to handle higher order processes, in other words, processes of processes. This paper offers a construction for a <semantics>*<annotation encoding="application/x-tex">\ast</annotation></semantics>-autonomous extension of a given compact closed category which allows one to reason about higher order processes in a non-trivial way.

We would like to thank Brendan Fong, Nina Otter, Joseph Hirsh and Tobias Heindel as well as the other participants for the discussions and feedback.

Preliminaries

We begin with a discussion about the types of categories which we will be working with, and the diagrammatic language we use to reason about these categories.

Diagrammatics

Recall the following diagrammatic language we use to reason about symmetric monoidal categories. Objects are represented by wires. Arrows can be graphically encoded as

Composition <semantics><annotation encoding="application/x-tex">\circ</annotation></semantics> and <semantics><annotation encoding="application/x-tex">\otimes</annotation></semantics> depicted vertically and horizontally

satisfying the proprties

and the interchange law

If <semantics>I<annotation encoding="application/x-tex">I</annotation></semantics> is the unit object, and <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> is an object, we call arrows <semantics>IA<annotation encoding="application/x-tex">I \to A</annotation></semantics> states, <semantics>AI<annotation encoding="application/x-tex">A \to I</annotation></semantics> effects, and <semantics>II<annotation encoding="application/x-tex">I \to I</annotation></semantics> numbers.

The identity morphism on an object is only displayed as a wire, and both <semantics>I<annotation encoding="application/x-tex">I</annotation></semantics> and its identity morphism are not displayed.

Compact Closed Categories

A symmetric monoidal category is compact closed if each object <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> has a dual object <semantics>A *<annotation encoding="application/x-tex">A^\ast</annotation></semantics> with arrows

<semantics>η A:IA *A,<annotation encoding="application/x-tex">\eta_A \colon I \to A^\ast \otimes A,</annotation></semantics>

and

<semantics>ϵ A:AA *I,<annotation encoding="application/x-tex">\epsilon_A \colon A \otimes A^\ast \to I,</annotation></semantics>

depicted as <semantics><annotation encoding="application/x-tex">\cup</annotation></semantics> and <semantics><annotation encoding="application/x-tex">\cap</annotation></semantics> and obeying the zigzag identities:

Given a process <semantics>f:AB<annotation encoding="application/x-tex">f \colon A \to B</annotation></semantics> in a compact closed category, we can construct a state <semantics>ρ f:IA *B<annotation encoding="application/x-tex">\rho_f \colon I \to A^\ast \otimes B</annotation></semantics> by defining

<semantics>ρ f=(1 A *f)η A.<annotation encoding="application/x-tex">\rho_f = (1_{A^\ast} \otimes f) \circ \eta_A.</annotation></semantics>

This gives a correspondence which is called “process-state duality”.

An Example

Let <semantics>Mat( +)<annotation encoding="application/x-tex">Mat(\mathbb{R}_+)</annotation></semantics> be the category in which objects are natural numbers, and morphisms <semantics>mn<annotation encoding="application/x-tex">m \to n</annotation></semantics> are <semantics>n×m<annotation encoding="application/x-tex">n\times m</annotation></semantics> <semantics> +<annotation encoding="application/x-tex">\mathbb{R}_+</annotation></semantics>-matrices with composition given by the usual multiplication of matrices. This category is made symmetric monoidal with tensor defined by <semantics>nm=nm<annotation encoding="application/x-tex">n \otimes m = nm</annotation></semantics> on objects and the Kronecker product of matrices on arrows, <semantics>(fg) ij kl=f i kg j l<annotation encoding="application/x-tex">(f \otimes g)^{kl}_{ij} = f^k_i g^l_j</annotation></semantics>. For example

<semantics>[1 2 3 4][0 5 6 7]=[1[0 5 6 7] 2[0 5 6 7] 3[0 5 6 7] 4[0 5 6 7]]=[0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28]<annotation encoding="application/x-tex"> \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \otimes \begin{bmatrix} 0 & 5 \\ 6 & 7 \end{bmatrix} = \begin{bmatrix} 1\cdot \begin{bmatrix} 0 & 5 \\ 6 & 7 \end{bmatrix} & 2\cdot \begin{bmatrix} 0 & 5 \\ 6 & 7 \end{bmatrix} \\ 3 \cdot \begin{bmatrix} 0 & 5 \\ 6 & 7 \end{bmatrix} &4\cdot \begin{bmatrix} 0 & 5 \\ 6 & 7 \end{bmatrix} \end{bmatrix} = \begin{bmatrix} 0 & 5 & 0 & 10 \\ 6 & 7 & 12 & 14 \\ 0 & 15 & 0 & 20 \\ 18 & 21 & 24 & 28 \end{bmatrix} </annotation></semantics>

The unit with respect to this tensor is <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics>. States in this category are column vectors, effects are row vectors, and numbers are <semantics>1×1<annotation encoding="application/x-tex">1\times 1</annotation></semantics> matrices, in other words, numbers. Composing a state <semantics>ρ:1n<annotation encoding="application/x-tex">\rho\colon 1 \to n</annotation></semantics> with an effect <semantics>π:n1<annotation encoding="application/x-tex">\pi\colon n \to 1</annotation></semantics>, is the dot product. To define a compact closed structure on this category, let <semantics>n *:=n<annotation encoding="application/x-tex">n^\ast := n</annotation></semantics>. Then <semantics>η n:1n 2<annotation encoding="application/x-tex">\eta_n \colon 1 \to n^2</annotation></semantics> and <semantics>ε n:n 21<annotation encoding="application/x-tex">\varepsilon_n \colon n^2 \to 1</annotation></semantics> are given by the Kronecker delta.

A categorical framework for causality

Encoding Causality

The main construction in this paper requires what is called a precausal category. In a precausal category, we demand that every system has a discard effect, which is a process <semantics>A:AI<annotation encoding="application/x-tex">_A \colon A \to I</annotation></semantics>. This collection of effects must be compatible with <semantics><annotation encoding="application/x-tex">\otimes</annotation></semantics>:

  • <semantics>AB=<annotation encoding="application/x-tex">_{A \otimes B} = </annotation></semantics><semantics>A<annotation encoding="application/x-tex">_A </annotation></semantics><semantics>B<annotation encoding="application/x-tex">_B</annotation></semantics>

  • <semantics>I=1<annotation encoding="application/x-tex">_I = 1</annotation></semantics>

A process <semantics>Φ:AB<annotation encoding="application/x-tex">\Phi \colon A \to B</annotation></semantics> is called causal if discarding <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> after having done <semantics>Φ<annotation encoding="application/x-tex">\Phi</annotation></semantics> is the same as just discarding <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics>.

If <semantics>A *<annotation encoding="application/x-tex">A^\ast</annotation></semantics> has discarding, we can produce a state for <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> by spawning an <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> and <semantics>A *<annotation encoding="application/x-tex">A^\ast</annotation></semantics> pair, then discarding the <semantics>A *<annotation encoding="application/x-tex">A^\ast</annotation></semantics>:

In the case of <semantics>Mat( +)<annotation encoding="application/x-tex">Mat(\mathbb{R}_+)</annotation></semantics>, the discard effect is given as row vector of <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics>’s: <semantics>1:=(11)<annotation encoding="application/x-tex">\mathbf{1}:=(1\cdots 1)</annotation></semantics>. Composing a matrix with the discard effect sums the entries of each column. So if a matrix is a causal process, then its column vectors have entries that sum to <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics>. Thus causal processes in <semantics>Mat( +)<annotation encoding="application/x-tex">Mat(\mathbb{R}_+)</annotation></semantics> are stochastic maps.

A process <semantics>Φ:ABAB<annotation encoding="application/x-tex">\Phi \colon A \otimes B \to A' \otimes B'</annotation></semantics> is one-way signalling with <semantics>AB<annotation encoding="application/x-tex">A \preceq B</annotation></semantics> if

and <semantics>BA<annotation encoding="application/x-tex">B \preceq A</annotation></semantics> if

and non-signalling if both <semantics>AB<annotation encoding="application/x-tex">A\preceq B</annotation></semantics> and <semantics>BA<annotation encoding="application/x-tex">B\preceq A</annotation></semantics>.

The intuition here is that <semantics>AB<annotation encoding="application/x-tex">A \preceq B</annotation></semantics> means <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> cannot signal to <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics>; the formal condition encodes the fact that had <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> influenced the transformation from <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> to <semantics>A<annotation encoding="application/x-tex">A'</annotation></semantics>, then it couldn’t have been discarded prior to it occurring.

Consider the following example: let <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> be a cup of tea, and <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> a glass of milk. Let <semantics>Φ<annotation encoding="application/x-tex">\Phi</annotation></semantics> the process of pouring half of <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> into <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> then mixing, to form <semantics>A<annotation encoding="application/x-tex">A'</annotation></semantics> milktea and <semantics>B<annotation encoding="application/x-tex">B'</annotation></semantics> half-glass of milk. Clearly this process would not be the same as if we start by discarding the milk. Our intuition is that the milk “signalled” to, or influenced, the tea, and hence intuitively we do not have <semantics>AB<annotation encoding="application/x-tex">A \preceq B</annotation></semantics>.

A compact closed category <semantics>C<annotation encoding="application/x-tex">\C</annotation></semantics> is precausal if

  1. Every system <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> has a discard process <semantics>A<annotation encoding="application/x-tex">_A</annotation></semantics>

  2. For every system <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics>, the dimension is invertible

  1. <semantics>C<annotation encoding="application/x-tex">\C</annotation></semantics> has enough causal states

  1. Second order processes factor

From the definition, we can begin to exclude certain causal situations from systems in precausal categories. In Theorem 3.12, we see that precausal categories do not admit ‘time-travel’.

Theorem   If there are systems <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics>, <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics>, <semantics>C<annotation encoding="application/x-tex">C</annotation></semantics> such that

then <semantics>AI<annotation encoding="application/x-tex">A \cong I</annotation></semantics>.

In precausal categories, we have processes that we call first order causal. However, higher order processes collapse into first order processes, because precausal categories are compact closed. For example, letting <semantics>AB:=A *B<annotation encoding="application/x-tex">A \Rightarrow B := A^\ast\otimes B</annotation></semantics>,

<semantics>(AB)C=(A *B)*CBAC<annotation encoding="application/x-tex"> (A\Rightarrow B)\Rightarrow C = (A^\ast\otimes B)\ast\otimes C \cong B\Rightarrow A\otimes C </annotation></semantics>

We can see this happens because of the condition <semantics>AB(A *B *) *<annotation encoding="application/x-tex">A \otimes B \cong (A^\ast \otimes B^\ast)^\ast</annotation></semantics>. Weakening this condition of compact closed categories yields <semantics>*<annotation encoding="application/x-tex">\ast</annotation></semantics>-autonomous categories. From a precausal category <semantics>C<annotation encoding="application/x-tex">\C</annotation></semantics>, we construct a category <semantics>Caus[C]<annotation encoding="application/x-tex">Caus[\C]</annotation></semantics> of higher order causal relations.

The category of higher order causal processes

Given a set of states <semantics>cC(I,C)<annotation encoding="application/x-tex">c \subseteq \C(I,C)</annotation></semantics> for a system <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics>, define its dual by

<semantics>c *={πA *|ρc,πρ=1}C(I,A *)<annotation encoding="application/x-tex"> c^\ast = \{\pi \in A^\ast |\, \forall \rho \in c ,\, \pi \circ \rho = 1\} \subseteq \C(I, A^\ast) </annotation></semantics>

Then we say a set of states <semantics>cC(I,A)<annotation encoding="application/x-tex">c \subseteq \C(I,A)</annotation></semantics> is closed if <semantics>c=c **<annotation encoding="application/x-tex">c=c^{\ast\ast}</annotation></semantics>, and flat if there are invertible scalars <semantics>λ<annotation encoding="application/x-tex">\lambda</annotation></semantics>, <semantics>μ<annotation encoding="application/x-tex">\mu</annotation></semantics> such that <semantics>λ<annotation encoding="application/x-tex">\lambda</annotation></semantics> <semantics>c<annotation encoding="application/x-tex">\in c</annotation></semantics>, <semantics>μ<annotation encoding="application/x-tex">\mu</annotation></semantics> <semantics>c *<annotation encoding="application/x-tex">\in c^\ast</annotation></semantics>.

Now we can define the category <semantics>Caus[C]<annotation encoding="application/x-tex">Caus[\C]</annotation></semantics>. Let the objects be pairs <semantics>(A,c A)<annotation encoding="application/x-tex">(A, c_A)</annotation></semantics> where <semantics>c A<annotation encoding="application/x-tex">c_A</annotation></semantics> is a closed and flat set of states of the system <semantics>AC<annotation encoding="application/x-tex">A \in \C</annotation></semantics>. A morphism <semantics>f:(A,c A)(B,c B)<annotation encoding="application/x-tex">f \colon (A, c_A) \to (B, c_B)</annotation></semantics> is a morphism <semantics>f:AB<annotation encoding="application/x-tex">f \colon A \to B</annotation></semantics> in <semantics>C<annotation encoding="application/x-tex">\C</annotation></semantics> such that if <semantics>ρc A<annotation encoding="application/x-tex">\rho \in c_A</annotation></semantics>, then <semantics>fρc B<annotation encoding="application/x-tex">f \circ \rho \in c_B</annotation></semantics>. This category is a symmetric monoidal category with <semantics>(A,c A)(B,c B)=(AB,c AB)<annotation encoding="application/x-tex">(A, c_A) \otimes (B, c_B) = (A \otimes B, c_{A \otimes B})</annotation></semantics>. Further, it’s <semantics>*<annotation encoding="application/x-tex">\ast</annotation></semantics>-autonomous, so higher order processes won’t necessarily collapse into first order.

A first order system in <semantics>Caus[C]<annotation encoding="application/x-tex">Caus[\C]</annotation></semantics> is one of the form <semantics>(A,{<annotation encoding="application/x-tex">(A, \{</annotation></semantics><semantics>A} *)<annotation encoding="application/x-tex">_A\}^\ast)</annotation></semantics>. First order systems are closed under <semantics><annotation encoding="application/x-tex">\otimes</annotation></semantics>. In fact, <semantics>C C<annotation encoding="application/x-tex">C_C</annotation></semantics> admits a full faithful monoidal embedding into <semantics>Caus[C]<annotation encoding="application/x-tex">Caus[\C]</annotation></semantics> by assigning systems to their corresponding first order systems <semantics>A(A,{<annotation encoding="application/x-tex">A \mapsto (A, \{</annotation></semantics><semantics>A})<annotation encoding="application/x-tex">_A\})</annotation></semantics>.

For an example of a higher order process in <semantics>Caus[Mat( +)]<annotation encoding="application/x-tex">Caus[Mat(\mathbb{R}_+)]</annotation></semantics>, consider a classical switch. Let

<semantics>ρ 0=[1 0],ρ 1=[0 1],ρ i=ρ i T,<annotation encoding="application/x-tex">\rho_0 = \begin{bmatrix} 1\\0 \end{bmatrix}, \quad \rho_1 = \begin{bmatrix} 0\\1 \end{bmatrix}, \quad \rho'_i = \rho_i^T,</annotation></semantics>

and let <semantics>s<annotation encoding="application/x-tex">s</annotation></semantics> be the second order process

This process is of type <semantics>XC(AB)(BA)C<annotation encoding="application/x-tex">X \otimes C \multimap (A \multimap B) \otimes (B \multimap A) \multimap C'</annotation></semantics>, where the two middle inputs take types <semantics>AB<annotation encoding="application/x-tex">A \multimap B</annotation></semantics> on the left and <semantics>BA<annotation encoding="application/x-tex">B \multimap A</annotation></semantics> on the right. Since <semantics>ρ iρ j<annotation encoding="application/x-tex">\rho'_i \circ \rho_j</annotation></semantics> is <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics> if <semantics>i=j<annotation encoding="application/x-tex">i=j</annotation></semantics> and <semantics>0<annotation encoding="application/x-tex">0</annotation></semantics> otherwise, then plugging in either <semantics>ρ 0<annotation encoding="application/x-tex">\rho_0</annotation></semantics> or <semantics>ρ 1<annotation encoding="application/x-tex">\rho_1</annotation></semantics> to the bottom left input switches the order that the <semantics>AB<annotation encoding="application/x-tex">A \multimap B</annotation></semantics> and <semantics>BA<annotation encoding="application/x-tex">B \multimap A</annotation></semantics> processes are composed in the final output process. This second order process is causal because

The authors go on to prove in Theorem 6.17 that a switch cannot be causally ordered, indicating that this process is genuinely second order.

by john (baez@math.ucr.edu) at January 22, 2018 09:36 PM

Axel Maas - Looking Inside the Standard Model

Finding - and curing - disagreements
The topic of grand-unified theories came up in the blog several times, most recently last year in January. To briefly recap, such theories, called GUTs for short, predict that all three forces between elementary particles emerge from a single master force. That would explain a lot of unconnected observations we have in particle physics. For example, why atoms are electrically neutral. The latter we can describe, but not yet explain.

However, if such a GUT exists, then it must not only explain the forces, but also somehow why we see the numbers and kinds of elementary particles we observe in nature. And now things become complicated. As discussed in the last entry on GUTs there maybe a serious issue in how we determine which particles are actually described by such a theory.

To understand how this issue comes about, I need to put together many different things my research partners and I have worked on during the last couple of years. All of these issues are actually put into an expert language in the review of which I talked in the previous entry. It is now finished, and if your interested, you can get it free from here. But it is very technical.

So, let me explain it less technically.

Particle physics is actually superinvolved. If we would like to write down a theory which describes what we see, and only what we see, it would be terribly complicated. It is much more simple to introduce redundancies in the description, so-called gauge symmetries. This makes life much easier, though still not easy. However, the most prominent feature is that we add auxiliary particles to the game. Of course, they cannot be really seen, as they are just auxiliary. Some of them are very obviously unphysical, called therefore ghosts. They can be taken care of comparatively simply. For others, this is less simple.

Now, it turns out that the weak interaction is a very special beast. In this case, there is a unique one-to-one identification between a really observable particle and an auxiliary particle. Thus, it is almost correct to identify both. But this is due to the very special structure of this part of particle physics.

Thus, a natural question is whether, even if it is special, it is justified to do the same for other theories. Well, in some cases, this seems to be the case. But we suspected that this may not be the case in general. And especially not in GUTs.

Now, recently we were going about this much more systematically. You can again access the (very, very technical) result for free here. There, we looked at a very generic class of such GUTs. Well, we actually looked at the most relevant part of them, and still by far not all of them. We also ignored a lot of stuff, e.g. what would become quarks and leptons, and concentrated only on the generalization of the weak interaction and the Higgs.

We then checked, based on our earlier experiences and methods, whether a one-to-one identification of experimentally accessible and auxiliary particles works. And it does essentially never. Visually, this result looks like


On the left, it is seen that everything works nicely with a one-to-one identification in the standard model. On the right, if one-to-one identification would work in a GUT, everything would still be nice. But a our more precise calculation shows that the actually situation, which would be seen in an experiment, is different. There is non one-to-one identification possible. And thus the prediction of the GUT differs from what we already see inn experiments. Thus, a previously good GUT candidate is no longer good.

Though more checks are needed, as always, this is a baffling, and at the same time very discomforting, result.

Baffling as we did originally expect to have problems under very special circumstances. It now appears that actually the standard model of particles is the very special case, and having problems is the standard.

It is discomforting because in the powerful method of perturbation theory the one-to-one identification is essentially always made. As this tool is widely used, this seems to question the validity of many predictions on GUTs. That could have far-reaching consequences. Is this the case? Do we need to forget everything about GUTs we learned so far?

Well, not really, for two reasons. One is that we also showed that methods almost as easily handleable as perturbation theory can be used to fix the problems. This is good, because more powerful methods, like the simulations we used before, are much more cumbersome. However, this leaves us with the problem of having made so far wrong predictions. Well, this we cannot change. But this is just normal scientific progress. You try, you check, you fail, you improve, and then you try again.

And, in fact, this does not mean that GUTs are wrong. Just that we need to consider somewhat different GUTs, and make the predictions more carefully next time. Which GUTs we need to look at we still need to figure out, and that will not be simple. But, fortunately, the improved methods mentioned beforehand can use much of what has been done so far, so most technical results are still unbelievable useful. This will help enormously in finding GUTs which are applicable, and yield a consistent picture, without the one-to-one identification. GUTs are not dead. They likely just need a bit of changing.

This is indeed a dramatic development. But one which fits logically and technically to the improved understanding of the theoretical structures underlying particle physics, which were developed over the last decades. Thus, we are confident that this is just the next logical step in our understanding of how particle physics works.

by Axel Maas (noreply@blogger.com) at January 22, 2018 04:54 PM

Emily Lakdawalla - The Planetary Society Blog

Here's our rolling list of space things affected by the U.S. government shutdown
The International Space Station stays open for business; everything else is at least somewhat affected.

January 22, 2018 04:46 PM

Peter Coles - In the Dark

Light, I know, treads the ten million stars

Light, I know, treads the ten million stars,
And blooms in the Hesperides. Light stirs
Out of the heavenly sea onto the moon’s shores.
Such light shall not illuminate my fears
And catch a turnip ghost in every cranny.
I have been frightened of the dark for years.
When the sun falls and the moon stares,
My heart hurls from my side and tears
Drip from my open eyes as honey
Drips from the humming darkness of the hive.
I am a timid child when light is dead.
Unless I learn the night I shall go mad.
It is night’s terrors I must learn to love,
Or pray for day to some attentive god
Who on his cloud hears all my wishes,
Hears and refuses.
Light walks the sky, leaving no print,
And there is always day, the shining of some sun,
In those high globes I cannot count,
And some shine for a second and are gone,
Leaving no print.
But lunar night will not glow in my blackness,
Make bright its corners where a skeleton
Sits back and smiles, A tiny corpse
Turns to the roof a hideous grimace,
Or mice play with an ivory tooth.
Stars’ light and sun’s light will not shine
As clearly as the light of my own brain,
Will only dim life, and light death.
I must learn night’s light or go mad.

by Dylan Thomas (1914-1953)

by telescoper at January 22, 2018 04:36 PM

Peter Coles - In the Dark

Newsflash: New Chair at STFC

As a quick piece of community service I thought I’d pass on the news of the appointment of a new Executive Chair for the Science and Technology Facilities Council (STFC), namely Professor Mark Thomson of the University of Cambridge. Developments at STFC will cease to be relevant to me after this summer as I’m moving to Ireland but this is potentially very important news for many readers of this blog.

Professor Thomson is an Experimental Particle Physicist whose home page at Cambridge describes his research in thuswise manner:

My main research interests are neutrino physics, the physics of the electroweak interactions, and the design of detectors at a future colliders. I am co-spokesperson of the DUNE collaboration, which consists of over 1000 scientiests and engineers from over 170 institutions in 31 nations across the globe. The Cambridge neutrino group splits its acivities between MicroBooNE and DUNE and is using advanced particle flow calorimetry techniques to interpret the images from large liquid argon TPC neutrino detector.

I’ve added a link to the DUNE collaboration for those of you who don’t know about it – it’s a very large neutrino physics experiment to be based in the USA.

On the announcement, Prof. Thomson stated:

I am passionate about STFC science, which spans the smallest scales of particle physics to the vast scales of astrophysics and cosmology, and it is a great honour be appointed to lead STFC as its new Executive Chair. The formation of UKRI presents exciting opportunities for STFC to further develop the UK’s world-leading science programme and to maximise the impact of the world-class facilities supported by STFC.

This appointment needs to be officially confirmed after a pre-appointment hearing by the House of Commons Science and Technology Committee but, barring a surprise offer of the position to Toby Young, he’s likely to take over the reins at STFC in April this year. He’ll have his work cut out trying to make the case for continued investment in fundamental science in the United Kingdom, in the face of numerous challenges, so I’d like to take this opportunity to wish him the very best of luck in his new role!

by telescoper at January 22, 2018 02:32 PM

Peter Coles - In the Dark

Hirsute cosmologist Peter Coles joins Beard of Winter poll after ‘write-in’ votes

I see that, despite popular demand, Keith Flett, on behalf of the Beard Liberation Front, has at the last minute decided to give me another chance to fail to win a beard award. I’m currently in second place in the poll as it enters its final week…

Kmflett's Blog

Beard Liberation Front

PRESS RELEASE 20th January

Contact Keith Flett      07803 167266

Hirsute cosmologist Peter Coles joins Beard of Winter poll after ‘write-in’ votes

The Beard Liberation Front, the informal network of beard wearers, has said that leading hirsute cosmologist Peter Coles has joined the Beard of Winter poll after write-in votes for him broke the threshold of 1% of the total poll

Mr Coles, based at Cardiff University and also at Maynooth in Ireland was a contender for Beard of the Year in 2014.

The Beard of Winter is the first of four seasonal awards that lead to the Beard of the Year Award in December 2018.

It focuses both on fuller organic beards, suitable for winter weather but also on beards that have made an early New Year impact in the public eye.

BLF Organiser Keith Flett said, Peter Coles has one of the most distinguished of scientific…

View original post 70 more words

by telescoper at January 22, 2018 10:55 AM

John Baez - Azimuth

Statebox: A Universal Language of Distributed Systems

guest post by Christian Williams

A short time ago, on the Croatian island of Zlarin, there gathered a band of bold individuals—rebels of academia and industry, whose everyday thoughts and actions challenge the separations of the modern world. They journeyed from all over to learn of the grand endeavor of another open mind, an expert functional programmer and creative hacktivist with significant mathematical knowledge: Jelle |yell-uh| Herold.

The Dutch computer scientist has devoted his life to helping our species and our planet: from consulting in business process optimization to winning a Greenpeace hackathon, from updating Netherlands telecommunications to creating a website to determine ways for individuals to help heal the earth, Jelle has gained a comprehensive perspective of the interconnected era. Through a diverse and innovative career, he has garnered crucial insights into software design and network computation—most profoundly, he has realized that it is imperative that these immense forces of global change develop thoughtful, comprehensive systematization.

Jelle understood that initiating such a grand ambition requires a massive amount of work, and the cooperation of many individuals, fluent in different fields of mathematics and computer science. Enter the Zlarin meeting: after a decade of consideration, Jelle has now brought together proponents of categories, open games, dependent types, Petri nets, string diagrams, and blockchains toward a singular end: a universal language of distributed systems—Statebox.

Statebox is a programming language formed and guided by fundamental concepts and principles of theoretical mathematics and computer science. The aim is to develop the canonical process language for distributed systems, and thereby elucidate the way these should actually be designed. The idea invokes the deep connections of these subjects in a novel and essential way, to make code simple, transparent, and concrete. Category theory is both the heart and pulse of this endeavor; more than a theory, it is a way of thinking universally. We hope the project helps to demonstrate the importance of this perspective, and encourages others to join.

The language is designed to be self-optimizing, open, adaptive, terminating, error-cognizant, composable, and most distinctively—visual. Petri nets are the natural representation of decentralized computation and concurrency. By utilizing them as program models, the entire language is diagrammatic, and this allows one to inspect the flow of the process executed by the program. While most languages only compile into illegible machine code, Statebox compiles directly into diagrams, so that the user immediately sees and understands the concrete realization of the abstract design. We believe that this immanent connection between the “geometric” and “algebraic” aspects of computation is of great importance.

Compositionality is a rightfully popular contemporary term, indicating the preservation of type under composition of systems or processes. This is essential to the universality of the type, and it is intrinsic to categories, which underpin the Petri net. A pertinent example is that composition allows for a form of abstraction in which programs do not require complete specification. This is parametricity: a program becomes executable when the functions are substituted with valid terms. Every term has a type, and one cannot connect pieces of code that have incompatible inputs and outputs—the compiler would simply produce an error. The intent is to preserve a simple mathematical structure that imposes as little as possible, and still ensure rationality of code. We can then more easily and reliably write tools providing automatic proofs of termination and type-correctness. Many more aspects will be explained as we go along, and in more detail in future posts.

Statebox is more than a specific implementation. It is an evolving aspiration, expressing an ideal, a source of inspiration, signifying a movement. We fully recognize that we are at the dawn of a new era, and do not assume that the current presentation is the best way to fulfill this ideal—but it is vital that this kind of endeavor gains the hearts and minds of these communities. By learning to develop and design by pure theory, we make a crucial step toward universal systems and knowledge. Formalisms are biased, fragile, transient—thought is eternal.

Thank you for reading, and thank you to John Baez—|bi-ez|, some there were not aware—for allowing me to write this post. Azimuth and its readers represent what scientific progress can and should be; it is an honor to speak to you. My name is Christian Williams, and I have just begun my doctoral studies with Dr. Baez. He received the invitation from Jelle and could not attend, and was generous enough to let me substitute. Disclaimer: I am just a young student with big dreams, with insufficient knowledge to do justice to this huge topic. If you can forgive some innocent confidence and enthusiasm, I would like to paint a big picture, to explain why this project is important. I hope to delve deeper into the subject in future posts, and in general to celebrate and encourage the cognitive revolution of Applied Category Theory. (Thank you also to Anton and Fabrizio for providing some of this writing when I was not well; I really appreciate it.)

Statebox Summit, Zlarin 2017, was awesome. Wish you could’ve been there. Just a short swim in the Adriatic from the old city of Šibenik |shib-enic|, there lies the small, green island of Zlarin |zlah-rin|, with just a few hundred kind inhabitants. Jelle’s friend, and part of the Statebox team, Anton Livaja and his family graciously allowed us to stay in their houses. Our headquarters was a hotel, one of the few places open in the fall. We set up in the back dining room for talks and work, and for food and sunlight we came to the patio and were brought platters of wonderful, wholesome Croatian dishes. As we burned the midnight oil, we enjoyed local beer, and already made history—the first Bitcoin transaction of the island, with a progressive bartender, Vinko.

Zlarin is a lovely place, but we haven’t gotten to the best part—the people. All who attended are brilliant, creative, and spirited. Everyone’s eyes had a unique spark to light. I don’t think I’ve ever met such a fascinating group in my life. The crew: Jelle, Anton, Emi Gheorghe, Fabrizio Genovese, Daniel van Dijk, Neil Ghani, Viktor Winschel, Philipp Zahn, Pawel Sobocinski, Jules Hedges, Andrew Polonsky, Robin Piedeleu, Alex Norta, Anthony di Franco, Florian Glatz, Fredrik Nordvall Forsberg. These innovators have provocative and complementary ideas in category theory, computer science, open game theory, functional programming, and the blockchain industry; and they came to share an important goal. These are people who work earnestly to better humanity, motivated by progress, not profit. Talking with them gave me hope, that there are enough intelligent, open-minded, and caring people to fix this mess of modern society. In our short time together, we connected—now, almost all continue to contribute and grow the endeavor.

Why is society a mess? The present human condition is absurd. We are in a cognitive renaissance, yet our world is in peril. We need to realize a deeper harmony of theory and practice—we need ideas that dare to dream big, that draw on the vast wealth of contemporary thought to guide and unite subjects in one mission. The way of the world is only a reflection of how we choose to think, and for more than a century we have delved endlessly into thought itself. If we truly learn from our thought, knowledge and application become imminently interrelated, not increasingly separate. It is imperative that we abandon preconception, pretense and prejudice, and ask with naive sincerity: “How should things be, really, and how can we make it happen?”

This pertains more generally to the irresponsibly ad hoc nature of society—we find ourselves entrenched in inadequate systems. Food, energy, medicine, finance, communications, media, governance, technology—our deepening dependence on centralization is our greatest vulnerability. Programming practice is the perfect example of the gradual failure of systems when their design is left to wander in abstraction. As business requirements evolved, technological solutions were created haphazardly, the priority being immediate return over comprehensive methodology, which resulted in ‘duct-taped’ systems, such as the Windows OS. Our entire world now depends on unsystematic software, giving rise to so much costly disorganization, miscommunication, and worse, bureaucracy. Statebox aims to close the gap between the misguided formalisms which came out of this type of degeneration, and design a language which corresponds naturally to essential mathematical concepts—to create systems which are rational, principled, universal. To explain why Statebox represents to us such an important ideal, we must first consider its closest relative, the elephant in the technological room: blockchain.

Often the best ideas are remarkably simple—in 2008, an unknown person under the alias Satoshi Nakamoto published the whitepaper Bitcoin: A Peer-to-Peer Electronic Cash System. In just a few pages, a protocol was proposed which underpins a new kind of computational network, called a blockchain, in which interactions are immediate, transparent, and permanent. This is a personal interpretation—the paper focuses on the application given in its title. In the original financial context, immediacy is one’s ability to directly transact with anyone, without intermediaries, such as banks; transparency is one’s right to complete knowledge of the economy in which one participates, meaning that each node owns a copy of the full history of the network; permanence is the irrevocability of one’s transactions. These core aspects are made possible by an elegant use of cryptography and game theory, which essentially removes the need for trusted third parties in the authorization, verification, and documentation of transactions. Per word, it’s almost peerless in modern influence; the short and sweet read is recommended.

The point of this simplistic explanation is that blockchain is about more than economics. The transaction could be any cooperation, the value could be any social good—when seen as a source of consensus, the blockchain protocol can be expanded to assimilate any data and code. After several years of competing cryptocurrencies, the importance of this deeper idea was gradually realized. There arose specialized tools to serve essential purposes in some broader system, and only recently have people dared to conceive of what this latter could be. In 2014, a wunderkind named Vitalik Buterin created Ethereum, a fully programmable blockchain. Solidity is a Turing-complete language of smart contracts, autonomous programs which enable interactions and enact rules on the network. With this framework, one can not only transact with others, but implement any kind of process; one can build currencies, websites, or organizations—decentralized applications, constructed with smart contracts, could be just about anything.

There is understandably great confidence and excitement for these ventures, and many are receiving massive public investment. Seriously, the numbers are staggering—but most of it is pure hype. There is talk of the first global computer, the internet of value, a single shared source of truth, and other speculative descriptions. But compared to the ambition, the actual theory is woefully underdeveloped. So far, implementations make almost no use of the powerful ideas of mathematics. There are still basic flaws in blockchain itself, the foundation of almost all decentralized technology. For example, the two viable candidates for transaction verification are called Proof of Work and Proof of Stake: the former requires unsustainable consumption of resources, namely hardware and electricity, and the latter is susceptible to centralization. Scalability is a major problem, thus also cost and speed of transactions. A major Ethereum dApp, Decentralized Autonomous Organization, was hacked.

These statements are absolutely not to disregard all of the great work of this community; it is primarily rhetoric to distinguish the high ideals of Statebox, and I lack the eloquence to make the point diplomatically, nor near the knowledge to give a real account of this huge endeavor. We now return to the rhetoric.

What seems to be lost in the commotion is the simple recognition that we do not yet really know what we should make, nor how to do so. The whole idea is simply too big—the space of possibility is almost completely unknown, because this innovation can open every aspect of society to reform. But as usual, people try to ignore their ignorance, imagining it will disappear, and millions clamor about things we do not yet understand. Most involved are seeing decentralization as an exciting business venture, rather than our best hope to change the way of this broken world; they want to cash in on another technological wave. Of the relatively few idealists, most still retain the assumptions and limitations of the blockchain.

For all this talk, there is little discussion of how to even work toward the ideal abstract design. Most mathematics associated to blockchain is statistical analysis of consensus, while we’re sitting on a mountain of powerful categorical knowledge of systems. At the summit, Prof. Neil Ghani said “it’s like we’re on the Moon, talking about going to Mars, while everyone back on Earth still doesn’t even have fire.” We have more than enough conceptual technology to begin developing an ideal and comprehensive system, if the right minds come together. Theory guides practice, practice motivates theory—the potential is immense.

Fortunately, there are those who have this big picture in mind. Long before the blockchain craze, Jelle saw the fundamental importance of both distributed systems and the need for academic-industrial symbiosis. In the mid-2000’s, he used Petri nets to create process tools for businesses. Employees could design and implement any kind of abstract workflow to more effectively communicate and produce. Jelle would provide consultation to optimize these processes, and integrate them into their existing infrastructure—as it executed, it would generate tasks, emails, forms and send them to designated individuals to be completed for the next iteration. Many institutions would have to shell out millions of dollars to IBM or Fujitsu for this kind of software, and his was more flexible and intuitive. This left a strong impression on Jelle, regarding the power of Petri nets and the impact of deliberate design.

Many experiences like this gradually instilled in Jelle a conviction to expand his knowledge and begin planning bold changes to the world of programming. He attended mathematics conferences, and would discuss with theorists from many relevant subjects. On the island, he told me that it was actually one of Baez’s talks about networks which finally inspired him to go for this huge idea. By sincerely and openly reaching out to the whole community, Jelle made many valuable connections. He invited these thinkers to share his vision—theorists from all over Europe, and some from overseas, gathered in Croatia to learn and begin to develop this project—and it was a great success.

By now you may be thinking, alright kid spill the beans already. Here they are, right into your brain—well, most will be in the next post, but we should at least have a quick overview of some of the main ideas not already discussed.

The notion of open system complements compositionality. The great difference between closure and openness, in society as well as theory, was a central theme in many of our conversations during the summit. Although we try to isolate and suspend life and cognition in abstraction, the real, concrete truth is what flows through these ethereal forms. Every system in Statebox is implicitly open, and this impels design to idealize the inner and outer connections of processes. Open systems are central to the Baez Network Theory research team. There are several ways to categorically formalize open systems; the best are still being developed, but the first main example can be found in The Algebra of Open and Interconnected Systems by Brendan Fong, an early member of the team.

Monoidal categories, as this blog knows well, represent systems with both series and parallel processes. One of the great challenge of this new era of interconnection is distributed computation—getting computers to work together as a supercomputer, and monoidal categories are essential to this. Here, objects are data types, and morphisms are computations, while composition is serial and tensor is parallel. As Dr. Baez has demonstrated with years of great original research, monoidal categories are essential to understanding the complexity of the world. If we can connect our knowledge of natural systems to social systems, we can learn to integrate valuable principles—a key example being complete resource cognizance.

Petri nets are presentations of free strict symmetric monoidal categories, and as such they are ideal models of “normal” computation, i.e. associative, unital, and commutative. Open Petri nets are the workhorses of Statebox. They are the morphisms of a category which is itself monoidal—and via openness it is even richer and more versatile. Most importantly it is compact closed, which introduces a simple but crucial duality into computation—input-output interchange—which is impossible in conventional cartesian closed computation, and actually brings the paradigm closer to quantum computation

Petri nets represent processes in an intuitive, consistent, and decentralized way. These will be multi-layered via the notion of operad and a resourceful use of Petri net tokens, representing the interacting levels of a system. Compositionality makes exploring their state space much easier: the state space of a big process can be constructed from those of smaller ones, a technique that more often than not avoids state space explosion, a long-standing problem in Petri net analysis. The correspondence between open Petri nets and a logical calculus, called place/transition calculus, allows the user to perform queries on the Petri net, and a revolutionary technique called information-gain computing greatly reduces response time.

Dependently typed functional programming is the exoskeleton of this beautiful beast; in particular, the underlying language is Idris. Dependent types arose out of both theoretical mathematics and computer science, and they are beginning to be recognized as very general, powerful, and natural in practice. Functional programming is a similarly pure and elegant paradigm for “open” computation. They are fascinating and inherently categorical, and deserve whole blog posts in the future.

Even economics has opened its mind to categories. Statebox is very fortunate to have several of these pioneers—open game theory is a categorical, compositional version of game theory, which allows the user to dynamically analyze and optimize code. Jules’ choice of the term “teleological category” is prescient; it is about more than just efficiency—it introduces the possibility of building principles into systems, by creating game-theoretical incentives which can guide people to cooperate for the greater good, and gradually lessen the influence of irrational, selfish priorities.

Categories are the language by which Petri nets, functional programming, and open games can communicate—and amazingly, all of these theories are unified in an elegant representation called string diagrams. These allow the user to forget the formalism, and reason purely in graphical terms. All the complex mathematics goes under the hood, and the user only needs to work with nodes and strings, which are guaranteed to be formally correct.

Category theory also models the data structures that are used by Statebox: Typedefs is a very lightweight—but also very expressive—data structure, that is at the very core of Statebox. It is based on initial F-algebras, and can be easily interpreted in a plethora of pre-existing solutions, enabling seamless integration with existing systems. One of the core features of Typedefs is that serialization is categorically internalized in the data structure, meaning that every operation involving types can receive a unique hash and be recorded on the blockchain public ledger. This is one of the many components that make Statebox fail-resistant: every process and event is accounted for on the public ledger, and the whole history of a process can be rolled back and analyzed thanks to the blockchain technology.

The Statebox team is currently working on a monograph that will neatly present how all the pertinent categorical theories work together in Statebox. This is a formidable task that will take months to complete, but will also be the cleanest way to understand how Statebox works, and which mathematical questions have still to be answered to obtain a working product. It will be a thorough document that also considers important aspects such as our guiding ethics.

The team members are devoted to creating something positive and different, explicitly and solely to better the world. The business paradigm is based on the principle that innovation should be open and collaborative, rather than competitive and exclusive. We want to share ideas and work with you. There are many blooming endeavors which share the ideals that have been described in this article, and we want them all to learn from each other and build off one another.

For example, Statebox contributor and visionary economist Viktor Winschel has a fantastic project called Oicos. The great proponent of applied category theory, David Spivak, has an exciting and impressive organization called Categorical Informatics. Mike Stay, a past student of Dr. Baez, has started a company called Pyrofex, which is developing categorical distributed computation. There are also somewhat related languages for blockchain, such as Simplicity, and innovative distributed systems such as Iota and RChain. Even Ethereum is beginning to utilize categories, with Casper. And of course there are research groups, such as Network Theory and Mathematically Structured Programming, as well as so many important papers, such as Algebraic Databases. This is just a slice of everything going on; as far as I know there is not yet a comprehensive account of all the great applied category theory and distributed innovations being developed. Inevitably these endeavors will follow the principle they share, and come together in a big way. Statebox is ready, willing, and able to help make this reality.

If you are interested in Statebox, you are welcomed with open arms. You can contact Jelle at jelle@statebox.io, Fabrizio at fabrizio@statebox.org, Emi at emi@statebox.io, Anton at anton@statebox.io; they can provide more information, connect you to the discussion, or anything else. There will be a second summit in 2018 in about six months, details to be determined. We hope to see you there. Future posts will keep you updated, and explain more of the theory and design of Statebox. Thank you very much for reading.

P.S. Found unexpected support in Šibenik! Great bar—once a reservoir.

by John Baez at January 22, 2018 03:12 AM

January 21, 2018

Christian P. Robert - xi'an's og

distributions for parameters [seminar]
Next Thursday, January 25, Nancy Reid will give a seminar in Paris-Dauphine on distributions for parameters that covers different statistical paradigms and bring a new light on the foundations of statistics. (Coffee is at 10am in the Maths department common room and the talk is at 10:15 in room A, second floor.)

Nancy Reid is University Professor of Statistical Sciences and the Canada Research Chair in Statistical Theory and Applications at the University of Toronto and internationally acclaimed statistician, as well as a 2014 Fellow of the Royal Society of Canada. In 2015, she received the Order of Canada, was elected a foreign associate of the National Academy of Sciences in 2016 and has been awarded many other prestigious statistical and science honours, including the Committee of Presidents of Statistical Societies (COPSS) Award in 1992.

Nancy Reid’s research focuses on finding more accurate and efficient methods to deduce and conclude facts from complex data sets to ultimately help scientists find specific solutions to specific problems.

There is currently some renewed interest in developing distributions for parameters, often without relying on prior probability measures. Several approaches have been proposed and discussed in the literature and in a series of “Bayes, fiducial, and frequentist” workshops and meeting sessions. Confidence distributions, generalized fiducial inference, inferential models, belief functions, are some of the terms associated with these approaches.  I will survey some of this work, with particular emphasis on common elements and calibration properties. I will try to situate the discussion in the context of the current explosion of interest in big data and data science. 

by xi'an at January 21, 2018 11:18 PM

Clifford V. Johnson - Asymptotia

Thinking…

(Process from the book.)

The book is appearing on bookshelves in the UK this week! Warehouses filling up for UK online shipping too.

-cvj Click to continue reading this post

The post Thinking… appeared first on Asymptotia.

by Clifford at January 21, 2018 07:41 PM

Peter Coles - In the Dark

Come Sunday

I can’t believe that I’ve been sharing music on this blog for almost a decade and haven’t yet posted this. It’s a beautiful Duke Ellington song Come Sunday, written for the extended concert suite Black, Brown and Beige, later appeared in the Duke Ellington concerts of sacred music, and eventually became a jazz standard. It was written for solo voice along with the full Ellington band, but this almost entirely a cappella version featuring the great gospel singer Mahalia Jackson with a few bits of Duke Ellington on piano is my favourite version. It’s a hauntingly elusive melody, but Mahalia Jackson fills it with her entire soul…

by telescoper at January 21, 2018 01:38 PM

January 20, 2018

Emily Lakdawalla - The Planetary Society Blog

Space Policy & Advocacy Program Quarterly Report - January 2018
As a service to our members and to promote transparency, The Planetary Society's Space Policy and Advocacy team publishes quarterly reports on their activities, actions, priorities, and goals in service of their efforts to promote space science and exploration in Washington, D.C.

January 20, 2018 01:37 AM

January 19, 2018

Lubos Motl - string vacua and pheno

Which stringy paper inspired Sheldon last night?
One week ago, Sheldon Cooper returned to string theory.



Even Edward Measure knew that dark matter was a bad fit.




While young Sheldon was finally bought a computer (while Missy got a plastic pony and their father bought some beer which almost destroyed his marriage), the old Sheldon took pictures of himself with his baby, some work in string theory.




I wonder how many of you can isolate the paper(s) that contain the same (or similar) diagrams and equations as Sheldon's new work. ;-)



Two screenshots are embedded in this blog post. Click at them to magnify them. A third one is here, too.

Incidentally, in order to focus on work, Sheldon had to hire his old bedroom where he no longer lives. He acted pleasantly which drove Leonard up the wall. Meanwhile, Rajesh met a blonde in the planetarium and had sex with her. It turned out she was married, her husband was unhappy about the relationship, Rajesh restored that broken marriage, and almost dated the husband afterwards. ;-)

by Luboš Motl (noreply@blogger.com) at January 19, 2018 07:10 PM

January 18, 2018

Emily Lakdawalla - The Planetary Society Blog

Planetary Society CEO Bill Nye to Attend the State of the Union Address
When a congressman and current nominee for NASA Administrator asks you to be his guest at the state of the union address in Washington, D.C., how do you respond? For us, the answer was easy. Yes, Bill would be there.

January 18, 2018 08:45 PM

Lubos Motl - string vacua and pheno

Multiverse and falsifiability: W*it vs Carroll
Sean Carroll has released an essay elaborating upon his talk at Why Trust a Theory, a late 2015 meeting organized by TRF guest blogger Richard Dawid:
Beyond Falsifiability: Normal Science in a Multiverse (arXiv, Jan 2018)
On his self-serving blog, Carroll promoted his own preprint.

Well, once I streamline them, his claims are straightforward. Even though we can't see outside the cosmic horizon – beyond the observable Universe – all the grand physical or cosmological theories still unavoidably have something to say about those invisible realms. These statements are scientifically interesting and they're believed to be more correct if the corresponding theories make correct predictions, are simpler or prettier explanations of the existing facts, and so on.




There's a clear risk that my endorsement could be misunderstood. Well, I think that Sean Carroll's actual papers about the Universe belong among the bad ones. So while I say it's right and legitimate to be intrigued by all these questions, propose potential answers, and claim that some evidence has strengthened some answers and weakened others, it doesn't mean that I actually like the way how Carroll is using this freedom.

In particular, his papers that depend on his completely wrong understanding of the probability calculus – and that promote as ludicrously wrong concepts as the Boltzmann brains – are rather atrocious as argued in dozens of TRF blog posts.




Peter W*it isn't quite hysterical but unsurprisingly, he still trashes Carroll's papers. According to W*it, Carroll is arguing against a straw man – the "naive Popperazism" – while he ignores the actual criticism which is that the evaluation of these multiverse and related theories (and even all of string theory, according to W*it) isn't just hard: it's impossible. The straw man is the claim that "things that can't be observed should never be discussed in science". W*it asserts that he has never made this claim; well, I would disagree because that's what he has said a few times and what he wanted his fans to believe very many times.

But let's ignore that the straw man isn't quite a straw man. Let's discuss W*it's claim that it's impossible to validate the multiverse-like cosmological theories even in principle. Is it impossible?

Well, it just isn't impossible. The literature is full of – correct and wrong – arguments saying that one theory or one model is more likely or less likely because it implies or follows from some results or assumptions that are rather empirically successful or unsuccessful. I found it necessary to say that the literature sometimes contains wrong claims of this type as well. But they're wrong claims of the right type. The authors are still trying to do science properly – and many other scientists do it properly and it's clearly possible to do it properly, even in the presence of the multiverse.

As Carroll correctly says, all this work still derives the scientific authority from abduction, Bayesian inference, and empirical success. For example, Jack Sarfatti has a great scientific authority because he was abducted by the extraterrestrial aliens. ;-) OK, no, this isn't the "abduction" that Carroll talks about. Carroll recommends "abduction" as a buzzword to describe the "scientific inference leading the scientists to the best explanation". So "abduction" is really a special kind of inference or induction combined with some other methods and considerations that are common in theoretical physics and a longer explanation may be needed – and there would surely be disagreements about details.

If you're a physics student who knows how to do physics properly, you don't need to know whether someone calls it inference, induction, or abduction!

But it's possible to do science even in the presence of unobservable objects and realms that are needed by the theory. The theory still deals with observable quantities as well. And if the agreement with the observed properties of the observable entities logically implies the need for some particular unobserved entities and their properties, well, then the latter are experimentally supported as well – although they are unobservable directly, they're indirectly supported because they seem necessary for the right explanation of the things that are observable.

Also, W*it observes that "some theoretical papers predict CMB patterns, others don't". But even if one proposes a new class of theories or a paradigm that makes no specific observable CMB or other predictions, it may still be a well-defined, new, clever class of theories and the particular models constructed within this new paradigm will make such CMB predictions. Because the discovery of the class or the paradigm or the idea is a necessary condition for finding the new models that make (new and perhaps more accurate) testable predictions, it's clearly a vital part of science as well – despite the particular papers' making no observable predictions!

Peter W*it has never done real science in his life so he can't even imagine that this indirect reasoning and "abduction" – activities that most of the deep enough theoretical physics papers were always all about – is possible at all. He's just a stupid, hostile layman leading an army of similar mediocre bitter jihadists in their holy war against modern physics.

There's another aspect of W*it's criticism I want to mention. At some moment, he addresses Carroll's "another definition of science":
Carroll: The best reason for classifying the multiverse as a straightforwardly scientific theory is that we don’t have any choice. This is the case for any hypothesis that satisfies two criteria:
  • It might be true.
  • Whether or not it is true affects how we understand what we observe.
Well, I am not 100% certain it's right to say that we can't avoid the multiverse. On the other hand, I understand the case for the multiverse and I surely agree that physics is full of situations in which "we don't have a choice" is the right conclusion.

Peter W*it doesn't like Carroll's quote above because it also allows "supreme beings" as a part of science. I don't see what those exact sentences have to do with "supreme beings" – why would an honest person who isn't a demagogue suddenly start to talk about "supreme beings" in this context. Nevertheless, I see a clear difference in W*it's recipe what science should look like and it's the following principle:
Whatever even remotely smells of "supreme beings", "God", or any other concept that has been labeled blasphemous ;-) by W*it has to be banned in science.
W*it hasn't articulated this principle clearly – because he doesn't have the skills to articulate any ideas clearly. But one may prove that he has actually applied this principle thousands of times. Apologies but this principle is incompatible with honest science that deals with deep questions.

Important discoveries in theoretical physics may totally contradict and be the "opposite" of stories about "supreme beings" (or any other "unpopular" concepts); but they may also resemble the stories about "supreme beings" in any way – in a way that simply cannot be "constrained" by any pre-existing assumptions. The correct theories of physics must really be allowed to be anything. Any idea, however compatible with Hitler's or Stalin's or W*it's ideology or political interests, must be allowed to "run" as a candidate for the laws of physics.

W*it clearly denies this basic point – that science is an open arena without taboos where all proposed ideas must compete fairly. He wants science to be just a servant that rationalizes answers that were predetermined by subpar pseudointellectuals such as himself and their not terribly intelligent prejudices.

That's not what real good science looks like. In real good science, answers are only determined once a spectrum of hypotheses is proposed, they are compared, and one of them succeeds in the empirical and logical tests much more impressively than others. Only when that's done, the big statements about "what properties the laws of physics should have" can be made authoritatively. W*it is making them from the beginning, before he actually does or learn any science, and that's not how a scientist may approach these matters.

If the vertices in the Feynman diagram were found to be generalized prayers to a supreme being, and if the corresponding scattering amplitudes could be interpreted as responses of the supreme being that generalize God's response to Christian prayers ;-), then it's something that physicists would have to seriously study. I don't really propose such a scenario seriously and my example is meant to be a satirical exaggeration (well, even if such an analogy were possible, I still think it would also be possible to ignore it and avoid the Christian jargon completely). But I am absolutely serious about the spirit. Whether something sounds unacceptable or ludicrous to people with lots of prejudices should never be used as an argument against a scientific model, theory, framework, or paradigm. (See Milton Friedman's F-twist for a strengthened version of that claim.)

That's why the influence of subpar pseudointellectuals such as W*it on science must remain strictly at zero if science is supposed to remain scientific – and avoid the deterioration into a slut whose task is to improve the image of a pre-selected master ideology or philosophy in the eyes of the laymen. Just like it was wrong for the Catholic church to demand that science serves the church or its God, it was also wrong to demand science to serve the Third Reich or the Aryan race or the communist regime, and it is wrong to demand that science must serve the fanatical atheists or West's leftists in general.



P.S.: In a comment, W*it wrote:
Rod Deyo,
Polchinski provided a reductio ad absurdum argument against the Bayesianism business in a paper for the same proceedings as the Carroll one. He calculated a Bayesian probability of “over 99.7%” for string theory, and 94% for the multiverse.
99.5% of this stuff written by W*it is composed of bullšit because Joe Polchinski was 97% serious.

Polchinski realizes that none of these values is "canonical" or "independent of various choices" and he likes to say (and explicitly wrote in his explanation of his usage of the Bayesian inference) that the Bayesian reasoning isn't the main point – physics is the main point – but he simply wanted to be quantitative about his beliefs and these are the actual fair subjective probabilities for the propositions he ended up with. That's completely analogous to the number 10% by which Polchinski once quantified his subjective belief that a cosmic string would be experimentally observed in some period of time (I forgot whether it was "ever" or "before some deadline"). I have repeatedly written similar numbers expressing my beliefs. It makes some sense. We don't need to talk in this way but we may and it's sometimes useful.

So Polchinski hasn't provided any argument against the Bayesian inference. He has pretty much seriously used the Bayesian inference in a somewhat unusual setup.

by Luboš Motl (noreply@blogger.com) at January 18, 2018 08:33 PM

Tommaso Dorigo - Scientificblogging

Accelerating Dark Matter Searches With Machine Learning
I am currently spending the week in Leiden (Netherlands), attending to a very interesting workshop that brought together computer scientists, astronomers, astrophysicists, and particle physicists to discuss how to apply the most cutting-edge tools in machine learning to improve our chances of discovering dark matter, the unknown non-luminous substance that vastly overweights luminous matter in the Universe.

read more

by Tommaso Dorigo at January 18, 2018 06:14 PM

Emily Lakdawalla - The Planetary Society Blog

Let's talk about NASA's latest commercial crew delay
SpaceX and Boeing might not be certified to carry astronauts to the International Space Station until 2019 or 2020.

January 18, 2018 04:19 AM

January 17, 2018

Sean Carroll - Preposterous Universe

Beyond Falsifiability

I have a backlog of fun papers that I haven’t yet talked about on the blog, so I’m going to try to work through them in reverse chronological order. I just came out with a philosophically-oriented paper on the thorny issue of the scientific status of multiverse cosmological models:

Beyond Falsifiability: Normal Science in a Multiverse
Sean M. Carroll

Cosmological models that invoke a multiverse – a collection of unobservable regions of space where conditions are very different from the region around us – are controversial, on the grounds that unobservable phenomena shouldn’t play a crucial role in legitimate scientific theories. I argue that the way we evaluate multiverse models is precisely the same as the way we evaluate any other models, on the basis of abduction, Bayesian inference, and empirical success. There is no scientifically respectable way to do cosmology without taking into account different possibilities for what the universe might be like outside our horizon. Multiverse theories are utterly conventionally scientific, even if evaluating them can be difficult in practice.

This is well-trodden ground, of course. We’re talking about the cosmological multiverse, not its very different relative the Many-Worlds interpretation of quantum mechanics. It’s not the best name, as the idea is that there is only one “universe,” in the sense of a connected region of space, but of course in an expanding universe there will be a horizon past which it is impossible to see. If conditions in far-away unobservable regions are very different from conditions nearby, we call the collection of all such regions “the multiverse.”

There are legitimate scientific puzzles raised by the multiverse idea, but there are also fake problems. Among the fakes is the idea that “the multiverse isn’t science because it’s unobservable and therefore unfalsifiable.” I’ve written about this before, but shockingly not everyone immediately agreed with everything I have said.

Back in 2014 the Edge Annual Question was “What Scientific Theory Is Ready for Retirement?”, and I answered Falsifiability. The idea of falsifiability, pioneered by philosopher Karl Popper and adopted as a bumper-sticker slogan by some working scientists, is that a theory only counts as “science” if we can envision an experiment that could potentially return an answer that was utterly incompatible with the theory, thereby consigning it to the scientific dustbin. Popper’s idea was to rule out so-called theories that were so fuzzy and ill-defined that they were compatible with literally anything.

As I explained in my short write-up, it’s not so much that falsifiability is completely wrong-headed, it’s just not quite up to the difficult task of precisely demarcating the line between science and non-science. This is well-recognized by philosophers; in my paper I quote Alex Broadbent as saying

It is remarkable and interesting that Popper remains extremely popular among natural scientists, despite almost universal agreement among philosophers that – notwithstanding his ingenuity and philosophical prowess – his central claims are false.

If we care about accurately characterizing the practice and principles of science, we need to do a little better — which philosophers work hard to do, while some physicists can’t be bothered. (I’m not blaming Popper himself here, nor even trying to carefully figure out what precisely he had in mind — the point is that a certain cartoonish version of his views has been elevated to the status of a sacred principle, and that’s a mistake.)

After my short piece came out, George Ellis and Joe Silk wrote an editorial in Nature, arguing that theories like the multiverse served to undermine the integrity of physics, which needs to be defended from attack. They suggested that people like me think that “elegance [as opposed to data] should suffice,” that sufficiently elegant theories “need not be tested experimentally,” and that I wanted to “to weaken the testability requirement for fundamental physics.” All of which is, of course, thoroughly false.

Nobody argues that elegance should suffice — indeed, I explicitly emphasized the importance of empirical testing in my very short piece. And I’m not suggesting that we “weaken” anything at all — I’m suggesting that we physicists treat the philosophy of science with the intellectual care that it deserves. The point is not that falsifiability used to be the right criterion for demarcating science from non-science, and now we want to change it; the point is that it never was, and we should be more honest about how science is practiced.

Another target of Ellis and Silk’s ire was Richard Dawid, a string theorist turned philosopher, who wrote a provocative book called String Theory and the Scientific Method. While I don’t necessarily agree with Dawid about everything, he does make some very sensible points. Unfortunately he coins the term “non-empirical theory confirmation,” which was an extremely bad marketing strategy. It sounds like Dawid is saying that we can confirm theories (in the sense of demonstrating that they are true) without using any empirical data, but he’s not saying that at all. Philosophers use “confirmation” in a much weaker sense than that of ordinary language, to refer to any considerations that could increase our credence in a theory. Of course there are some non-empirical ways that our credence in a theory could change; we could suddenly realize that it explains more than we expected, for example. But we can’t simply declare a theory to be “correct” on such grounds, nor was Dawid suggesting that we could.

In 2015 Dawid organized a conference on “Why Trust a Theory?” to discuss some of these issues, which I was unfortunately not able to attend. Now he is putting together a volume of essays, both from people who were at the conference and some additional contributors; it’s for that volume that this current essay was written. You can find other interesting contributions on the arxiv, for example from Joe Polchinski, Eva Silverstein, and Carlo Rovelli.

Hopefully with this longer format, the message I am trying to convey will be less amenable to misconstrual. Nobody is trying to change the rules of science; we are just trying to state them accurately. The multiverse is scientific in an utterly boring, conventional way: it makes definite statements about how things are, it has explanatory power for phenomena we do observe empirically, and our credence in it can go up or down on the basis of both observations and improvements in our theoretical understanding. Most importantly, it might be true, even if it might be difficult to ever decide with high confidence whether it is or not. Understanding how science progresses is an interesting and difficult question, and should not be reduced to brandishing bumper-sticker mottos to attack theoretical approaches to which we are not personally sympathetic.

by Sean Carroll at January 17, 2018 04:44 PM

Tommaso Dorigo - Scientificblogging

On The Qualifications Of Peer Reviewers For Scientific Papers
Peer review is the backbone of high quality scientific publications. Although the idea that only articles that are approved by a set of anonymous nitpickers can ever see the light of publication on "serious" journals is old and perfectible, there is currently no valid alternative to identify verified, rigorous scientific work, and to filter out unsubstantiated claims, and methodologically unsound results - the scientific analogue of "fake news".

read more

by Tommaso Dorigo at January 17, 2018 02:33 PM

Emily Lakdawalla - The Planetary Society Blog

Dawn Journal: 4 Billion Miles
Permanently in residence at dwarf planet Ceres, Dawn is now preparing to add some finishing touches to its mission.

January 17, 2018 01:00 PM

January 16, 2018

CERN Bulletin

Happy New Year 2018!

Early in the new year, the Staff Association wishes you and your loved ones its best wishes for a happy and healthy New Year 2018, as well as individual and collective success. May it be filled with satisfaction in both your professional and private life.

A Difficult start

The results of the election of the new Staff Council were published on 20th November 2017 in Echo N° 281.

The process of renewing the Staff Council proceeded very well: candidates in numbers, from all departments, ranks and categories (staff, fellows and associates); and the turnout rate in this election is up compared to previous elections ... something to be celebrated and congratulated.

In accordance with the statutes of the Staff Association, the new Staff Council shall, at its first meeting, elect an Executive Committee comprising a “Bureau” with four statutory posts: President, Vice-President, Secretary and Treasurer.

However, while the composition of the Executive Committee was easily established, the appointment of the “Bureau” proved to be more complicated, for too few delegates were ready to get involved at this level in the current context.

Chief among the reasons put forth, is the impossibility for many delegates to devote at least 50 % of their work time in the Staff Association; considering the workload linked to professional activities and the chronic shortage of personnel in many services at CERN, this argument is well known within the Staff Association.

Secondly, fears were raised about putting one’s career “on-hold” by spending more time with the Staff Association. More time devoted to the Staff Association can mean a degraded recognition of the merit. This argument is new and worrying to say the least.

Finally, and we are down to the heart of the issue, several delegates consider that they cannot trust our interlocutors in the consultation process (“Concertation” in French). This feeling certainly follows the difficult management of some issues, but also stems from a difference in the understanding, between the Management and the Staff Association, of the principle of consultation which governs our relations.

Executive Crisis Committee

The Staff Council, in its meeting of 5 December 2017, finally elected and established a Crisis Executive Committee for a three months period ending 31 March 2018, with the main objective to find a resolution for the "Nursery and School" issue, (see Echo N° 282 of 11/12/2017).

The composition of this interim crisis committee is as follows:

A year of challenges

2018 is therefore from the start a year of challenges for the Staff Association, and the consultation process (“Concertation”) with the Management.

The Staff Association reaffirms its will to work in a climate of trust and good faith, two necessary elements of a fruitful and constructive consultation process.

We wish you once again a happy new year 2018!

January 16, 2018 03:01 PM

CERN Bulletin

Entretien avec M. Bernard Dormy, Président du TREF, courant novembre 2017

M. Bernard Dormy, Président du TREF (Tripartite Employment Conditions Forum) (voir Echo n° 242) a terminé son mandat à la fin de l'année 2017.

L’Association du personnel a souhaité s’entretenir avec lui sur le CERN et son personnel et, entre autres, sur le modèle de concertation.

Cette publication est également l’occasion pour l’Association du personnel de saluer M. Dormy pour l’engagement dont il a fait preuve depuis 2003, année où il a débuté au TREF comme délégué français. Il faut croire que son mandat au sein de ce forum lui a particulièrement plu, puisque M. Dormy a occupé les fonctions de Vice-président du TREF de 2007 à 2011, puis de Président de 2012 à 2017.

Sous sa présidence, M. Dormy a toujours veillé à ce que la concertation se déroule dans les meilleures conditions possibles, dans un esprit constructif et de respect mutuel. Il a également mis un accent tout particulier sur la diversité au CERN; pas une seule réunion du TREF sans un point sur la diversité et sur les avancées dans ce domaine.

Avant de passer aux questions-réponses, l’Association du personnel tient à remercier M. Dormy qui a, il nous l’a souvent dit, un profond respect pour le personnel du CERN. À notre tour, nous lui adressons nos très respectueuses et chaleureuses salutations.

M. Dormy, quels ont été vos premiers contacts avec le CERN et quels souvenirs en avez-vous ?

J’ai rejoint la délégation française au TREF du CERN il y a quinze ans, et son comité des finances quelques années plus tard. Mais mon premier contact avec le CERN est bien plus ancien. Au lycée, un professeur de physique doué pour éveiller des vocations en recherche fondamentale nous avait persuadés de visiter le CERN si nous nous trouvions à Genève; le moment venu, je n’ai malheureusement pas réussi à le faire, et j’ai dû me rabattre sur une carte postale achetée en ville. Quant à ma vocation, malgré un goût certain pour les sciences, j’ai dû plus tard me contenter d’HEC et de l’ENA.

Mon second contact remonte aux années 80. Le directeur scientifique des humanités du CNRS, dont j’étais l’adjoint, relatait un échange avec un célèbre physicien lors d’un Conseil du CNRS :

  • Monsieur N., avec une miette de vos accélérateurs, je fais vivre tous les laboratoires de mon secteur.
  • Monsieur P., je suis bien élevé, je ne laisse jamais tomber de miettes.

Amusant certes, mais j’ai heureusement trouvé plus d’ouverture d’esprit chez les scientifiques que j’ai ensuite fréquentés, fussent-ils utilisateurs de très grands instruments comme le CERN.

Un troisième contact enfin a été l’écho d’un dialogue entre un ministre semblant intéressé par la seule recherche appliquée (si possible en entreprise) et un autre grand physicien. À la question « Monsieur le ministre, savez-vous qui a inventé le web ? », celui-ci aurait répondu « Bill Gates, bien sûr ». De quoi vous donner l’envie de connaître enfin le berceau du web de l’intérieur !

Qu’avez-vous « découvert » au CERN ?

Ma découverte concrète du CERN au sein du TREF a été d’abord celle du multilatéralisme. On le décrit souvent comme l’art du compromis entre des positions différentes, ce qui est un peu réducteur, car il arrive souvent, je l’ai constaté au TREF ou au Comité des finances, qu’un accord unanime se fasse sans qu’il soit nécessaire de construire une position médiane acceptable par tous. Mais, pour moi, le multilatéralisme, c’est avant tout la découverte que les modes de pensée et surtout d’expression sont parfois assez différents selon les États, même si ceux-ci partagent une même vision du monde. Ce qui apparaît comme une formulation un peu brutale aux yeux de certains d’entre eux peut en même temps être vu comme peu clair et alambiqué par d’autres. Ce qui est aujourd’hui intégré au comportement de tous les jours dans certains pays, comme, par exemple, la place faite aux femmes dans la société demande encore une politique volontariste dans d’autres. Cela rend la présidence du TREF passionnante, et, je le pense sincèrement, conduit rapidement à vivre cette diversité plus comme un enrichissement que comme une contrainte.

Comment pourriez-vous définir la concertation au CERN ?

La consultation des personnels dans les grandes organisations publiques ou privées prend ou a pris des formes très diverses dans les divers États membres du CERN, allant du recueil d’un simple avis dont le poids dans la décision finale est souvent modeste, jusqu’à la cogestion. La concertation telle qu’elle est pratiquée au CERN me paraît marquer un équilibre entre ces deux extrêmes. Pour la résumer, je la décrirais volontiers comme la recherche d’une position commune entre l’Association du personnel, l’administration du CERN et ses États membres, chacun étant préalablement et loyalement informé des divers aspects du dossier et ayant eu la possibilité de confronter sa position à celle des autres. Tout comme le Comité de Concertation Permanent (CCP), le TREF joue un rôle non négligeable dans ce processus.

J’ai été frappé lors de mes premières séances comme délégué au TREF par la diversité des origines professionnelles des délégués, qui, au-delà de leurs spécialisations personnelles, ne partageaient pas tous le même socle de connaissances de base. Je suis moi-même arrivé dans un monde presque entièrement à découvrir, armé de mes seuls souvenirs du droit de la fonction publique internationale appris à la fac et, je l’espère, d’un peu de bon sens. Le partage d’un même corpus d’informations en amont des débats est pourtant essentiel au bon fonctionnement du TREF. C’est pourquoi, avec Jean-Marc Saint-Viteux, nous avons décidé de présenter le CERN et son environnement (notamment économique), les modes de fonctionnement du TREF et l’historique de ses décisions, ainsi que les conditions juridiques du processus de concertation, notamment lors de la Revue quinquennale des salaires et des conditions d’emploi. Cette information de base est depuis six ans offerte à tous les nouveaux délégués des États membres, afin de faire en sorte que tous partagent un même niveau d’information commune avant les séances du TREF.

Comment compareriez-vous la concertation au CERN avec les processus en place dans les autres Organisations ?

Les comparaisons sont tentantes, surtout pour quelqu’un qui, comme moi, a eu la chance de présider les comités administratifs et financiers de deux autres grandes infrastructures de recherche, actuellement en construction à Darmstadt et à Lund. À l’expérience, je pense qu’il faut s’en garder.

Le CERN est une organisation internationale, dont le Conseil fixe le droit applicable à ses personnels, sous le contrôle du juge international. En Allemagne et en Suède, j’ai rencontré des organisations où les personnels sont régis par des conventions collectives nationales. Le rôle des organes de consultation des personnels y est donc limité à l’application des règles, à l’exclusion de leur élaboration. Leurs liens avec Conseil et Comité des finances existent certes, mais sont par nature plus limités qu’ils ne le sont au CERN.

Le CERN est donc singulier à cet égard, comme toute organisation internationale. Mais il est lui-même une organisation internationale singulière, la plus ancienne des grandes infrastructures scientifiques internationales en exploitation. D’où cette culture d’organisme que l’on ne rencontre pas ailleurs à un tel niveau de développement. Un lieu où les personnels se disent-ils plus « cernois » que français, allemands, polonais... ne se rencontre pas tous les jours. Comme nos voisins vaudois, les gens du CERN pourraient dire « y’en a pas comme nous ».

Que pensez-vous du personnel du CERN et du travail fait par l’Association du personnel ?

C’est une banalité de dire que le CERN et les investissements considérables qui y ont été et y seront faits ne seraient rien sans les femmes et sans les hommes qui le composent. Mais il est bon parfois de répéter des banalités, car les délégués aux divers organes du CERN ont sans cesse à composer entre les exigences d‘une maîtrise des budgets alloués par les divers États membres et celles d’une politique du personnel permettant d’attirer les meilleurs et de leur offrir de bonnes conditions de travail.

Dans ce cadre, j’ai envie de répondre à la question « À quoi sert l’Association du personnel ? » par une simple phrase : « elle sert à rendre service aux membres du personnel ». Un exemple concret de ses services, qui contribue à l’attractivité du CERN : en entrant sur le campus, on laisse à droite une crèche et un jardin d’enfants, qui sont gérés par l’Association du personnel. Un autre exemple : par sa simple présence au TREF, l’Association du personnel aide chacun à ne pas oublier que le personnel ne doit pas être vu comme un simple « coût », à se souvenir qu’il y a de vrais hommes et de vraies femmes derrière l’appellation générale de « personnel ».

Selon vous, quel est l’avenir du CERN ?

J’ai bien envie de répondre par la boutade attribuée à Niels Bohr (et reprise par Pierre Dac), « en matière scientifique, il est difficile de prévoir, surtout l’avenir ». Qui aurait pu prévoir que les principes du web, imaginés pour faciliter l’accès commun aux données de laboratoires de recherche, allaient conduire à des modifications si profondes de nos sociétés contemporaines. Soyons modestes, et faisons confiance à la recherche, y compris la plus fondamentale, qui est un peu, pour employer le vocabulaire des économistes, le capital-risque de nos États.

Ceci n’interdit pas de faire des souhaits. En ce qui concerne le personnel, les membres du TREF savent combien j’espère voir se développer le rôle des femmes dans la science, et particulièrement dans les grandes infrastructures de recherche. Il fallu quelque 60 ans pour voit une femme Directrice générale du CERN, plus de vingt ans pour qu’une femme préside le TREF et à peine moins pour y voir une femme au sein de la délégation de l’Association du personnel. Le fait que l’on souligne ces élections montre bien qu’on les considère comme des événements sortant de l’ordinaire. Pour en faire dans l’avenir des nouvelles banales, un changement de mentalités devrait s’opérer. Je suis persuadé que l’on ne l’obtiendra pas par la contrainte, et j’approuve totalement le CERN d’avoir rejeté les politiques dites de discrimination positive, qui jouent à terme contre celles qu’elles souhaitent aider. Je pense au contraire qu’une pédagogie continue peut aider chacun à considérer comme normal de choisir ses collaborateurs en fonction de leur seule compétence. C’est pourquoi j’ai demandé qu’une communication sur la place des femmes dans l’Organisation, et, plus largement, sur la politique de diversité, soit faite à chaque réunion du TREF. On en revient à mon propos initial : cette forme de diversité est elle aussi une chance pour tous, non une contrainte.

January 16, 2018 02:01 PM

January 15, 2018

Lubos Motl - string vacua and pheno

Superfluid dark matter, an example of excessive hype
Presenting such papers as revolutions in physics is a full-blown scam

The most recent text on Backreaction is titled Superfluid dark matter gets seriously into business. At this moment, this popular text celebrates a November 2017 preprint by Justin Khoury and two co-authors which added some technicalities to Khoury's program that's been around for some three years.

Justin Khoury is a cosmologist who is well-known for his work on colliding branes cosmologies, chameleon fields, and a few other topics. You should also search Google Scholar for Justin Khoury superfluid. You will find several papers – the most famous of which has 62 citations at this moment. That's fine but much fewer than Khoury's most famous papers that are safely above 1,000 citations. The "revolutionary" November 2017 paper on the "superfluid dark matter" only has one self-citation so far.




Hossenfelder's popular text ends up with this short paragraph:
I consider this one of the most interesting developments in the foundations of physics I have seen in my lifetime. Superfluid dark matter is without doubt a pretty cool idea.
These are big words. Is there some substance for such big words? Well, I could imagine there could be and 1% of the time, I could get slightly excited about the idea. But 99% of the time, I feel certain that there is no conceivable justification for such big words, and not even a justification for words that would be 90% smaller.




Superfluid dark matter is supposed to be a hybrid of the "cold dark matter" paradigm which is the standard way to explain the anomalies in the rotation curves of galaxies and "corresponding" aspects of the expansion of the Universe; and the "modified gravity" which tries to modify the equations of gravity, fails to provide us with a satisfactory picture of physics and cosmology, but could be a "simpler" theory that intriguingly explains some universal phenomenological laws that seem to be obeyed even though "cold dark matter" has no explanation for them.

OK, according to superfluid dark matter, the Universe is filled with some low-viscosity fluid, a superfluid, and it acts like dark matter. But a standardized description of the dynamics within this fluid may also be interpreted as "modified gravity".

It seems like a plausible combination of approaches but the devil is in the details. However, what I find extremely weird is the idea that this rough paradigm is enough for a revolution in cosmology or physics. You know, the "anomalous" galactic rotation curves are either explained with the help of some new matter – which may carry some variable entropy density and which is assumed not to be visible in the telescopes – or without it. This is a Yes/No question. So if there's some extra matter which is a superfluid, it's still some extra matter – in other words, it must be considered an example of dark matter. After all, even superfluid dark matter has to have some microscopic behavior which may be studied by local experiments – it must be composed of some (probably new) particle species.

The Universe must still allow the idealized "empty space" phenomena that have been measured extremely accurately and incorporated into the state-of-the-art theories of particle physics. For this reason, whether or not someone (e.g. Erik Verlinde) gets completely lost in vague, almost religious musings saying that the "spacetime might be a fluid", any "dark matter superfluid" or anything of that sort simply has to be some extra matter added on top of the things we know to exist. Any such dark matter may also be captured by some macroscopic, "hydrodynamic or aerodynamic" equations, and if the dark matter is a superfluid, they may have some special features.

(The empty space might in principle be a "fluid" but if the entropy density were nonzero and variable, the conflict with the tests of relativity would be almost unavoidable because such a fluid would be nothing else than a variation of the aether even though, in this case, it wouldn't be the luminiferous aether but rather the lumo-prohibiting aether. Lumo is light, not only in Esperanto, just to be sure. The entropy density, along with an entropy flux, is a 4-vector and its nonzero value breaks the Lorentz invariance. So any matter with some entropy density does so which is bad. A Lorentz-covariant spacetime fluid could in principle exist but it would have to be a new dual description of string/M-theory and it's clearly hopeless to dream about any Lorentz-covariant "fluid" without a glimpse of evidence of such a connection to string/M-theory.)

But because every dark matter model has such emergent, "hydrodynamic" field equations, I think it's just wrong to sell the "dark matter superfluid" as a totally new paradigm. These authors still add dark matter; and they must still decide whether Einstein's equations hold at the fundamental classical level. One may spread lots of hype about a "revolution" but at the end, it's just another technical model of dark matter, like e.g. the ultralight axion model by Witten et al.

Note that Witten et al. have employed an extremely modest, technical language – which is appropriate despite the fact that their proposal is clever and attractive. This approach is so different from the approach of Ms Hossenfelder.

I don't think that the "superfluid dark matter" papers contain something that would make their reading irresistible. But I find the "framing" of these superfluid dark matter papers in the media and the blogosphere – and the "framing" of many other papers – more important and highly problematic. It seems utterly inconceivable to me that an honest yet competent physicist could consider these papers "one of the most interesting developments in her lifetime".

When you look at the response (followups) by the other physicists and cosmologists, these papers don't even make it to top 100 in the year when they were published. Especially because I know quite something about Ms Hossenfelder, it seems vastly more likely that she has a completely different agenda when she overhypes such papers. What is it?

She has written at least one paper about these MOND and Verlinde issues – the 300th most important derivative paper commenting on the 101st most influential paper in a year ;-) – and she simply has personal incentives to make the world think that this kind of work is very interesting even though it is not. She is working on similar things because she doesn't have the skills (and vitality) needed to work on more interesting and deeper things. She says "it is most interesting and cool" but she really means "its fame is beneficial for her".

The financial news servers (e.g. SeekingAlpha) usually require the authors to disclose their positions in assets that they discuss in their articles. That has good reasons. Someone's being long or short may reduce his or her integrity and encourage him or her to write positive or negative things about the asset. The readers have the right not to be scammed in an easy way – which is why fair publishers insist on informing the readers whether there could be a clash of interests. One should expect the scientific integrity to be much deeper than the integrity of the journalists in the financial media. Sadly, it isn't so these days. Self-serving scammers such as Ms Hossenfelder face no restrictions – they are free to fool and delude everybody and lots of the people in the media want to be fooled and be parts of this scam because they're as unethical as Ms Hossenfelder herself.

Readers should learn how to use Google Scholar to acquire some rough clue about the importance of a paper or idea as evaluated by the body of the genuine scientists. If the folks learned how to use this "simple branch of Google", they could instantly figure out that 99% of the hype is probably rubbish (well, of course, this method isn't waterproof so there would be false positives as well as false negatives). It's too bad that almost no laymen – and, in fact, almost no journalists – are doing these things. So they're constantly drowning in hype and in a superfluid of fairy-tales that overhype papers that are either average or totally wrong.

Self-serving, fake scientists such as Sabine Hossenfelder are obviously the main drivers that propagate this fog and misinformation.

P.S.: In an older popular article about the topic, one at Aeon.CO, Hossenfelder emphasized the point that superfluidity represents a quantum behavior across the Universe. This assertion – which is just another way to add the hype – is really a deep distortion of the issues. A superfluid is nicely described by a classical field theory. Some of the fields seem to behave like the wave function but because this is a macroscopic limit of many particles in the same state, it is really a classical limit, with no minimal uncertainty etc., so the function of the spacetime coordinates isn't a wave function and shouldn't be called a wave function. It is a classical field. The classical limit isn't really any different in the case of a superfluid and in the case of electromagnetism or any other pair of a quantum field theory and its corresponding classical field theory!

by Luboš Motl (noreply@blogger.com) at January 15, 2018 05:39 PM

CERN Bulletin

Exhibition

Cosmos

KOLI

Du 15 au 26 janvier 2018
CERN Meyrin, Main Building


(Nébuleuse d'Orion- KOLI)

KOLI,

Artiste confirmé, diplômé de l’Académie de Beaux Arts de Tirana, depuis 26 ans en Suisse, où il a participé à maintes expositions collectives et organisé 10 expositions privées avec  beaucoup de succès, s’exprime actuellement dans un bonheur de couleur et de matières qui côtoient des hautes sphères… le cosmos !

Gagnant d’un premier prix lors d’une exposition collective organisée par le consulat Italien, il s’est installé au bord du lac dans le canton de Vaud où il vit depuis maintenant déjà 13 ans.

www.kolicreation.com

Pour plus d’informations et demandes d’accès : staff.association@cern.ch | Tél: 022 766 37 38

January 15, 2018 11:01 AM

CERN Bulletin

The Yacthing Club CERN celebrates its 50th anniversary this year

YCC 50th anniversary & Swiss SU Championship 2018, there’s a lot going on in the club!

For those of you that wonder how the YCC operates at CERN the simple answer is that it is made of passionate members that care about the club’s operations. The YCC has reached almost 400 members as of the closing of 2017 and it’s looking forward to bring more members onboard to experience the adrenaline of winds! YCC is not only is the a place to learn how to sail, but it is also a community of international people that gathers during the year through other social events.

There’s nothing better than spending the summer on the lake, learning how to rig and sail a boat, getting to know different people during YCC practices and getting a tan before gathering for a drink in the port! 

When you’re on a boat you need to trust your crew no matter how big it is, especially during strong-wind conditions. It is thanks to this that relationships and friendships begins at YCC, at least this is my personal experience. I had the pleasure to meet exceptional people that are now becoming not only sailing partners, but also friends!

The YCC community is ever-growing and ever-evolving, this year we celebrate the YCC 50th anniversary! It’s been 50 years the club has been founded and boats getting out of Versoix port populating the lake all summer long. The committee is planning several events to celebrate and advertise it!

In order to share this great news there will be a brand new logo for the YCC designed by one of the members! Stay tuned for the reveal!

One of the most important events in the radar is the 2018 Swiss SU Championship that will be organized by the YCC in collaboration with CNV (club Nautique d Versoix). This is the utmost event for all SU categories across Switzerland. There’s a strong team at YCC taking care of the organization of the championship that will be commencing on the 07th of July. The YCC will not only ensure the championship is organized with attention to details, but it will also participate with few SU boats! All the YCC members are welcome to take part.

We hope you’re looking forward the beginning of the season, for those that are missing the lake already there are few winter regattas coming up (SU, J) – in the external regattas section of the website), don’t hesitate to signup and participate!

January 15, 2018 11:01 AM

CERN Bulletin

Cine club

Wednesday 17 January 2018 at 20:00
CERN Council Chamber

Memories of Murder

Directed by Joon-ho Bong
South Korea, 2003, 132 minutes

In a small Korean province in 1986, three detectives struggle with the case of multiple young women being found raped and murdered by an unknown culprit.

Original version Korean; English subtitles

January 15, 2018 10:01 AM

The n-Category Cafe

On the Magnitude Function of Domains in Euclidean Space, I

Guest post by Heiko Gimperlein and Magnus Goffeng.

The magnitude of a metric space was born, nearly ten years ago, on this blog, although it went by the name of cardinality back then. There has been much development since (for instance, see Tom Leinster and Mark Meckes’ survey of what was known in 2016). Basic geometric questions about magnitude, however, remain open even for compact subsets of <semantics> n<annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics>: Tom Leinster and Simon Willerton suggested that magnitude could be computed from intrinsic volumes, and the algebraic origin of magnitude created hopes for an inclusion-exclusion principle.

In this sequence of three posts we would like to discuss our recent article, which is about asymptotic geometric content in the magnitude function and also how it relates to scattering theory.

For “nice” compact domains in <semantics> n<annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics> we prove an asymptotic variant of Leinster and Willerton’s conjecture, as well as an asymptotic inclusion-exclusion principle. Starting from ideas by Juan Antonio Barceló and Tony Carbery, our approach connects the magnitude function with ideas from spectral geometry, heat kernels and the Atiyah-Singer index theorem.

We will also address the location of the poles in the complex plane of the magnitude function. For example, here is a plot of the poles and zeros of the magnitude function of the <semantics>21<annotation encoding="application/x-tex">21</annotation></semantics>-dimensional ball.

poles and zeros of the magnitude function of the 21-dim ball

We thank Simon for inviting us to write this post and also for his paper on the magnitude of odd balls as the computations in it rescued us from some tedious combinatorics.

The plan for the three café posts is as follows:

  1. State the recent results on the asymptotic behaviour as a metric space is scaled up and on the meromorphic extension of the magnitude function.

  2. Discuss the proof in the toy case of a compact domain <semantics>X<annotation encoding="application/x-tex">X\subseteq \mathbb{R}</annotation></semantics> and indicate how it generalizes to arbitrary odd dimension.

  3. Consider the relationship of the methods to geometric analysis and potential ramifications; also state some open problems that could be interesting.

Asymptotic results

As you may recall, the magnitude <semantics>mag(X)<annotation encoding="application/x-tex">\mathrm{mag}(X)</annotation></semantics> of a finite subset <semantics>X n<annotation encoding="application/x-tex">X\subseteq \mathbb{R}^n</annotation></semantics> is easy to define: let <semantics>w:X<annotation encoding="application/x-tex">w:X\to \mathbb{R}</annotation></semantics> be a function which satisfies

<semantics> yXe d(x,y)w(y)=1for allxX.<annotation encoding="application/x-tex"> \sum_{y\in X} \mathrm{e}^{-\mathrm{d}(x,y)} w(y) = 1 \quad \text{for all}\ x \in X. </annotation></semantics>

Such a function is called a weighting. Then the magnitude is defined as the sum of the weights:

<semantics>mag(X)= xXw(x).<annotation encoding="application/x-tex"> \mathrm{mag}(X) = \sum_{x \in X} w(x). </annotation></semantics>

For a compact subset <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics> of <semantics> n<annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics>, Mark Meckes shows that all reasonable definitions of magnitude are equal to what you get by taking the supremum of the magnitudes of all finite subsets of <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics>:

<semantics>mag(X)=sup{mag(Ξ):ΞXfinite}.<annotation encoding="application/x-tex"> \mathrm{mag}(X) = \sup \{\mathrm{mag}(\Xi) : \Xi \subset X \ \text{finite} \} . </annotation></semantics>

Unfortunately, few explicit computations of the magnitude for a compact subset of <semantics> n<annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics> are known.

Instead of the magnitude of an individual set <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics>, it proves fruitful to study the magnitude of dilates <semantics>RX<annotation encoding="application/x-tex">R\cdot X</annotation></semantics> of <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics>, for <semantics>R>0<annotation encoding="application/x-tex">R\gt 0</annotation></semantics>. Here the dilate <semantics>RX<annotation encoding="application/x-tex">R\cdot X</annotation></semantics> means the space <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics> with the metric scaled by a factor of <semantics>R<annotation encoding="application/x-tex">R</annotation></semantics>. We can vary <semantics>R<annotation encoding="application/x-tex">R</annotation></semantics> and this gives rise to the magnitude function of <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics>:

<semantics> X:(0,); X(R):=mag(RX)for R>0.<annotation encoding="application/x-tex"> \mathcal{M}_X\colon (0,\infty)\to \mathbb{R};\quad\mathcal{M}_X(R) := \mathrm{mag}(R\cdot X)\quad\text{for }\ R \gt 0. </annotation></semantics>

Tom and Simon conjectured a relation between the magnitude function of <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics> and its intrinsic volumes <semantics>(V i(X)) i=0 n<annotation encoding="application/x-tex">(V_i(X))_{i=0}^n</annotation></semantics>. The intrinsic volumes of subsets of <semantics> n<annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics> generalize notions such as volume, surface area and Euler characteristic, with <semantics>V n(X)=vol n(X)<annotation encoding="application/x-tex">V_n(X)=\text{vol}_n(X)</annotation></semantics> and <semantics>V 0(X)=χ(X)<annotation encoding="application/x-tex">V_0(X)=\chi(X)</annotation></semantics>.

Convex Magnitude Conjecture. Suppose <semantics>X n<annotation encoding="application/x-tex">X \subseteq \mathbb{R}^n</annotation></semantics> is compact and convex, then the magnitude function is a polynomial whose coefficients involve the intrinsic volumes:
<semantics> X(R)= i=0 nV i(X)i!ω iR n,<annotation encoding="application/x-tex"> \mathcal{M}_X(R) = \sum_{i=0}^n \frac{V_i(X)}{i! \,\omega_i} R^n, </annotation></semantics> where <semantics>V i(X)<annotation encoding="application/x-tex">V_i(X)</annotation></semantics> is the <semantics>i<annotation encoding="application/x-tex">i</annotation></semantics>-th intrinsic volume of <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics> and <semantics>ω i<annotation encoding="application/x-tex">\omega_i</annotation></semantics> the volume of the <semantics>i<annotation encoding="application/x-tex">i</annotation></semantics>-dimensional ball.

The conjecture was disproved by Barceló and Carbery (see also this post). They computed the magnitude function of the <semantics>5<annotation encoding="application/x-tex">5</annotation></semantics>-dimensional ball <semantics>B 5<annotation encoding="application/x-tex">B_5</annotation></semantics> to be the rational function <semantics> B 5(R)=R 55!+3R 5+27R 4+105R 3+216R+7224(R+3).<annotation encoding="application/x-tex"> \mathcal{M}_{B_5}(R)=\frac{R^5}{5!} +\frac{3R^5+27R^4+105R^3+216R+72}{24(R+3)}. </annotation></semantics> In particular, the magnitude function is not even a polynomial for <semantics>B 5<annotation encoding="application/x-tex">B_5</annotation></semantics>. Also, the coefficient of <semantics>R 4<annotation encoding="application/x-tex">R^4</annotation></semantics> in the asymptotic expansion of <semantics> B 5(R)<annotation encoding="application/x-tex">\mathcal{M}_{B_5}(R)</annotation></semantics> as <semantics>R<annotation encoding="application/x-tex">R \to \infty</annotation></semantics> does not agree with the conjecture.

Nevertheless, for any smooth, compact domain in odd-dimensional Euclidean space, <semantics>X n<annotation encoding="application/x-tex">X\subseteq\mathbb{R}^n</annotation></semantics> (in other words, the closure of an open bounded subset with smooth boundary), for <semantics>n=2m1<annotation encoding="application/x-tex">n=2m-1</annotation></semantics>, our main result shows that a modified form of the conjecture holds asymptotically as <semantics>R<annotation encoding="application/x-tex">R \to \infty</annotation></semantics>.

Theorem A. Suppose <semantics>n=2m1<annotation encoding="application/x-tex">n=2m-1</annotation></semantics> and that <semantics>X n<annotation encoding="application/x-tex">X\subseteq \mathbb{R}^n</annotation></semantics> is a smooth domain.

  1. There exists an asymptotic expansion of the magnitude function: <semantics> X(R)1n!ω n j=0 c j(X)R njas R<annotation encoding="application/x-tex"> \mathcal{M}_X(R) \sim \frac{1}{n!\omega_n}\sum_{j=0}^\infty c_j(X) R^{n-j} \quad \text{as }\ R\to \infty </annotation></semantics> with coefficients <semantics>c j(X)<annotation encoding="application/x-tex">c_j(X)\in\mathbb{R}</annotation></semantics> for <semantics>j=0,1,2,<annotation encoding="application/x-tex">j=0,1,2,\ldots</annotation></semantics>.

  2. The first three coefficients are given by <semantics>c 0(X) =vol n(X), c 1(X) =mvol n1(X), c 2(X) =m 22(n1) XHdS,<annotation encoding="application/x-tex"> \begin{aligned} c_0(X)&=\text{vol}_n(X),\\ c_1(X)&=m\text{vol}_{n-1}(\partial X),\\ c_2(X)&=\frac{m^2}{2} (n-1)\int_{\partial X} H \ \mathrm{d}S, \end{aligned} </annotation></semantics> where <semantics>H<annotation encoding="application/x-tex">H</annotation></semantics> is the mean curvature of <semantics>X<annotation encoding="application/x-tex">\partial X</annotation></semantics>. (The coefficient <semantics>c 0<annotation encoding="application/x-tex">c_0</annotation></semantics> was computed by Barceló and Carbery.) If <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics> is convex, these coefficients are proportional to the intrinsic volumes <semantics>V n(X)<annotation encoding="application/x-tex">V_{n}(X)</annotation></semantics>, <semantics>V n1(X)<annotation encoding="application/x-tex">V_{n-1}(X)</annotation></semantics> and <semantics>V n2(X)<annotation encoding="application/x-tex">V_{n-2}(X) </annotation></semantics> respectively.

  3. For <semantics>j1<annotation encoding="application/x-tex">j\geq 1</annotation></semantics>, the coefficient <semantics>c j(X)<annotation encoding="application/x-tex">c_j(X)</annotation></semantics> is determined by the second fundamental form <semantics>L<annotation encoding="application/x-tex">L</annotation></semantics> and covariant derivative <semantics> X<annotation encoding="application/x-tex">\nabla_{\partial X}</annotation></semantics> of <semantics>X<annotation encoding="application/x-tex">\partial X</annotation></semantics>: <semantics>c j(X)<annotation encoding="application/x-tex">c_j(X)</annotation></semantics> is an integral over <semantics>X<annotation encoding="application/x-tex">\partial X</annotation></semantics> of a universal polynomial in the entries of <semantics> X kL<annotation encoding="application/x-tex">\nabla_{\partial X}^k L</annotation></semantics>, <semantics>0kj2<annotation encoding="application/x-tex">0 \leq k\leq j-2</annotation></semantics>. The total number of covariant derivatives appearing in each term of the polynomial is <semantics>j2<annotation encoding="application/x-tex">j-2</annotation></semantics>.

  4. The previous part implies an asymptotic inclusion-exclusion principle: if <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics>, <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> and <semantics>AB n<annotation encoding="application/x-tex">A \cap B \subset \mathbb{R}^n</annotation></semantics> are smooth, compact domains, <semantics> AB(R) A(R) B(R)+ AB(R)0as R<annotation encoding="application/x-tex"> \mathcal{M}_{A \cup B}(R) - \mathcal{M}_A(R) - \mathcal{M}_B(R) + \mathcal{M}_{A \cap B}(R) \to 0 \quad \text{as }\ R \to \infty </annotation></semantics> faster than <semantics>R N<annotation encoding="application/x-tex">R^{-N}</annotation></semantics> for all <semantics>N<annotation encoding="application/x-tex">N</annotation></semantics>.

If you’re not familiar with the second fundamental form, you should think of it as the container for curvature information of the boundary relative to the ambient Euclidean space. Since Euclidean space is flat, any curvature invariant of the boundary (satisfying reasonable symmetry conditions) will only depend on the second fundamental form and its derivatives.

Note that part 4 of the theorem does not imply that an asymptotic inclusion-exclusion principle holds for all <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> and <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics>, even if <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> and <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> are smooth, since the intersection <semantics>AB<annotation encoding="application/x-tex">A \cap B</annotation></semantics> usually is not smooth. In fact, it seems unlikely that an asymptotic inclusion-exclusion principle holds for general <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> and <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics> without imposing curvature conditions, for example by means of assuming convexity of <semantics>A<annotation encoding="application/x-tex">A</annotation></semantics> and <semantics>B<annotation encoding="application/x-tex">B</annotation></semantics>.

The key idea of the short proof relates the computation of the magnitude to classical techniques from geometric and semiclassical analysis, applied to a reformulated problem already studied by Meckes and by Barceló and Carbery. Meckes proved that the magnitude can be computed from the solution to a partial differential equation in the exterior domain <semantics> nX<annotation encoding="application/x-tex">\mathbb{R}^n\setminus X</annotation></semantics> with prescribed values in <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics>. A careful analysis by Barceló and Carbery refined Meckes’ results and expressed the magnitude by means of the solution to a boundary value problem. We refer to this boundary value problem as the “Barceló-Carbery boundary value problem” below.

Meromorphic extension of the magnitude function

Intriguingly, we find that the magnitude function <semantics> X<annotation encoding="application/x-tex">\mathcal{M}_X</annotation></semantics> extends meromorphically to complex values of the scaling factor <semantics>R<annotation encoding="application/x-tex">R</annotation></semantics>. The meromorphic extension was noted by Tom for finite metric spaces and was observed in all previously known examples.

Theorem B. Suppose <semantics>n=2m1<annotation encoding="application/x-tex">n=2m-1</annotation></semantics> and that <semantics>X n<annotation encoding="application/x-tex">X\in \mathbb{R}^n</annotation></semantics> is a smooth domain.

  • The magnitude function <semantics> X<annotation encoding="application/x-tex">\mathcal{M}_X</annotation></semantics> admits a meromorphic continuation to the complex plane.

  • The poles of <semantics> X<annotation encoding="application/x-tex">\mathcal{M}_X</annotation></semantics> are generalized scattering resonances, and each sector <semantics>{z:|arg(z)|<θ}<annotation encoding="application/x-tex">\{z : |\arg(z)| \lt \theta \}</annotation></semantics> with <semantics>θ<π2<annotation encoding="application/x-tex">\theta \lt \frac{\pi}{2}</annotation></semantics> contains at most finitely many of them (all of them outside <semantics>{z:|arg(z)|<πn+1}<annotation encoding="application/x-tex">\{z : |\arg(z)|\lt {\textstyle \frac{\pi}{n+1}}\}</annotation></semantics>).

The ordinary notion of scattering resonances comes from studying waves scattering at a compact obstacle <semantics>X n<annotation encoding="application/x-tex">X\subseteq \mathbb{R}^n</annotation></semantics>. A scattering resonance is a pole of the meromorphic extension of the solution operator <semantics>(R 2Δ) 1<annotation encoding="application/x-tex">(R^2-\Delta)^{-1}</annotation></semantics> to the Helmholtz equation on <semantics> nX<annotation encoding="application/x-tex">\mathbb{R}^n\setminus X</annotation></semantics>, with suitable boundary conditions. These resonances determine the long-time behaviour of solutions to the wave equation and are well studied in geometric analysis as well as throughout physics. The Barceló-Carbery boundary value problem is a higher order version of this problem and studies solutions to <semantics>(R 2Δ) mu=0<annotation encoding="application/x-tex">(R^2-\Delta)^{m}u=0</annotation></semantics> outside <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics>. In dimension <semantics>n=1<annotation encoding="application/x-tex">n=1</annotation></semantics> (i.e. <semantics>m=1<annotation encoding="application/x-tex">m=1</annotation></semantics>), the Barceló-Carbery problem coincides with the Helmholtz problem, and the poles of the magnitude function are indeed scattering resonances. As in scattering theory, one might hope to find detailed structure in the location of the poles. A plot of the poles and zeros in case of the <semantics>21<annotation encoding="application/x-tex">21</annotation></semantics>-dimensional ball is given at the top of the post.

The second part of this theorem is sharp. In fact, the poles do not need to lie in any half plane. Using the techniques of Barceló and Carbery we observe that the magnitude function of the 3-dimensional spherical shell <semantics>X=(2B 3)B 3 <annotation encoding="application/x-tex">X=(2B_3)\setminus B_3^\circ</annotation></semantics> is not rational and contains an infinite discrete sequence of poles which approaches the curve <semantics>Re(R)=log(|Im(R)|)<annotation encoding="application/x-tex">\mathrm{Re}(R)= \log(|\mathrm{Im}(R)|)</annotation></semantics> as <semantics>Re(R)<annotation encoding="application/x-tex">\mathrm{Re}(R) \to \infty</annotation></semantics>. Here’s a plot of the poles of <semantics> X<annotation encoding="application/x-tex">\mathcal{M}_{X}</annotation></semantics> with <semantics>|Im(R)|<30<annotation encoding="application/x-tex">|\mathrm{Im}(R)|\lt 30</annotation></semantics>.

poles 3-d spherical shell

The magnitude function of <semantics>X<annotation encoding="application/x-tex">X</annotation></semantics> is given by <semantics> X(R)=(7/6)R 3+5R 2+2R+2+e 2R(R 2+1)+2R 33R 2+2R1sinh2R2R.<annotation encoding="application/x-tex">\mathcal{M}_X(R)=(7/6)R^3 +5R^2 + 2R + 2 + \frac{e^{-2R}(R^2+1)+2R^3-3R^2+2R-1}{\sinh 2R -2R}.</annotation></semantics>

[EDIT: The above formula has been corrected, following comments below.]

Our techniques extend to compact domains with a <semantics>C k<annotation encoding="application/x-tex">C^k</annotation></semantics> boundary, as long as <semantics>k<annotation encoding="application/x-tex">k</annotation></semantics> is large enough. In this case, the asymptotic inclusion-exclusion principle takes the form that <semantics> AB(R) A(R) B(R)+ AB(R)0asR<annotation encoding="application/x-tex">\mathcal{M}_{A \cup B}(R) - \mathcal{M}_A(R) - \mathcal{M}_B(R) + \mathcal{M}_{A \cap B}(R) \to 0 \quad \text{as}\ R \to \infty </annotation></semantics> faster than <semantics>R N<annotation encoding="application/x-tex">R^{-N}</annotation></semantics> for an <semantics>N=N(k)<annotation encoding="application/x-tex">N=N(k)</annotation></semantics>.

by willerton (S.Willerton@sheffield.ac.uk) at January 15, 2018 03:35 AM

January 14, 2018

John Baez - Azimuth

Retrotransposons

This article is very interesting:

• Ed Yong, Brain cells share information with virus-like capsules, Atlantic, January 12, 2018.

Your brain needs a protein called Arc. If you have trouble making this protein, you’ll have trouble forming new memories. The neuroscientist Jason Shepherd noticed something weird:

He saw that these Arc proteins assemble into hollow, spherical shells that look uncannily like viruses. “When we looked at them, we thought: What are these things?” says Shepherd. They reminded him of textbook pictures of HIV, and when he showed the images to HIV experts, they confirmed his suspicions. That, to put it bluntly, was a huge surprise. “Here was a brain gene that makes something that looks like a virus,” Shepherd says.

That’s not a coincidence. The team showed that Arc descends from an ancient group of genes called gypsy retrotransposons, which exist in the genomes of various animals, but can behave like their own independent entities. They can make new copies of themselves, and paste those duplicates elsewhere in their host genomes. At some point, some of these genes gained the ability to enclose themselves in a shell of proteins and leave their host cells entirely. That was the origin of retroviruses—the virus family that includes HIV.

It’s worth pointing out that gypsy is the name of a specific kind of retrotransposon. A retrotransposon is a gene that can make copies of itself by first transcribing itself from DNA into RNA and then converting itself back into DNA and inserting itself at other places in your chromosomes.

About 40% of your genes are retrotransposons! They seem to mainly be ‘selfish genes’, focused on their own self-reproduction. But some are also useful to you.

So, Arc genes are the evolutionary cousins of these viruses, which explains why they produce shells that look so similar. Specifically, Arc is closely related to a viral gene called gag, which retroviruses like HIV use to build the protein shells that enclose their genetic material. Other scientists had noticed this similarity before. In 2006, one team searched for human genes that look like gag, and they included Arc in their list of candidates. They never followed up on that hint, and “as neuroscientists, we never looked at the genomic papers so we didn’t find it until much later,” says Shepherd.

I love this because it confirms my feeling that viruses are deeply entangled with our evolutionary past. Computer viruses are just the latest phase of this story.

As if that wasn’t weird enough, other animals seem to have independently evolved their own versions of Arc. Fruit flies have Arc genes, and Shepherd’s colleague Cedric Feschotte showed that these descend from the same group of gypsy retrotransposons that gave rise to ours. But flies and back-boned animals co-opted these genes independently, in two separate events that took place millions of years apart. And yet, both events gave rise to similar genes that do similar things: Another team showed that the fly versions of Arc also sends RNA between neurons in virus-like capsules. “It’s exciting to think that such a process can occur twice,” says Atma Ivancevic from the University of Adelaide.

This is part of a broader trend: Scientists have in recent years discovered several ways that animals have used the properties of virus-related genes to their evolutionary advantage. Gag moves genetic information between cells, so it’s perfect as the basis of a communication system. Viruses use another gene called env to merge with host cells and avoid the immune system. Those same properties are vital for the placenta—a mammalian organ that unites the tissues of mothers and babies. And sure enough, a gene called syncytin, which is essential for the creation of placentas, actually descends from env. Much of our biology turns out to be viral in nature.

Here’s something I wrote in 1998 when I was first getting interested in this business:

RNA reverse transcribing viruses

RNA reverse transcribing viruses are usually called retroviruses. They have a single-stranded RNA genome. They infect animals, and when they get inside the cell’s nucleus, they copy themselves into the DNA of the host cell using reverse transcriptase. In the process they often cause tumors, presumably by damaging the host’s DNA.

Retroviruses are important in genetic engineering because they raised for the first time the possibility that RNA could be transcribed into DNA, rather than the reverse. In fact, some of them are currently being deliberately used by scientists to add new genes to mammalian cells.

Retroviruses are also important because AIDS is caused by a retrovirus: the human immunodeficiency virus (HIV). This is part of why AIDS is so difficult to treat. Most usual ways of killing viruses have no effect on retroviruses when they are latent in the DNA of the host cell.

From an evolutionary viewpoint, retroviruses are fascinating because they blur the very distinction between host and parasite. Their genome often contains genetic information derived from the host DNA. And once they are integrated into the DNA of the host cell, they may take a long time to reemerge. In fact, so-called endogenous retroviruses can be passed down from generation to generation, indistinguishable from any other cellular gene, and evolving along with their hosts, perhaps even from species to species! It has been estimated that up to 1% of the human genome consists of endogenous retroviruses! Furthermore, not every endogenous retrovirus causes a noticeable disease. Some may even help their hosts.

It gets even spookier when we notice that once an endogenous retrovirus lost the genes that code for its protein coat, it would become indistinguishable from a long terminal repeat (LTR) retrotransposon—one of the many kinds of “junk DNA” cluttering up our chromosomes. Just how much of us is made of retroviruses? It’s hard to be sure.

For my whole article, go here:

Subcellular life forms.

It’s about the mysterious subcellular entities that stand near the blurry border between the living and the non-living—like viruses, viroids, plasmids, satellites, transposons and prions. I need to update it, since a lot of new stuff is being discovered!

Jason Shepherd’s new paper has a few other authors:

• Elissa D. Pastuzyn, Cameron E. Day, Rachel B. Kearns, Madeleine Kyrke-Smith, Andrew V. Taibi, John McCormick, Nathan Yoder, David M. Belnap, Simon Erlendsson, Dustin R. Morado, John A.G. Briggs, Cédric Feschotte and Jason D. Shepherd, The neuronal gene Arc encodes a repurposed retrotransposon gag protein that mediates intercellular RNA transfer, Cell 172 (2018), 275–288.

by John Baez at January 14, 2018 06:28 PM

ZapperZ - Physics and Physicists

Table-Top Elementary Particle Experiment
I love reading articles like this one, where it shows that one can do quite useful research in elementary particles using experimental setup that is significantly smaller (and cheaper) than large particle colliders.

Now, he’s suddenly moving from the fringes of physics to the limelight. Northwestern University in Evanston, Illinois, is about to open a first-of-its-kind research institute dedicated to just his sort of small-scale particle physics, and Gabrielse will be its founding director.

The move signals a shift in the search for new physics. Researchers have dreamed of finding subatomic particles that could help them to solve some of the thorniest remaining problems in physics. But six years’ worth of LHC data have failed to produce a definitive detection of anything unexpected.

More physicists are moving in Gabrielse’s direction, with modest set-ups that can fit in standard university laboratories. Instead of brute-force methods such as smashing particles, these low-energy experimentalists use precision techniques to look for extraordinarily subtle deviations in some of nature’s most fundamental parameters. The slightest discrepancy could point the way to the field’s future. 

Again, I salute very much this type of endeavor, but I dislike the tone of the title of the article, and I'll tell you why.

In science, and especially physics, there is seldom something that has been verified, found, or discovered using just ONE experimental technique or detection method. For example, in the discovery of the Top quark, both CDF and D0 detectors at Fermilab had to agree. In the discovery of the Higgs, both ATLAS and CMS had to agree. In trying to show that something is a superconductor, you not only measure the resistivity, but also magnetic susceptibility.

In other words, you require many different types of verification, and the more the better or the more convincing it becomes.

While these table-top experiments are very ingenious, they will NOT replace the big colliders. No one in their right mind will tell CERN to "step aside", other than the author of this article. There are discoveries or parameters of elementary particles that these table-top experiments can study more efficiently than the LHC, but there are also plenty of the parameter phase space that the LHC can probe that can't be easily reached by these table-top experiments. They all are complimenting each other!

People who don't know any better, or don't know the intricacies of how experiments are done or how knowledge is gathered, will get the impression that because of these table-top experiments, facilities like the LHC will no longer be needed. I hate to think that this is the "take-home" message that many people will get.

Zz.

by ZapperZ (noreply@blogger.com) at January 14, 2018 03:25 PM

January 13, 2018

Clifford V. Johnson - Asymptotia

Process for The Dialogues

Over on instagram (@asymptotia – and maybe here too, not sure) I’ll be posting some images of developmental drawings I did for The Dialogues, sometimes alongside the finished panels in the book. It is often very interesting to see how a finished thing came to be, and so that’s why … Click to continue reading this post

The post Process for The Dialogues appeared first on Asymptotia.

by Clifford at January 13, 2018 12:55 AM

January 12, 2018

Lubos Motl - string vacua and pheno

Only string theory excites Sheldon, after all
In Demons, Sunday School and Prime Numbers (S01E11), Young Sheldon's mother finds out he plays a demonic game, Dungeons and Dragons, and he is fooled into attending the Sunday School. He reads and learns the Bible and other religions and ultimately establishes his own, math-based religion that teaches that prime numbers make you feel good. He has one (stupid kid) follower. I was actually teaching similar religions at that age.



Meanwhile, in the S11E13 episode of The Big Bang Theory, The Solo Oscillation, Howard is almost replaced by geologist Bert in the rock band (also featuring Rajesh). The folks discuss various projects and it turns out that Sheldon has nothing serious to work on. Recall that almost 4 years ago, Sheldon Cooper left string theory. But you can't really leave string theory.




The wives got replaced, Leonard and Amy were revisiting their school projects, and Sheldon spent some quality pizza time with Penny. Penny figured out that Sheldon's recent projects didn't excite him. They were largely about dark matter. He didn't start to work on dark matter because it excited him. Instead, it was everywhere (e.g. in this fake news on Hossenfelder's blog yesterday), and it served as "rebound science", using Penny's jargon. The term describes someone whom she has dated to feel pretty again.




So instead of calculating the odds that Sheldon's mother meets an old friend (that theme was clearly analogous to Richard Feynman's calculations of the wobbling plates), Sheldon was explaining string theory to a collaborative Penny again. Different elementary particle species are different energy eigenstates of a string vibrating in a higher-dimensional space, Penny was de facto taught.

Note the beautiful whiteboard posted at the top – click the picture to zoom it in. It contains a picture of standing waves; the Nambu-Goto action; the transformation to the Polyakov action at the world sheet, the superstringy fermions are included. Then he writes down the beta-function on the world sheet, finding out that its vanishing imposes Einstein's equations in spacetime. The beta-function even has the stress-energy tensor from the Maxwell field so it's not just the vacuum Einstein's equations that are derived.

What are the odds that you could derive the correct spacetime field equations from a totally different starting point like vibrating strings, Penny is asked? Even Penny understands it can't be a coincidence. Which other TV show discusses the beta-function on the world sheet, its spacetime consequences, and the implications of these consequences for the rational assessment of the validity of string theory? Even 99% of science writers who claim to have some understanding of theoretical physics don't have a clue what the world sheet beta-function is and means!

Penny introduces a natural idea – from the layman's viewpoint – about "what can be done with strings". You can make knots out of strings. But lines may only get "knitted" in 3 dimensions. 4 dimensions is already too much, Sheldon warns her. "Unless..." Penny pretends that she's on verge of discovering a loophole, and thus forces Sheldon to discover the loophole by himself. Unless you consider the "knots" involving the whole world sheets (and/or membranes and other branes or their world volumes).

Well, just to be sure, I think that "knots involving fundamental strings or branes" still don't play an important role in string theory – and there are good reasons for that. To calculate the knot invariants, you need to know the shape of the curve (strings in this case) very accurately, and this sort of goes against the basic principles and lore of string theory as a theory of quantum gravity (in some sense, there is roughly a Planck length uncertainty about all locations, so all finite-energy states of nearby vibrating strings are almost unavoidably weird superpositions of states with different knot invariants).

But that doesn't mean that I haven't been fascinated by "knots" made out of strings and branes and their physical meaning. For example, I and Ori Ganor wrote a paper about knitting of fivebranes. If you have two parallel M5-branes in M-theory (it's not a coincidence that their dimension is about 1/2 of the spacetime dimension!), you may knit them in a new way and the knot invariant may be interpreted as the number of membranes (M2-branes) stretched in between the two (or more) M5-branes. It's a higher-dimensional extension of the "Skyrmions".

The TV show makes it sound as if the discovery of some knitting of strings and branes might become "the next revolution" in string theory and therefore physics and science, too. Well, I have some doubts. Topological solitons like that are important – but they have become a part of the toolkit that is being applied in many different ways. They're not a shocking revolutionary idea that is likely to change everything.

On the other hand, knot theory and lots of geometric and physical effects that have nice interpretations are often naturally embedded within string theory. String theory has provided us with a perfectly physically consistent and complete incarnation of so many physical effects that were known to exist in geometry or condensed matter physics or other fields that a brilliant person simply cannot fail to be excited. And I do think that some proper treatment of monodromy operators in the stringy spacetime does hold the key to the next revolution in quantum gravity.

It will be good if The Big Bang Theory ends this pointless intellectual castration of Sheldon who was supposed to work, like some average astrophysicists and cosmologists, on some uninteresting and often fishy ad hoc dark matter projects in recent almost 4 years. It's natural for Sheldon to get back to serious work – to look for a theory of everything – and I sincerely hope that Penny will continue to be helpful in these efforts. (Well, I still think that the claims that Sheldon will "have to" share his Nobel with Penny are exaggerated LOL.)

Good luck to him and her – and to me, too.

by Luboš Motl (noreply@blogger.com) at January 12, 2018 09:00 AM

Clifford V. Johnson - Asymptotia

Nice to be Back…

Back where? In front of a classroom teaching quantum field theory, that is. It is a wonderful, fascinating, and super-important subject, and it has been a while since I've taught it. I actually managed to dig out some pretty good notes for the last time I taught it. (Thank you, my inner pack rat for keeping those notes and putting them where I could find them!) They'll be a helpful foundation. (Aren't they beautiful by the way? Those squiggly diagrams are called Feynman diagrams.)

Important? Quantum field theory (QFT) is perhaps one of the most remarkable [...] Click to continue reading this post

The post Nice to be Back… appeared first on Asymptotia.

by Clifford at January 12, 2018 06:07 AM

January 11, 2018

ZapperZ - Physics and Physicists

How Do We Know Blackholes Exist?
If you don't care to read in detail on the physics, and have the attention span of a 2-year old, this is Minute Physics's attempt at convincing you that blackholes exist.



Zz.

by ZapperZ (noreply@blogger.com) at January 11, 2018 08:58 PM

January 10, 2018

Clifford V. Johnson - Asymptotia

An LA Times Piece…

It seems appropriate somehow that there's an extensive interview with me in the LA Times with Deborah Netburn about my work on the book. Those of you who have read it might have recognised some of the landscape in one of the stories as looking an awful lot like downtown Los Angeles, and if you follow the conversation and pay attention to your surroundings, you see that they pass a number of LA Landmarks, ultimately ending up very close to the LA Times Building, itself a landmark!

(In the shot above, you see a bit of the Angel's Flight railway.)

Anyway, I hope you enjoy the interview! We talk a lot about the motivations for making the book, about drawing, and - most especially - the issue of science being for everyone...

[For those of you trying to get the book, note that although it is showing out of stock at Amazon, go ahead and place your order. Apparently they are getting the book and shipping it out constantly, even though it might not stop showing as out of stock. Also, check your local bookstores... Several Indys and branches of Barnes and Noble do have copies on their shelves. (I've checked.) Or they can order it for you. Also, the publisher's site is another source. They are offering a 50% discount as thank you for being patient while they restock. There's a whole new batch of books being printed and that will soon help make it easier to grab.]

-cvj Click to continue reading this post

The post An LA Times Piece… appeared first on Asymptotia.

by Clifford at January 10, 2018 04:21 AM

January 09, 2018

Tommaso Dorigo - Scientificblogging

J4BnUHZMGRqHevJ7QB72vtaM
The title of this post is the password code required to connect to my wireless network at home. The service, provided by the Italian internet provider Fastweb, has been discontinued as I moved to Padova from Venice, so you can't really do much with the code itself. On the other hand, it should give you hints on a couple of things.

read more

by Tommaso Dorigo at January 09, 2018 03:16 PM

Lubos Motl - string vacua and pheno

An Indian interview with Nathan Seiberg
While the U.S.-based Quanta Magazine dedicated its pages to a crackpot's diatribe about a fictitious research on a non-existent alternative theory of quantum gravity (another crackpot, Pentcho Valev, contributed the only comment so far), the media in India still keep some quality and attractiveness for intelligent readers.

The Wire India has interviewed Princeton's string theorist Nati Seiberg who is just visiting India:
Interview: ‘There’s No Conflict Between Lack of Evidence of String Theory and Work Being Done on It’
They cover lots of questions and the interview is rather interesting.




Spoilers: beware.

The interview took place at Bengaluru. They explain Seiberg is an important theoretical physicist – e.g. a 2016 Dirac medal laureate. Sandhya Ramesh asks him to define string theory – Seiberg says it's a theory meant to be a TOE that keeps on transforming, it will probably be transforming, and the progress is very exciting.




Seiberg is asked the question from the title: How should you reconcile the absence of an experimental proof with the work on string theory? There is nothing to reconcile, the latter doesn't need the former. There are numerous reasons why people keep on researching string theory, e.g. its consequences for adjacent fields.

He is also asked how he imagines higher-dimensional objects. It's hard for him, too. When answering a question about the role of interdisciplinary research, Seiberg importantly says that there is no "string theory approach to climate science" but sometimes the collaboration on the borders of disciplines is fruitful. SUSY could have been found, it wasn't found, and it may be useful to build bigger colliders. Seiberg knows nothing about politics of begging for the big funds.

Seiberg is asked about alternative contenders running against string/M-theory and his answer is that he doesn't know of any.

Suddenly the journalist asks about the recent results on gauge theories and global symmetries and their implications on the paradigms in condensed matter physics. So unsurprisingly, Seiberg is surprised because the question betrays someone's IQ that is some 40 IQ points above the average journalist. The roles get reversed, Seiberg asks: Where did you get this question? The answer is that the journalist got it from his editor. Seiberg is impressed, and so am I. Maybe the editor just read The Reference Frame recently to improve the questions his colleagues ask. ;-)

Yesterday, Seiberg gave a talk in India that was about related questions but he didn't recommend the "public" to attend the talk because it would be a somewhat technical, although not too technical, talk. OK, he said some basic things about symmetries of faces, supersymmetry, and supersymmetry's diverse implications aside from the discovery of superpartner particles (that hasn't materialized yet).

He praises Indian string theorists – I agree with those sentiments. Seiberg rejects recommendations to give advises what people should work on and to deal with the public more often – because "he's not good at it". He addresses another great question, one about naturalness, and says that the strongest "around the corner" edition of naturalness has been disproved by the LHC null results and the assumptions that went into it have to be reassessed.

Also, Seiberg doesn't know where the work will be done. LIGO is interesting. When asked about the number of string theorists, he says that it's small enough for everyone to know everybody else and it's wonderful. He was offered the meme that India has a good weather and it's a reason to visit the country but he visits India because of the colleagues.

by Luboš Motl (noreply@blogger.com) at January 09, 2018 05:49 AM

January 07, 2018

John Baez - Azimuth

The Kepler Problem (Part 1)

Johannes Kepler loved geometry, so of course he was fascinated by Platonic solids. His early work Mysterium Cosmographicum, written in 1596, includes pictures showing how the 5 Platonic solids correspond to the 5 elements:

Five elements? Yes, besides earth, air, water and fire, he includes a fifth element that doesn’t feel the Earth’s gravitational pull: the ‘quintessence’, or ‘aether’, from which heavenly bodies are made.

In the same book he also tried to use the Platonic solids to explain the orbits of the planets:

The six planets are Mercury, Venus, Earth, Mars, Jupiter and Saturn. And the tetrahedron and cube, in case you’re wondering, sit outside the largest sphere shown above. You can see them another picture from Kepler’s book:

These ideas may seem goofy now, but studying the exact radii of the planets’ orbits led him to discover that these orbits aren’t circular: they’re ellipses! By 1619 this led him to what we call Kepler’s laws of planetary motion. And those, in turn, helped Newton verify Hooke’s hunch that the force of gravity goes as the inverse square of the distance between bodies!

In honor of this, the problem of a particle orbiting in an inverse square force law is called the Kepler problem.

So, I’m happy that Greg Egan, Layra Idarani and I have come across a solid mathematical connection between the Platonic solids and the Kepler problem.

But this involves a detour into the 4th dimension!

It’s a remarkable fact that the Kepler problem has not just the expected conserved quantities—energy and the 3 components of angular momentum—but also 3 more: the components of the Runge–Lenz vector. To understand those extra conserved quantities, go here:

• Greg Egan, The ellipse and the atom.

Noether proved that conserved quantities come from symmetries. Energy comes from time translation symmetry. Angular momentum comes from rotation symmetry. Since the group of rotations in 3 dimensions, called SO(3), is itself 3-dimensional, it gives 3 conserved quantities, which are the 3 components of angular momentum.

None of this is really surprising. But if we take the angular momentum together with the Runge–Lenz vector, we get 6 conserved quantities—and these turn out to come from the group of rotations in 4 dimensions, SO(4), which is itself 6-dimensional. The obvious symmetries in this group just rotate a planet’s elliptical orbit, while the unobvious ones can also squash or stretch it, changing the eccentricity of the orbit.

(To be precise, all this is true only for the ‘bound states’ of the Kepler problem: the circular and elliptical orbits, not the parabolic or hyperbolic ones, which work in a somewhat different way. I’ll only be talking about bound states in this post!)

Why should the Kepler problem have symmetries coming from rotations in 4 dimensions? This is a fascinating puzzle—we know a lot about it, but I doubt the last word has been spoken. For an overview, go here:

• John Baez, Mysteries of the gravitational 2-body problem.

This SO(4) symmetry applies not only to the classical mechanics of the inverse square force law, but also the quantum mechanics! Nobody cares much about the quantum mechanics of two particles attracting gravitationally via an inverse square force law—but people care a lot about the quantum mechanics of hydrogen atoms, where the electron and proton attract each other via their electric field, which also obeys an inverse square force law.

So, let’s talk about hydrogen. And to keep things simple, let’s pretend the proton stays fixed while the electron orbits it. This is a pretty good approximation, and experts will know how to do things exactly right. It requires only a slight correction.

It turns out that wavefunctions for bound states of hydrogen can be reinterpreted as functions on the 3-sphere, S3 The sneaky SO(4) symmetry then becomes obvious: it just rotates this sphere! And the Hamiltonian of the hydrogen atom is closely connected to the Laplacian on the 3-sphere. The Laplacian has eigenspaces of dimensions n2 where n = 1,2,3,…, and these correspond to the eigenspaces of the hydrogen atom Hamiltonian. The number n is called the principal quantum number, and the hydrogen atom’s energy is proportional to -1/n2.

If you don’t know all this jargon, don’t worry! All you need to know is this: if we find an eigenfunction of the Laplacian on the 3-sphere, it will give a state where the hydrogen atom has a definite energy. And if this eigenfunction is invariant under some subgroup of SO(4), so will this state of the hydrogen atom!

The biggest finite subgroup of SO(4) is the rotational symmetry group of the 600-cell, a wonderful 4-dimensional shape with 120 vertices and 600 dodecahedral faces. The rotational symmetry group of this shape has a whopping 7,200 elements! And here is a marvelous moving image, made by Greg Egan, of an eigenfunction of the Laplacian on S3 that’s invariant under this 7,200-element group:

We’re seeing the wavefunction on a moving slice of the 3-sphere, which is a 2-sphere. This wavefunction is actually real-valued. Blue regions are where this function is positive, yellow regions where it’s negative—or maybe the other way around—and black is where it’s almost zero. When the image fades to black, our moving slice is passing through a 2-sphere where the wavefunction is almost zero.

For a full explanation, go here:

• Greg Egan, In the chambers with seven thousand symmetries, 2 January 2018.

Layra Idarani has come up with a complete classification of all eigenfunctions of the Laplacian on S3 that are invariant under this group… or more generally, eigenfunctions of the Laplacian on a sphere of any dimension that are invariant under the even part of any Coxeter group. For the details, go here:

• Layra Idarani, SG-invariant polynomials, 4 January 2018.

All that is a continuation of a story whose beginning is summarized here:

• John Baez, Quantum mechanics and the dodecahedron.

So, there’s a lot of serious math under the hood. But right now I just want to marvel at the fact that we’ve found a wavefunction for the hydrogen atom that not only has a well-defined energy, but is also invariant under this 7,200-element group. This group includes the usual 60 rotational symmetries of a dodecahedron, but also other much less obvious symmetries.

I don’t have a good picture of what these less obvious symmetries do to the wavefunction of a hydrogen atom. I understand them a bit better classically—where, as I said, they squash or stretch an elliptical orbit, changing its eccentricity while not changing its energy.

We can have fun with this using the old quantum theory—the approach to quantum mechanics that Bohr developed with his colleague Sommerfeld from 1920 to 1925, before Schrödinger introduced wavefunctions.

In the old Bohr–Sommerfeld approach to the hydrogen atom, the quantum states with specified energy, total angular momentum and angular momentum about a fixed axis were drawn as elliptical orbits. In this approach, the symmetries that squash or stretch elliptical orbits are a bit easier to visualize:

This picture by Pieter Kuiper shows some orbits at the 5th energy level, n = 5: namely, those with different eigenvalues of the total angular momentum, ℓ.

While the old quantum theory was superseded by the approach using wavefunctions, it’s possible to make it mathematically rigorous for the hydrogen atom. So, we can draw elliptical orbits that rigorously correspond to a basis of wavefunctions for the hydrogen atom. So, I believe we can draw the orbits corresponding to the basis elements whose linear combination gives the wavefunction shown as a function on the 3-sphere in Greg’s picture above!

We should get a bunch of ellipses forming a complicated picture with dodecahedral symmetry. This would make Kepler happy.

As a first step in this direction, Greg drew the collection of orbits that results when we take a circle and apply all the symmetries of the 600-cell:

For more details, read this:

• Greg Egan, Kepler orbits with the symmetries of the 600-cell.

Postscript

To do this really right, one should learn a bit about ‘old quantum theory’. I believe people have been getting it a bit wrong for quite a while—starting with Bohr and Sommerfeld!

If you look at the ℓ = 0 orbit in the picture above, it’s a long skinny ellipse. But I believe it really should be a line segment straight through the proton: that’s what’s an orbit with no angular momentum looks like.

There’s a paper about this:

• Manfred Bucher, Rise and fall of the old quantum theory.

Matt McIrvin had some comments on this:

This paper from 2008 is a kind of thing I really like: an exploration of an old, incomplete theory that takes it further than anyone actually did at the time.

It has to do with the Bohr-Sommerfeld “old quantum theory”, in which electrons followed definite orbits in the atom, but these were quantized–not all orbits were permitted. Bohr managed to derive the hydrogen spectrum by assuming circular orbits, then Sommerfeld did much more by extending the theory to elliptical orbits with various shapes and orientations. But there were some problems that proved maddeningly intractable with this analysis, and it eventually led to the abandonment of the “orbit paradigm” in favor of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics, what we know as modern quantum theory.

The paper argues that the old quantum theory was abandoned prematurely. Many of the problems Bohr and Sommerfeld had came not from the orbit paradigm per se, but from a much simpler bug in the theory: namely, their rejection of orbits in which the electron moves entirely radially and goes right through the nucleus! Sommerfeld called these orbits “unphysical”, but they actually correspond to the s orbital states in the full quantum theory, with zero angular momentum. And, of course, in the full theory the electron in these states does have some probability of being inside the nucleus.

So Sommerfeld’s orbital angular momenta were always off by one unit. The hydrogen spectrum came out right anyway because of the happy accident of the energy degeneracy of certain orbits in the Coulomb potential.

I guess the states they really should have been rejecting as “unphysical” were Bohr’s circular orbits: no radial motion would correspond to a certain zero radial momentum in the full theory, and we can’t have that for a confined electron because of the uncertainty principle.

by John Baez at January 07, 2018 11:00 PM

January 06, 2018

Jon Butterworth - Life and Physics

Atom Land: A Guided Tour Through the Strange (and Impossibly Small) World of Particle Physics

Book review in Publishers’ Weekly.

Butterworth (Most Wanted Particle), a CERN alum and professor of physics at University College London, explains everything particle physics from antimatter to Z bosons in this charming trek through a landscape of “the otherwise invisible.” His accessible narrative cleverly relates difficult concepts, such as wave-particle duality or electron spin, in bite-size bits. Readers become explorers on Butterworth’s metaphoric map… Read more.

by Jon Butterworth at January 06, 2018 05:13 PM

Jon Butterworth - Life and Physics

January 05, 2018

ZapperZ - Physics and Physicists

Determining The Hubble Constant
Ethan Siegel has a nice article on the pitfalls in determining one of the most important constants in our universe, the Hubble constant. The article describes why this constant is so important, and all the ramifications that come from it.

As you read this, notice all the "background knowledge" that one must have to be able to know how well certain things are known, and what are the assumptions and uncertainties in each of the methods and values that we use. All of these need to be known, and people using them must be aware of them.

Compare that to the decision we make everyday on things we accept in social policies and politics.

Zz.

by ZapperZ (noreply@blogger.com) at January 05, 2018 09:01 PM

ZapperZ - Physics and Physicists

Why Did Matter Matter?
Ethan Siegel has yet another nice article. This time, he tackles on why we have an abundant of matter in our universe, but hardly any antimatter, when all our physics seems to indicate that there should be equal amount of both, or simply a universe filled with no matter.

I have highlighted a number of CP-violation experiments on here, which is something mentioned in the article. But it is nice to have a layman-type summary of the baryo-lepton-genesis ideas that are floating out there.

Zz.

by ZapperZ (noreply@blogger.com) at January 05, 2018 09:01 PM

January 03, 2018

Clifford V. Johnson - Asymptotia

Press Play!

Happy New Year!

Yesterday, the NPR affiliate KCRW's Press Play broadcast an interview with me. I spoke with the host Madeleine Brand about my non-fiction graphic novel about science, and several other things that came up on the spur of the moment. Rather like one of the wide-ranging conversations in the book itself, come to think of it...

This was a major interview for me because I've been a huge fan of Madeleine for many years, going back to her NPR show Day to Day (which I still [...] Click to continue reading this post

The post Press Play! appeared first on Asymptotia.

by Clifford at January 03, 2018 11:54 PM

January 02, 2018

Tommaso Dorigo - Scientificblogging

Venice --> Padova
I was born and have lived in Venice for over 51 years now (omitting to mention some 2 years of interruption when I worked for Harvard University, 18 years ago), but this has come to an end on December 31st, when I concluded a rather complex move to Padova, 35 kilometers west. 
Venice is a wonderful city and quite a special place, if you ask me. A city with a millenary history, crammed with magnificent palaces and churches. A place where one could write a book about every stone. Walking through the maze of narrow streets or making one's way through a tight network of canals is an unforgettable experience, but living there for decades is something else - it makes you a part of it. I feel I own the place, in some way. So why did I leave it?

read more

by Tommaso Dorigo at January 02, 2018 12:20 PM

January 01, 2018

The n-Category Cafe

Entropy Modulo a Prime (Continued)

In the comments last time, a conversation got going about <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>-adic entropy. But here I’ll return to the original subject: entropy modulo <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>. I’ll answer the question:

Given a “probability distribution” mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>, that is, a tuple <semantics>π=(π 1,,π n)(/p) n<annotation encoding="application/x-tex"> \pi = (\pi_1, \ldots, \pi_n) \in (\mathbb{Z}/p\mathbb{Z})^n </annotation></semantics> summing to <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics>, what is the right definition of its entropy <semantics>H p(π)/p?<annotation encoding="application/x-tex"> H_p(\pi) \in \mathbb{Z}/p\mathbb{Z}? </annotation></semantics>

How will we know when we’ve got the right definition? As I explained last time, the acid test is whether it satisfies the chain rule

<semantics>H p(γ(π 1,,π n))=H p(γ)+ i=1 nγ iH p(π i).<annotation encoding="application/x-tex"> H_p(\gamma \circ (\pi^1, \ldots, \pi^n)) = H_p(\gamma) + \sum_{i = 1}^n \gamma_i H_p(\pi^i). </annotation></semantics>

This is supposed to hold for all <semantics>γ=(γ 1,,γ n)Π n<annotation encoding="application/x-tex">\gamma = (\gamma_1, \ldots, \gamma_n) \in \Pi_n</annotation></semantics> and <semantics>π i=(π 1 i,,π k i i)Π k i<annotation encoding="application/x-tex">\pi^i = (\pi^i_1, \ldots, \pi^i_{k_i}) \in \Pi_{k_i}</annotation></semantics>, where <semantics>Π n<annotation encoding="application/x-tex">\Pi_n</annotation></semantics> is the hyperplane

<semantics>Π n={(π 1,,π n)(/p) n:π 1++π n=1},<annotation encoding="application/x-tex"> \Pi_n = \{ (\pi_1, \ldots, \pi_n) \in (\mathbb{Z}/p\mathbb{Z})^n : \pi_1 + \cdots + \pi_n = 1\}, </annotation></semantics>

whose elements we’re calling “probability distributions” mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>. And if God is smiling on us, <semantics>H p<annotation encoding="application/x-tex">H_p</annotation></semantics> will be essentially the only quantity that satisfies the chain rule. Then we’ll know we’ve got the right definition.

Black belts in functional equations will be able to use the chain rule and nothing else to work out what <semantics>H p<annotation encoding="application/x-tex">H_p</annotation></semantics> must be. But the rest of us might like an extra clue, and we have one in the definition of real Shannon entropy:

<semantics>H (π)= i:π i0π ilogπ i.<annotation encoding="application/x-tex"> H_\mathbb{R}(\pi) = - \sum_{i: \pi_i \neq 0} \pi_i \log \pi_i. </annotation></semantics>

Now, we saw last time that there is no logarithm mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>; that is, there is no group homomorphism

<semantics>(/p) ×/p.<annotation encoding="application/x-tex"> (\mathbb{Z}/p\mathbb{Z})^\times \to \mathbb{Z}/p\mathbb{Z}. </annotation></semantics>

But there is a next-best thing: a homomorphism

<semantics>(/p 2) ×/p.<annotation encoding="application/x-tex"> (\mathbb{Z}/p^2\mathbb{Z})^\times \to \mathbb{Z}/p\mathbb{Z}. </annotation></semantics>

This is called the Fermat quotient <semantics>q p<annotation encoding="application/x-tex">q_p</annotation></semantics>, and it’s given by

<semantics>q p(n)=n p11p/p.<annotation encoding="application/x-tex"> q_p(n) = \frac{n^{p - 1} - 1}{p} \in \mathbb{Z}/p\mathbb{Z}. </annotation></semantics>

Let’s go through why this works.

The elements of <semantics>/p 2<annotation encoding="application/x-tex">\mathbb{Z}/p^2\mathbb{Z}</annotation></semantics> are the congruence classes mod <semantics>p 2<annotation encoding="application/x-tex">p^2</annotation></semantics> of the integers not divisible by <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>. Fermat’s little theorem says that whenever <semantics>n<annotation encoding="application/x-tex">n</annotation></semantics> is not divisible by <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>,

<semantics>n p11p<annotation encoding="application/x-tex"> \frac{n^{p - 1} - 1}{p} </annotation></semantics>

is an integer. This, or rather its congruence class mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>, is the Fermat quotient. The congruence class of <semantics>n<annotation encoding="application/x-tex">n</annotation></semantics> mod <semantics>p 2<annotation encoding="application/x-tex">p^2</annotation></semantics> determines the congruence class of <semantics>n p11<annotation encoding="application/x-tex">n^{p - 1} - 1</annotation></semantics> mod <semantics>p 2<annotation encoding="application/x-tex">p^2</annotation></semantics>, and it therefore determines the congruence class of <semantics>(n p11)/p<annotation encoding="application/x-tex">(n^{p - 1} - 1)/p</annotation></semantics> mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>. So, <semantics>q p<annotation encoding="application/x-tex">q_p</annotation></semantics> defines a function <semantics>(/p 2) ×/p<annotation encoding="application/x-tex">(\mathbb{Z}/p^2\mathbb{Z})^\times \to \mathbb{Z}/p\mathbb{Z}</annotation></semantics>. It’s a pleasant exercise to show that it’s a homomorphism. In other words, <semantics>q p<annotation encoding="application/x-tex">q_p</annotation></semantics> has the log-like property

<semantics>q p(mn)=q p(m)+q p(n)<annotation encoding="application/x-tex"> q_p(m n) = q_p(m) + q_p(n) </annotation></semantics>

for all integers <semantics>m,n<annotation encoding="application/x-tex">m, n</annotation></semantics> not divisible by <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>.

In fact, it’s essentially unique as such. Any other homomorphism <semantics>(/p 2) ×/p<annotation encoding="application/x-tex">(\mathbb{Z}/p^2\mathbb{Z})^\times \to \mathbb{Z}/p\mathbb{Z}</annotation></semantics> is a scalar multiple of <semantics>q p<annotation encoding="application/x-tex">q_p</annotation></semantics>. (This follows from the classical theorem that the group <semantics>(/p 2) ×<annotation encoding="application/x-tex">(\mathbb{Z}/p^2\mathbb{Z})^\times</annotation></semantics> is cyclic.) It’s just like the fact that up to a scalar multiple, the real logarithm is the unique measurable function <semantics>log:(0,)R<annotation encoding="application/x-tex">\log : (0, \infty) \to \R</annotation></semantics> such that <semantics>log(xy)=logx+logy<annotation encoding="application/x-tex">\log(x y) = \log x + \log y</annotation></semantics>, but here there’s nothing like measurability complicating things.

So: <semantics>q p<annotation encoding="application/x-tex">q_p</annotation></semantics> functions as a kind of logarithm. Given a mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> probability distribution <semantics>π=Π n<annotation encoding="application/x-tex">\pi = \in \Pi_n</annotation></semantics>, we might therefore guess that the right definition of its entropy is

<semantics> i:π i0π iq p(a i),<annotation encoding="application/x-tex"> - \sum_{i : \pi_i \neq 0} \pi_i q_p(a_i), </annotation></semantics>

where <semantics>a i<annotation encoding="application/x-tex">a_i</annotation></semantics> is an integer representing <semantics>π i/p<annotation encoding="application/x-tex">\pi_i \in \mathbb{Z}/p\mathbb{Z}</annotation></semantics>.

However, this doesn’t work. It depends on the choice of representatives <semantics>a i<annotation encoding="application/x-tex">a_i</annotation></semantics>.

To get the right answer, we’ll look at real entropy in a slightly different way. Define <semantics> :[0,1]<annotation encoding="application/x-tex">\partial_\mathbb{R}: [0, 1] \to \mathbb{R}</annotation></semantics> by

<semantics> (x)={xlogx if x0, 0 if x=0..<annotation encoding="application/x-tex"> \partial_\mathbb{R}(x) = \begin{cases} - x \log x &if&nbsp; x \neq 0, \\ 0 &if&nbsp; x = 0. \end{cases}. </annotation></semantics>

Then <semantics> <annotation encoding="application/x-tex">\partial_\mathbb{R}</annotation></semantics> has the derivative-like property

<semantics> (xy)=x (y)+ (x)y.<annotation encoding="application/x-tex"> \partial_\mathbb{R}(x y) = x \partial_\mathbb{R}(y) + \partial_\mathbb{R}(x) y. </annotation></semantics>

A linear map with this property is called a derivation, so it’s reasonable to call <semantics> <annotation encoding="application/x-tex">\partial_\mathbb{R}</annotation></semantics> a nonlinear derivation.

The observation that <semantics> <annotation encoding="application/x-tex">\partial_\mathbb{R}</annotation></semantics> is a nonlinear derivation turns out to be quite useful. For instance, real entropy is given by

<semantics>H (π)= i=1 n (π i)<annotation encoding="application/x-tex"> H_\mathbb{R}(\pi) = \sum_{i = 1}^n \partial_\mathbb{R}(\pi_i) </annotation></semantics>

(<semantics>πΠ n<annotation encoding="application/x-tex">\pi \in \Pi_n</annotation></semantics>), and verifying the chain rule for <semantics>H <annotation encoding="application/x-tex">H_\mathbb{R}</annotation></semantics> is done most neatly using the derivation property of <semantics> <annotation encoding="application/x-tex">\partial_\mathbb{R}</annotation></semantics>.

An equivalent formula for real entropy is

<semantics>H (π)= i=1 n (π i) ( i=1 nπ i).<annotation encoding="application/x-tex"> H_\mathbb{R}(\pi) = \sum_{i = 1}^n \partial_\mathbb{R}(\pi_i) - \partial_\mathbb{R}\biggl( \sum_{i = 1}^n \pi_i \biggr). </annotation></semantics>

This is a triviality: <semantics>π i=1<annotation encoding="application/x-tex">\sum \pi_i = 1</annotation></semantics>, so <semantics> (π i)=0<annotation encoding="application/x-tex">\partial_\mathbb{R}\bigl( \sum \pi_i \bigr) = 0</annotation></semantics>, so this is the same as the previous formula. But it’s also quite suggestive: <semantics>H (π)<annotation encoding="application/x-tex">H_\mathbb{R}(\pi)</annotation></semantics> measures the extent to which the nonlinear derivation <semantics> <annotation encoding="application/x-tex">\partial_\mathbb{R}</annotation></semantics> fails to preserve the sum <semantics>π i<annotation encoding="application/x-tex">\sum \pi_i</annotation></semantics>.

Now let’s try to imitate this in <semantics>/p<annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z}</annotation></semantics>. Since <semantics>q p<annotation encoding="application/x-tex">q_p</annotation></semantics> plays a similar role to <semantics>log<annotation encoding="application/x-tex">\log</annotation></semantics>, it’s natural to define

<semantics> p(n)=nq p(n)=nn pp<annotation encoding="application/x-tex"> \partial_p(n) = -n q_p(n) = \frac{n - n^p}{p} </annotation></semantics>

for integers <semantics>n<annotation encoding="application/x-tex">n</annotation></semantics> not divisible by <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>. But the last expression makes sense even if <semantics>n<annotation encoding="application/x-tex">n</annotation></semantics> is divisible by <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>. So, we can define a function

<semantics> p:/p 2/p<annotation encoding="application/x-tex"> \partial_p : \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} </annotation></semantics>

by <semantics> p(n)=(nn p)/p<annotation encoding="application/x-tex">\partial_p(n) = (n - n^p)/p</annotation></semantics>. (This is called a <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>-derivation.) It’s easy to check that <semantics> p<annotation encoding="application/x-tex">\partial_p</annotation></semantics> has the derivative-like property

<semantics> p(mn)=m p(n)+ p(m)n.<annotation encoding="application/x-tex"> \partial_p(m n) = m \partial_p(n) + \partial_p(m) n. </annotation></semantics>

And now we arrive at the long-awaited definition. The entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> of <semantics>π=(π 1,,π n)<annotation encoding="application/x-tex">\pi = (\pi_1, \ldots, \pi_n)</annotation></semantics> is

<semantics>H p(π)= i=1 n p(a i) p( i=1 na i),<annotation encoding="application/x-tex"> H_p(\pi) = \sum_{i = 1}^n \partial_p(a_i) - \partial_p\biggl( \sum_{i = 1}^n a_i \biggr), </annotation></semantics>

where <semantics>a i<annotation encoding="application/x-tex">a_i \in \mathbb{Z}</annotation></semantics> represents <semantics>π i/p<annotation encoding="application/x-tex">\pi_i \in \mathbb{Z}/p\mathbb{Z}</annotation></semantics>. This is independent of the choice of representatives <semantics>a i<annotation encoding="application/x-tex">a_i</annotation></semantics>. And when you work it out explicitly, it gives

<semantics>H p(π)=1p(1 i=1 na i p).<annotation encoding="application/x-tex"> H_p(\pi) = \frac{1}{p} \biggl( 1 - \sum_{i = 1}^n a_i^p \biggr). </annotation></semantics>

Just as in the real case, <semantics>H p<annotation encoding="application/x-tex">H_p</annotation></semantics> satisfies the chain rule, which is most easily shown using the derivation property of <semantics> p<annotation encoding="application/x-tex">\partial_p</annotation></semantics>.

Before I say any more, let’s have some examples.

  • In the real case, the uniform distribution <semantics>u n=(1/n,,1/n)<annotation encoding="application/x-tex">u_n = (1/n, \ldots, 1/n)</annotation></semantics> has entropy <semantics>logn<annotation encoding="application/x-tex">\log n</annotation></semantics>. Mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>, this distribution only makes sense if <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> does not divide <semantics>n<annotation encoding="application/x-tex">n</annotation></semantics> (otherwise <semantics>1/n<annotation encoding="application/x-tex">1/n</annotation></semantics> is undefined); but assuming that, we do indeed have <semantics>H p(u n)=q p(n)<annotation encoding="application/x-tex">H_p(u_n) = q_p(n)</annotation></semantics>, as we’d expect.

  • When we take our prime <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> to be <semantics>2<annotation encoding="application/x-tex">2</annotation></semantics>, a probability distribution <semantics>π<annotation encoding="application/x-tex">\pi</annotation></semantics> is just a sequence of bits like <semantics>(0,0,1,0,1,1,1,0,1)<annotation encoding="application/x-tex">(0, 0, 1, 0, 1, 1, 1, 0, 1)</annotation></semantics> with an odd number of <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics>s. Its entropy <semantics>H 2(π)/2<annotation encoding="application/x-tex">H_2(\pi) \in \mathbb{Z}/2\mathbb{Z}</annotation></semantics> turns out to be <semantics>0<annotation encoding="application/x-tex">0</annotation></semantics> if the number of <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics>s is congruent to <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics> mod <semantics>4<annotation encoding="application/x-tex">4</annotation></semantics>, and <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics> if the number of <semantics>1<annotation encoding="application/x-tex">1</annotation></semantics>s is congruent to <semantics>3<annotation encoding="application/x-tex">3</annotation></semantics> mod <semantics>4<annotation encoding="application/x-tex">4</annotation></semantics>.

  • What about distributions on two elements? In other words, let <semantics>α/p<annotation encoding="application/x-tex">\alpha \in \mathbb{Z}/p\mathbb{Z}</annotation></semantics> and put <semantics>π=(α,1α)<annotation encoding="application/x-tex">\pi = (\alpha, 1 - \alpha)</annotation></semantics>. What is <semantics>H p(π)<annotation encoding="application/x-tex">H_p(\pi)</annotation></semantics>?

    It takes a bit of algebra to figure this out, but it’s not too hard, and the outcome is that for <semantics>p2<annotation encoding="application/x-tex">p \neq 2</annotation></semantics>, <semantics>H p(α,1α)= r=1 p1α rr.<annotation encoding="application/x-tex"> H_p(\alpha, 1 - \alpha) = \sum_{r = 1}^{p - 1} \frac{\alpha^r}{r}. </annotation></semantics> This function was, in fact, the starting point of Kontsevich’s note, and it’s what he called the <semantics>112<annotation encoding="application/x-tex">1\tfrac{1}{2}</annotation></semantics>-logarithm.

We’ve now succeeded in finding a definition of entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> that satisfies the chain rule. That’s not quite enough, though. In principle, there could be loads of things satisfying the chain rule, in which case, what special status would ours have?

But in fact, up to the inevitable constant factor, our definition of entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> is the one and only definition satisfying the chain rule:

Theorem   Let <semantics>(I:Π n/p)<annotation encoding="application/x-tex">(I: \Pi_n \to \mathbb{Z}/p\mathbb{Z})</annotation></semantics> be a sequence of functions. Then <semantics>I<annotation encoding="application/x-tex">I</annotation></semantics> satisfies the chain rule if and only if <semantics>I=cH p<annotation encoding="application/x-tex">I = c H_p</annotation></semantics> for some <semantics>c/p<annotation encoding="application/x-tex">c \in \mathbb{Z}/p\mathbb{Z}</annotation></semantics>.

This is precisely analogous to the characterization theorem for real entropy, except that in the real case some analytic condition on <semantics>I<annotation encoding="application/x-tex">I</annotation></semantics> has to be imposed (continuity in Faddeev’s theorem, and measurability in the stronger theorem of Lee). So, this is excellent justification for calling <semantics>H p<annotation encoding="application/x-tex">H_p</annotation></semantics> the entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>.

I’ll say nothing about the proof except the following. In Faddeev’s theorem over <semantics><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics>, the tricky part of the proof involves the fact that the sequence <semantics>(logn) n1<annotation encoding="application/x-tex">(\log n)_{n \geq 1}</annotation></semantics> is not uniquely characterized up to a constant factor by the equation <semantics>log(mn)=logm+logn<annotation encoding="application/x-tex">\log(m n) = \log m + \log n</annotation></semantics>; to make that work, you have to introduce some analytic condition. Over <semantics>/p<annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z}</annotation></semantics>, the tricky part involves the fact that the domain of the “logarithm” (Fermat quotient) is not <semantics>/p<annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z}</annotation></semantics>, but <semantics>/p 2<annotation encoding="application/x-tex">\mathbb{Z}/p^2\mathbb{Z}</annotation></semantics>. So, analytic difficulties are replaced by number-theoretic difficulties.

Kontsevich didn’t actually write down a definition of entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> in his two-and-a-half page note. He did exactly enough to show that there must be a unique sensible such definition… and left it there! Of course he could have worked it out if he’d wanted to, and maybe he even did, but he didn’t write it up here.

Anyway, let’s return to the quotation from Kontsevich that I began my first post with:

Conclusion: If we have a random variable <semantics>ξ<annotation encoding="application/x-tex">\xi</annotation></semantics> which takes finitely many values with all probabilities in <semantics><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics> then we can define not only the transcendental number <semantics>H(ξ)<annotation encoding="application/x-tex">H(\xi)</annotation></semantics> but also its “residues modulo <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>” for almost all primes <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> !

In the notation of these posts, he’s saying the following. Let

<semantics>π=(π 1,,π n)<annotation encoding="application/x-tex"> \pi = (\pi_1, \ldots, \pi_n) </annotation></semantics>

be a real probability distribution in which each <semantics>π i<annotation encoding="application/x-tex">\pi_i</annotation></semantics> is rational. There are only finitely many primes that divide one or more of the denominators of <semantics>π 1,,π n<annotation encoding="application/x-tex">\pi_1, \ldots, \pi_n</annotation></semantics>. For primes <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> not belonging to this exceptional set, we can interpret <semantics>π<annotation encoding="application/x-tex">\pi</annotation></semantics> as a probability distribution in <semantics>/p<annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z}</annotation></semantics>. We can therefore take its mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> entropy, <semantics>H p(π)<annotation encoding="application/x-tex">H_p(\pi)</annotation></semantics>.

Kontsevich is playfully suggesting that we view <semantics>H p(π)/p<annotation encoding="application/x-tex">H_p(\pi) \in \mathbb{Z}/p\mathbb{Z}</annotation></semantics> as the residue class mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> of <semantics>H (π)<annotation encoding="application/x-tex">H_\mathbb{R}(\pi) \in \mathbb{R}</annotation></semantics>.

There is more to this than meets the eye! Different real probability distributions can have the same real entropy, so there’s a question of consistency. Kontsevich’s suggestion only makes sense if

<semantics>H (π)=H (π)H p(π)=H p(π).<annotation encoding="application/x-tex"> H_\mathbb{R}(\pi) = H_\mathbb{R}(\pi') \implies H_p(\pi) = H_p(\pi'). </annotation></semantics>

And this is true! I have a proof, though I’m not convinced it’s optimal. Does anyone see an easy argument for this?

Let’s write <semantics> (p)<annotation encoding="application/x-tex">\mathcal{H}^{(p)}</annotation></semantics> for the set of real numbers of the form <semantics>H (π)<annotation encoding="application/x-tex">H_\mathbb{R}(\pi)</annotation></semantics>, where <semantics>π<annotation encoding="application/x-tex">\pi</annotation></semantics> is a real probability distribution whose probabilities <semantics>π i<annotation encoding="application/x-tex">\pi_i</annotation></semantics> can all be expressed as fractions with denominator not divisible by <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>. We’ve just seen that there’s a well-defined map

<semantics>[.]: (p)/p<annotation encoding="application/x-tex"> [.] : \mathcal{H}^{(p)} \to \mathbb{Z}/p\mathbb{Z} </annotation></semantics>

defined by

<semantics>[H (π)]=H p(π).<annotation encoding="application/x-tex"> [H_\mathbb{R}(\pi)] = H_p(\pi). </annotation></semantics>

For <semantics>x (p)<annotation encoding="application/x-tex">x \in \mathcal{H}^{(p)} \subseteq \mathbb{R}</annotation></semantics>, we view <semantics>[x]<annotation encoding="application/x-tex">[x]</annotation></semantics> as the congruence class mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> of <semantics>x<annotation encoding="application/x-tex">x</annotation></semantics>. This notion of “congruence class” even behaves something like the ordinary notion, in the sense that <semantics>[.]<annotation encoding="application/x-tex">[.]</annotation></semantics> preserves addition.

(We can even go a bit further. Accompanying the characterization theorem for entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>, there is a characterization theorem for information loss mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>, strictly analogous to the theorem that John Baez, Tobias Fritz and I proved over <semantics><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics>. I won’t review that stuff here, but the point is that an information loss is a difference of entropies, and this enables us to define the congruence class mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> of the difference of two elements of <semantics> (p)<annotation encoding="application/x-tex">\mathcal{H}^{(p)}</annotation></semantics>. The same additivity holds.)

There’s just one more thing. In a way, the definition of entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> is unsatisfactory. In order to define it, we had to step outside the world of <semantics>/p<annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z}</annotation></semantics> by making arbitrary choices of representing integers, and then we had to show that the definition was independent of those choices. Can’t we do it directly?

In fact, we can. It’s a well-known miracle about finite fields <semantics>K<annotation encoding="application/x-tex">K</annotation></semantics> that any function <semantics>KK<annotation encoding="application/x-tex">K \to K</annotation></semantics> is a polynomial. It’s a slightly less well-known miracle that any function <semantics>K nK<annotation encoding="application/x-tex">K^n \to K</annotation></semantics>, for any <semantics>n0<annotation encoding="application/x-tex">n \geq 0</annotation></semantics>, is also a polynomial.

Of course, multiple polynomials can induce the same function. For instance, the polynomials <semantics>x p<annotation encoding="application/x-tex">x^p</annotation></semantics> and <semantics>x<annotation encoding="application/x-tex">x</annotation></semantics> induce the same function <semantics>/p/p<annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}</annotation></semantics>. But it’s still possible to make a uniqueness statement. Given a function <semantics>F:K nK<annotation encoding="application/x-tex">F : K^n \to K</annotation></semantics>, there’s a unique polynomial <semantics>fK[x 1,,x n]<annotation encoding="application/x-tex">f \in K[x_1, \ldots, x_n]</annotation></semantics> that induces <semantics>F<annotation encoding="application/x-tex">F</annotation></semantics> and is of degree less than the order of <semantics>K<annotation encoding="application/x-tex">K</annotation></semantics> in each variable separately.

So, there must be a polynomial representing entropy, of order less than <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> in each variable. And as it turns out, it’s this one:

<semantics>H p(π 1,,π n)= 0r 1,,r n<p: r 1++r n=pπ 1 r 1π n r nr 1!r n!.<annotation encoding="application/x-tex"> H_p(\pi_1, \ldots, \pi_n) = - \sum_{\substack{0 \leq r_1, \ldots, r_n \lt p:\\r_1 + \cdots + r_n = p}} \frac{\pi_1^{r_1} \cdots \pi_n^{r_n}}{r_1! \cdots r_n!}. </annotation></semantics>

You can check that when <semantics>n=2<annotation encoding="application/x-tex">n = 2</annotation></semantics>, this is in fact the same polynomial <semantics> r=1 p1π 1 r/r<annotation encoding="application/x-tex">\sum_{r = 1}^{p - 1} \pi_1^r/r</annotation></semantics> as we met before — Kontsevich’s <semantics>112<annotation encoding="application/x-tex">1\tfrac{1}{2}</annotation></semantics>-logarithm.

It’s striking that this direct formula for entropy modulo a prime looks quite unlike the formula for real entropy,

<semantics>H (π)= i:π i0π ilogπ i.<annotation encoding="application/x-tex"> H_\mathbb{R}(\pi) = - \sum_{i : \pi_i \neq 0} \pi_i \log \pi_i. </annotation></semantics>

It’s also striking that in the case <semantics>n=2<annotation encoding="application/x-tex">n = 2</annotation></semantics>, the formula for real entropy is

<semantics>H (α,1α)=αlogα(1α)log(1α),<annotation encoding="application/x-tex"> H_\mathbb{R}(\alpha, 1 - \alpha) = - \alpha \log \alpha - (1 - \alpha) \log(1 - \alpha), </annotation></semantics>

whereas mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>, we get

<semantics>H p(α,1α)= r=1 p1α rr,<annotation encoding="application/x-tex"> H_p(\alpha, 1 - \alpha) = \sum_{r = 1}^{p - 1} \frac{\alpha^r}{r}, </annotation></semantics>

which is a truncation of the Taylor series of <semantics>log(1α)<annotation encoding="application/x-tex">-\log(1 - \alpha)</annotation></semantics>. And yet, the characterization theorems for entropy over <semantics><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics> and over <semantics>/p<annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z}</annotation></semantics> are strictly analogous.

As I see it, there are two or three big open questions:

  • Entropy over <semantics><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics> can be understood, interpreted and applied in many ways. How can we understand, interpret or apply entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics>?

  • Entropy over <semantics><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics> and entropy mod <semantics>p<annotation encoding="application/x-tex">p</annotation></semantics> are defined in roughly analogous ways, and uniquely characterized by strictly analogous theorems. Is there a common generalization? That is, can we unify the two definitions and characterization theorems, perhaps proving a theorem about entropy over suitable fields?

by leinster (Tom.Leinster@gmx.com) at January 01, 2018 01:12 PM

December 31, 2017

John Baez - Azimuth

Quantum Mechanics and the Dodecahedron

This is an expanded version of my G+ post, which was a watered-down version of Greg Egan’s G+ post and the comments on that. I’ll start out slow, and pick up speed as I go.

Quantum mechanics meets the dodecahedron

In quantum mechanics, the position of a particle is not a definite thing: it’s described by a ‘wavefunction’. This says how probable it is to find the particle at any location… but it also contains other information, like how probable it is to find the particle moving at any velocity.

Take a hydrogen atom, and look at the wavefunction of the electron.

Question 1. Can we make the electron’s wavefunction have all the rotational symmetries of a dodecahedron—that wonderful Platonic solid with 12 pentagonal faces?

Yes! In fact it’s too easy: you can make the wavefunction look like whatever you want.

So let’s make the question harder. Like everything else in quantum mechanics, angular momentum can be uncertain. In fact you can never make all 3 components of angular momentum take definite values simultaneously! However, there are lots of wavefunctions where the magnitude of an electron’s angular momentum is completely definite.

This leads naturally to the next question, which was first posed by Gerard Westendorp:

Question 2. Can an electron’s wavefunction have a definite magnitude for its angular momentum while having all the rotational symmetries of a dodecahedron?

Yes! And there are infinitely many ways for this to happen! This is true even if we neglect the radial dependence of the wavefunction—that is, how it depends on the distance from the proton. Henceforth I’ll always do that, which lets us treat the wavefunction as a function on a sphere. And by the way, I’m also ignoring the electron’s spin! So, whenever I say ‘angular momentum’ I mean orbital angular momentum: the part that depends only on the electron’s position and velocity.

Question 2 has a trivial solution that’s too silly to bother with. It’s the spherically symmetric wavefunction! That’s invariant under all rotations. The real challenge is to figure out the simplest nontrivial solution. Egan figured it out, and here’s what it looks like:

The rotation here is just an artistic touch. Really the solution should be just sitting there, or perhaps changing colors while staying the same shape.

In what sense is this the simplest nontrivial solution? Well, the magnitude of the angular momentum is equal to

\hbar^2 \sqrt{\ell(\ell+1)}

where the number \ell is quantized: it can only take values 0, 1, 2, 3,… and so on.

The trivial solution to Question 2 has \ell = 0. The first nontrivial solution has \ell = 6. Why 6? That’s where things get interesting. We can get it using the 6 lines connecting opposite faces of the dodecahedron!

I’ll explain later how this works. For now, let’s move straight on to a harder question:

Question 3. What’s the smallest choice of \ell where we can find two linearly independent wavefunctions that both have the same \ell and both have all the rotational symmetries of a dodecahedron?

It turns out to be \ell = 30. And Egan created an image of a wavefunction oscillating between these two possibilities!

But we can go a lot further:

Question 4. For each \ell, how many linearly independent functions on the sphere have that value of \ell and all the rotational symmetries of a dodecahedron?

For \ell ranging from 0 to 29 there are either none or one. There are none for these numbers:

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29

and one for these numbers:

0, 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28

The pattern continues as follows. For \ell ranging from 30 to 59 there are either one or two. There is one for these numbers:

31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 53, 59

and two for these numbers:

30, 36, 40, 42, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58

The numbers in these two lists are just 30 more than the numbers in the first two lists! And it continues on like this forever: there’s always one more linearly independent solution for \ell + 30 than there is for \ell.

Question 5. What’s special about these numbers from 0 to 29?

0, 6, 10, 12, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28

You don’t need to know tons of math to figure this out—but I guess it’s a sort of weird pattern-recognition puzzle unless you know which patterns are likely to be important here. So I’ll give away the answer.

Here’s the answer: these are the numbers below 30 that can be written as sums of the numbers 6, 10 and 15.

But the real question is why? Also: what’s so special about the number 30?

The short, cryptic answer is this. The dodecahedron has 6 axes connecting the centers of opposite faces, 10 axes connecting opposite vertices, and 15 axes connecting the centers of opposite edges. The least common multiple of these numbers is 30.

But this requires more explanation!

For this, we need more math. You may want to get off here. But first, let me show you the solutions for \ell = 6, \ell = 10, and \ell = 15, as drawn by Greg Egan. I’ve already showed you \ell = 6, which we could call the quantum dodecahedron:

Here is \ell = 10, which looks like a quantum icosahedron:

And here is \ell = 15:

Maybe this deserves to be called a quantum Coxeter complex, since the Coxeter complex for the group of rotations and reflections of the dodecahedron looks like this:

Functions with icosahedral symmetry

The dodecahedron and icosahedron have the same symmetries, but for some reason people talk about the icosahedron when discussing symmetry groups, so let me do that.

So far we’ve been looking at the rotational symmetries of the icosahedron. These form a group called \mathrm{A}_5, or \mathrm{I} for short, with 60 elements. We’ve been looking for certain functions on the sphere that are invariant under the action of this group. To get them all, we’ll first get ahold of all polynomials on \mathbb{R}^3 that are invariant under the action of this group Then we’ll restrict these to the sphere.

To save time, we’ll use the work of Claude Chevalley. He looked at rotation and reflection symmetries of the icosahedron. These form the group \mathrm{I} \times \mathbb{Z}/2, also known as \mathrm{H}_3, but let’s call it \hat{\mathrm{I}} for short. It has 120 elements, but never confuse it with two other groups with 120 elements: the symmetric group on 5 letters, and the binary icosahedral group.

Chevalley found all polynomials on \mathbb{R}^3 that are invariant under the action of this bigger group \hat{\mathrm{I}}. These invariant polynomials form an algebra, and Chevalley showed that this algebra is freely generated by 3 homogeneous polynomials:

P(x,y,z) = x^2 + y^2 + z^2, of degree 2.

Q(x,y,z), of degree 6. To get this we take the dot product of (x,y,z) with each of the 6 vectors joining antipodal vertices of the icosahedron, and multiply them together.

R(x,y,z), of degree 10. To get this we take the dot product of (x,y,z) with each of the 10 vectors joining antipodal face centers of the icosahedron, and multiply them together.

So, linear combinations of products of these give all polynomials on \mathbb{R}^3 invariant under all rotation and reflection symmetries of the icosahedron.

But we want the polynomials that are invariant under just rotational symmetries of the icosahedron! To get all these, we need an extra generator:

S(x,y,z), of degree 15. To get this we take the dot product of (x,y,z) with each of the 15 vectors joining antipodal edge centers of the icosahedron, and multiply them together.

You can check that this is invariant under rotational symmetries of the icosahedron. But unlike our other polynomials, this one is not invariant under reflection symmetries! Because 15 is an odd number, S switches sign under ‘total inversion’—that is, replacing (x,y,z) with -(x,y,z). This is a product of three reflection symmetries of the icosahedron.

Thanks to Egan’s extensive computations, I’m completely convinced that P,Q,R and S generate the algebra of all \mathrm{I}-invariant polynomials on \mathbb{R}^3. I’ll take this as a fact, even though I don’t have a clean, human-readable proof. But someone must have proved it already—do you know where?

Since we now have 4 polynomials on \mathbb{R}^3, they must obey a relation. Egan figured it out:

S^2 = 500 P^9 Q^2 - 2275 P^6 Q^3 + 3440 P^3 Q^4 - 1728 Q^5 + 200 P^7 Q R
- 795 P^4 Q^2 R + 720 P Q^3 R + 4 P^5 R^2 -65 P^2 Q R^2 - R^3

The exact coefficients depend on some normalization factors used in defining Q,R and S. Luckily the details don’t matter much. All we’ll really need is that this relation expresses S^2 in terms of the other generators. And this fact is easy to see without any difficult calculations!

How? Well, we’ve seen S is unchanged by rotations, while it changes sign under total inversion. So, the most any rotation or reflection symmetry of the icosahedron can do to S is change its sign. This means that S^2 is invariant under all these symmetries. So, by Chevalley’s result, it must be a polynomial in P, Q, and R.

So, we now have a nice description of the \mathrm{I}-invariant polynomials on \mathbb{R}^3, in terms of generators and relations. Each of these gives an \mathrm{I}-invariant function on the sphere. And Leo Stein, a postdoc at Caltech who has a great blog on math and physics, has kindly created some images of these.

The polynomial P is spherically symmetric so it’s too boring to draw. The polynomial Q, of degree 6, looks like this when restricted to the sphere:

Since it was made by multiplying linear functions, one for each axis connecting opposite vertices of an icosahedron, it shouldn’t be surprising that we see blue blobs centered at these vertices.

The polynomial R, of degree 10, looks like this:

Here the blue blobs are centered on the icosahedron’s 20 faces.

Finally, here’s S, of degree 15:

This time the blue blobs are centered on the icosahedron’s 30 edges.

Now let’s think a bit about functions on the sphere that arise from polynomials on \mathbb{R}^3. Let’s call them algebraic functions on the sphere. They form an algebra, and it’s just the algebra of polynomials on \mathbb{R}^3 modulo the relation P = 1, since the sphere is the set \{P = 1\}.

It makes no sense to talk about the ‘degree’ of an algebraic function on the sphere, since the relation P = 1 equates polynomials of different degree. What makes sense is the number \ell that I was talking about earlier!

The group \mathrm{SO}(3) acts by rotation on the space of algebraic functions on the sphere, and we can break this space up into irreducible representations of \mathrm{SO}(3). It’s a direct sum of irreps, one of each ‘spin’ \ell = 0, 1, 2, \dots.

So, we can’t talk about the degree of a function on the sphere, but we can talk about its \ell value. On the other hand, it’s very convenient to work with homogeneous polynomials on \mathbb{R}^3, which have a definite degree—and these restrict to functions on the sphere. How can we relate the degree and the quantity \ell?

Here’s one way. The polynomials on \mathbb{R}^3 form a graded algebra. That means it’s a direct sum of vector spaces consisting of homogeneous polynomials of fixed degree, and if we multiply two homogeneous polynomials their degrees add. But the algebra of polynomials restricted to the sphere is merely filtered algebra.

What does this mean? Let F be the algebra of all algebraic functions on the sphere, and let F_\ell \subset F consist of those that are restrictions of polynomials of degree \le \ell. Then:

1) F_\ell \subseteq F_{\ell + 1}

and

2) \displaystyle{ F = \bigcup_{\ell = 0}^\infty F_\ell }

and

3) if we multiply a function in F_\ell by one in F_m, we get one in F_{\ell + m}.

That’s what a filtered algebra amounts to.

But starting from a filtered algebra, we can get a graded algebra! It’s called the associated graded algebra.

To do this, we form

G_\ell = F_\ell / F_{\ell - 1}

and let

\displaystyle{ G = \bigoplus_{\ell = 0}^\infty G_\ell }

Then G has a product where multiplying a guy in G_\ell and one in G_m gives one in G_{\ell + m}. So, it’s indeed a graded algebra! For the details, see Wikipedia, which manages to make it look harder than it is. The basic idea is that we multiply in F and then ‘ignore terms of lower degree’. That’s what G_\ell = F_\ell / F_{\ell - 1} is all about.

Now I want to use two nice facts. First, G_\ell is the spin-\ell representation of \mathrm{SO}(3). Second, there’s a natural map from any filtered algebra to its associated graded algebra, which is an isomorphism of vector spaces (though not of algebras). So, we get an natural isomorphism of vector spaces

\displaystyle{  F \cong G = \bigoplus_{\ell = 0}^\infty G_\ell }

from the algebraic functions on the sphere to the direct sum of all the spin-\ell representations!

Now to the point: because this isomorphism is natural, it commutes with symmetries, so we can also use it to study algebraic functions on the sphere that are invariant under a group of linear transformations of \mathbb{R}^3.

Before tackling the group we’re really interested in, let’s try the group of rotation and reflection symmetries of the icosahedron, \hat{\mathrm{I}}. As I mentioned, Chevalley worked out the algebra of polynomials on \mathbb{R}^3 that are invariant under this bigger group. It’s a graded commutative algebra, and it’s free on three generators: P of degree 2, Q of degree 6, and R of degree 10.

Starting from here, to get the algebra of \hat{\mathrm{I}}-invariant algebraic functions on the sphere, we mod out by the relation P = 1. This gives a filtered algebra which I’ll call F^{\hat{\mathrm{I}}}. (It’s common to use a superscript with the name of a group to indicate that we’re talking about the stuff that’s invariant under some action of that group.) From this we can form the associated graded algebra

\displaystyle{ G^{\hat{\mathrm{I}}} = \bigoplus_{\ell = 0}^\infty G_\ell^{\hat{\mathrm{I}}} }

where

G_\ell^{\hat{\mathrm{I}}} = F_\ell^{\hat{\mathrm{I}}} / F_{\ell - 1}^{\hat{\mathrm{I}}}

If you’ve understood everything I’ve been trying to explain, you’ll see that G_\ell^{\hat{\mathrm{I}}} is the space of all functions on the sphere that transform in the spin-\ell representation and are invariant under the rotation and reflection symmetries of the icosahedron.

But now for the fun part: what is this space like? By the work of Chevalley, the algebra F^{\hat{\mathrm{I}}} is spanned by products

P^p Q^q R^r

but since we have the relation P = 1, and no other relations, it has a basis given by products

Q^q R^r

So, the space F_\ell^{\hat{\mathrm{I}}} has a basis of products like this whose degree is \le \ell, meaning

6 q + 10 r \le \ell

Thus, the space we’re really interested in:

G_\ell^{\hat{\mathrm{I}}} = F_\ell^{\hat{\mathrm{I}}} / F_{\ell - 1}^{\hat{\mathrm{I}}}

has a basis consisting of equivalence classes

[Q^q R^r]

where

6 q + 10 r = \ell

So, we get:

Theorem 1. The dimension of the space of functions on the sphere that lie in the spin-\ell representation of \mathrm{SO}(3) and are invariant under the rotation and reflection symmetries of the icosahedron equals the number of ways of writing \ell as an unordered sum of 6’s and 10’s.

Let’s see how this goes:

\ell = 0: dimension 1, with basis [1]

\ell = 1: dimension 0

\ell = 2: dimension 0

\ell = 3: dimension 0

\ell = 4: dimension 0

\ell = 5: dimension 0

\ell = 6: dimension 1, with basis [Q]

\ell = 7: dimension 0

\ell = 8: dimension 0

\ell = 9: dimension 0

\ell = 10: dimension 1, with basis [R]

\ell = 11: dimension 0

\ell = 12: dimension 1, with basis [Q^2]

\ell = 13: dimension 0

\ell = 14: dimension 0

\ell = 15: dimension 0

\ell = 16: dimension 1, with basis [Q R]

\ell = 17: dimension 0

\ell = 18: dimension 1, with basis [Q^3]

\ell = 19: dimension 0

\ell = 20: dimension 1, with basis [R^2]

\ell = 21: dimension 0

\ell = 22: dimension 1, with basis [Q^2 R]

\ell = 23: dimension 0

\ell = 24: dimension 1, with basis [Q^4]

\ell = 25: dimension 0

\ell = 26: dimension 1, with basis [Q R^2]

\ell = 27: dimension 0

\ell = 28: dimension 1, with basis [Q^3 R]

\ell = 29: dimension 0

\ell = 30: dimension 2, with basis [Q^5], [R^3]

So, the story starts out boring, with long gaps. The odd numbers are completely uninvolved. But it heats up near the end, and reaches a thrilling climax at \ell = 30. At this point we get two linearly independent solutions, because 30 is the least common multiple of the degrees of Q and R.

It’s easy to see that from here on the story ‘repeats’ with period 30, with the dimension growing by 1 each time:

\mathrm{dim}(G_{\ell+30}^{\hat{\mathrm{I}}}) = \mathrm{dim}(G_{\ell}^{\hat{\mathrm{I}}}) + 1

Now, finally, we are to tackle Question 4 from the first part of this post: for each \ell, how many linearly independent functions on the sphere have that value of \ell and all the rotational symmetries of a dodecahedron?

We just need to repeat our analysis with \mathrm{I}, the group of rotational symmetries of the dodecahedron, replacing the bigger group \hat{\mathrm{I}}.

We start with algebra of polynomials on \mathbb{R}^3 that are invariant under \mathrm{I}. As we’ve seen, this is a graded commutative algebra with four generators: P,Q,R as before, but also S of degree 15. To make up for this extra generator there’s an extra relation, which expresses S^2 in terms of the other generators.

Starting from here, to get the algebra of \mathrm{I}-invariant algebraic functions on the sphere, we mod out by the relation P = 1. This gives a filtered algebra I’ll call F^{\mathrm{I}}. Then we form the associated graded algebra

\displaystyle{ G^{\mathrm{I}} = \bigoplus_{\ell = 0}^\infty G_\ell^{\mathrm{I}} }

where

G_\ell^{\mathrm{I}} = F_\ell^{\mathrm{I}} / F_{\ell - 1}^{\mathrm{I}}

What we really want to know is the dimension of G_\ell^{\mathrm{I}}, since this is the space of functions on the sphere that transform in the spin-\ell representation and are invariant under the rotational symmetries of the icosahedron.

So, what’s this space like? The algebra F^{\mathrm{I}} is spanned by products

P^p Q^q R^r S^t

but since we have the relation P = 1, and a relation expressing S^2 in terms of other generators, it has a basis given by products

Q^q R^r S^s where s = 0, 1

So, the space F_\ell^{\mathrm{I}} has a basis of products like this whose degree is \le \ell, meaning

6 q + 10 r + 15 s \le \ell and s = 0, 1

Thus, the space we’re really interested in:

G_\ell^{\mathrm{I}} = F_\ell^{\mathrm{I}} / F_{\ell - 1}^{\mathrm{I}}

has a basis consisting of equivalence classes

[Q^q R^r S^s]

where

6 q + 10 r + 15 s = \ell and s = 0, 1

So, we get:

Theorem 2. The dimension of the space of functions on the sphere that lie in the spin-\ell representation of \mathrm{SO}(3) and are invariant under the rotational symmetries of the icosahedron equals the number of ways of writing \ell as an unordered sum of 6’s, 10’s and at most one 15.

Let’s work out these dimensions explicitly, and see how the extra generator S changes the story! Since it has degree 15, it contributes some solutions for odd values of \ell. But when we reach the magic number 30, this extra generator loses its power: S^2 has degree 30, but it’s a linear combination of other things.

\ell = 0: dimension 1, with basis [1]

\ell = 1: dimension 0

\ell = 2: dimension 0

\ell = 3: dimension 0

\ell = 4: dimension 0

\ell = 5: dimension 0

\ell = 6: dimension 1, with basis [Q]

\ell = 7: dimension 0

\ell = 8: dimension 0

\ell = 9: dimension 0

\ell = 10: dimension 1, with basis [R]

\ell = 11: dimension 0

\ell = 12: dimension 1, with basis [Q^2]

\ell = 13: dimension 0

\ell = 14: dimension 0

\ell = 15: dimension 1, with basis [S]

\ell = 16: dimension 1, with basis [Q R]

\ell = 17: dimension 0

\ell = 18: dimension 1, with basis [Q^3]

\ell = 19: dimension 0

\ell = 20: dimension 1, with basis [R^2]

\ell = 21: dimension 1, with basis [Q S]

\ell = 22: dimension 1, with basis [Q^2 R]

\ell = 23: dimension 0

\ell = 24: dimension 1, with basis [Q^4]

\ell = 25: dimension 1, with basis [R S]

\ell = 26: dimension 1, with basis [Q R^2]

\ell = 27: dimension 1, with basis [Q^2 S]

\ell = 28: dimension 1, with basis [Q^3 R]

\ell = 29: dimension 0

\ell = 30: dimension 2, with basis [Q^5], [R^3]

From here on the story ‘repeats’ with period 30, with the dimension growing by 1 each time:

\mathrm{dim}(G_{\ell+30}^{\mathrm{I}}) = \mathrm{dim}(G_{\ell}^{\mathrm{I}}) + 1

So, we’ve more or less proved everything that I claimed in the first part. So we’re done!

Postscript

But I can’t resist saying a bit more.

First, there’s a very different and somewhat easier way to compute the dimensions in Theorems 1 and 2. It uses the theory of characters, and Egan explained it in a comment on the blog post on which this is based.

Second, if you look in these comments, you’ll also see a lot of material about harmonic polynomials on \mathbb{R}^3—that is, those obeying the Laplace equation. These polynomials are very nice when you’re trying to decompose the space of functions on the sphere into irreps of \mathrm{SO}(3). The reason is that the harmonic homogeneous polynomials of degree \ell, when restricted to the sphere, give you exactly the spin-\ell representation!

If you take all homogeneous polynomials of degree \ell and restrict them to the sphere you get a lot of ‘redundant junk’. You get the spin-\ell rep, plus the spin-(\ell-2) rep, plus the spin-(\ell-4) rep, and so on. The reason is the polynomial

P = x^2 + y^2 + z^2

and its powers: if you have a polynomial living in the spin-\ell rep and you multiply it by P, you get another one living in the spin-\ell rep, but you’ve boosted the degree by 2.

Layra Idarani pointed out that this is part of a nice general theory. But I found all this stuff slightly distracting when I was trying to prove Theorems 1 and 2 assuming that we had explicit presentations of the algebras of \hat{\mathrm{I}}– and \mathrm{I}-invariant polynomials on \mathbb{R}^3. So, instead of introducing facts about harmonic polynomials, I decided to use the ‘associated graded algebra’ trick. This is a more algebraic way to ‘eliminate the redundant junk’ in the algebra of polynomials and chop the space of functions on the sphere into irreps of \mathrm{SO}(3).

Also, Egan and Idarani went ahead and considered what happens when we replace the icosahedron by another Platonic solid. It’s enough to consider the cube and tetrahedron. These cases are actually subtler than the icosahedron! For example, when we take the dot product of (x,y,z) with each of the 10 vectors joining antipodal face centers of the cube, and multiply them together, we get a polynomial that’s not invariant under rotations of the cube! Up to a constant it’s just x y z, and this changes sign under some rotations.

People call this sort of quantity, which gets multiplied by a number under transformations instead of staying the same, a semi-invariant. The reason we run into semi-invariants for the cube and tetrahedron is that their rotational symmetry groups, \mathrm{S}_4 and \mathrm{A}_4, have nontrivial abelianizations, namely \mathbb{Z}/2 and \mathbb{Z}/3. The abelianization of \mathrm{I} \cong \mathrm{A}_5 is trivial.

Egan summarized the story as follows:

Just to sum things up for the cube and the tetrahedron, since the good stuff has ended up scattered over many comments:

For the cube, we define:

A of degree 4 from the cube’s vertex-axes, a full invariant
B of degree 6 from the cube’s edge-centre-axes, a semi-invariant
C of degree 3 from the cube’s face-centre-axes, a semi-invariant

We have full invariants:

A of degree 4
C2 of degree 6
BC of degree 9

B2 can be expressed in terms of A, C and P, so we never use it, and we use BC at most once.

So the number of copies of the trivial rep of the rotational symmetry group of the cube in spin ℓ is the number of ways to write ℓ as an unordered sum of 4, 6 and at most one 9.

For the tetrahedron, we embed its vertices as four vertices of the cube. We then define:

V of degree 4 from the tet’s vertices, a full invariant
E of degree 3 from the tet’s edge-centre axes, a full invariant

And the B we defined for the embedding cube serves as a full invariant of the tet, of degree 6.

B2 can be expressed in terms of V, E and P, so we use B at most once.

So the number of copies of the trivial rep of the rotational symmetry group of the tetrahedron in spin ℓ is the number of ways to write ℓ as a sum of 3, 4 and at most one 6.

All of this stuff reminds me of a baby version of the theory of modular forms. For example, the algebra of modular forms is graded by ‘weight’, and it’s the free commutative algebra on a guy of weight 4 and a guy of weight 6. So, the dimension of the space of modular forms of weight k is the number of ways of writing k as an unordered sum of 4’s and 6’s. Since the least common multiple of 4 and 6 is 12, we get a pattern that ‘repeats’, in a certain sense, mod 12. Here I’m talking about the simplest sort of modular forms, based on the group \mathrm{SL}_2(\mathbb{Z}). But there are lots of variants, and I have the feeling that this post is secretly about some sort of variant based on finite subgroups of \mathrm{SL}(2,\mathbb{C}) instead of infinite discrete subgroups.

There’s a lot more to say about all this, but I have to stop or I’ll never stop. Please ask questions and if you want me to say more!

by John Baez at December 31, 2017 01:00 AM

December 30, 2017

Cormac O’Raifeartaigh - Antimatter (Life in a puzzling universe)

A week’s research and a New Year resolution

If anyone had suggested a few years ago that I would forgo a snowsports holiday in the Alps for a week’s research, I would probably not have believed them. Yet here I am, sitting comfortably in the library of the Dublin Institute for Advanced Studies.

ew3          The School of Theoretical Physics at the Dublin Institute for Advanced Studies

It’s been a most satisfying week. One reason is that a change truly is as good as a rest – after a busy teaching term, it’s very enjoyable to spend some time in a quiet spot, surrounded by books on the history of physics. Another reason is that one can accomplish an astonishing amount in one week’s uninterrupted study. That said, I’m not sure I could do this all year round, I’d miss the teaching!

As regards a resolution for 2018, I’ve decided to focus on getting a book out this year. For some time, I have been putting together a small introductory book on the big bang theory, based on a public lecture I give to diverse audiences, from amateur astronomers to curious taxi drivers. The material is drawn from a course I teach at both Waterford Institute of Technology and University College Dublin and is almost in book form already. The UCD experience is particularly useful, as the module is aimed at first-year students from all disciplines.

Of course, there are already plenty of books out there on this topic. My students have a comprehensive reading list, which includes classics such as A Brief History of Time (Hawking), The First Three Minutes (Weinberg) and The Big Bang (Singh). However, I regularly get feedback to the effect that the books are too hard (Hawking) or too long (Singh) or out of date (Weinberg). So I decided a while ago to put together my own effort; a useful exercise if nothing else comes of it.

In particular, I intend to take a historical approach to the story. I’m a great believer in the ‘how-we-found-out’ approach to explaining scientific theories (think for example of that great BBC4 documentary on the discovery of oxygen). My experience is that a historical approach allows the reader to share the excitement of discovery and makes technical material much easier to understand. In addition, much of the work of the early pioneers remains relevant today. The challenge will be to present a story that is also concise – that’s the hard part!

by cormac at December 30, 2017 04:26 PM

December 28, 2017

Life as a Physicist

Christmas Project

Every Christmas I try to do some sort of project. Something new. Sometimes it turns into something real, and last for years. Sometimes it goes no where. Normally, I have an idea of what I’m going to attempt – usually it has been bugging me for months and I can’t wait till break to get it started. This year, I had none.

But, I arrived home at my parent’s house in New Jersey and there it was waiting for me. The house is old – more 200 yrs old – and the steam furnace had just been replaced. For those of you unfamiliar with this method of heating a house: it is noisy! The furnace boils water, and the steam is forced up through the pipes to cast iron radiators. The radiators hiss through valves as the air is forced up – an iconic sound from my childhood. Eventually, after traveling sometimes four floors, the super hot steam reaches the end of a radiator and the valve shuts off. The valves are cool – heat sensitive! The radiator, full of hot steam, then warms the room – and rather effectively.

The bane of this system, however, is that it can leak. And you have no idea where the leak is in the whole house! The only way you know: the furnace reservoir needs refilling too often. So… the problem: how to detect the reservoir needs refilling? Especially with this new modern furnace which can automatically refill its resevoir.

Me: Oh, look, there is a little LED that comes on when the automatic refilling system comes on! I can watch that! Dad: Oh, look, there is a little light that comes on when the water level is low. We can watch that.

Dad’s choice of tools: a wifi cam that is triggered by noise. Me: A Raspberry Pi 3, a photo-resistor, and a capacitor. Hahahaha. Game on!

IMG_20171227_030002What’s funny? Neither of us have detected a water-refill since we started this project. The first picture at the right you can see both of our devices – in the foreground taped to the gas input line is the CAM watching the water refill light through a mirror, and in the background (look for the yellow tape) is the Pi taped to the refill controller (and the capacitor and sensor hanging down looking at the LED on the bottom of the box).

I chose the Pi because I’ve used it once before – for a Spotify end-point. But never for anything that it is designed for. An Arduino is almost certainly better suited to this – but I wasn’t confident that I could get it up and running in the 3 days I had to make this (including time for ordering and shipping of all parts from Amazon). It was a lot of fun! And consumed a bunch of time. “Hey, where is Gordon? He needs to come for Christmas dinner!” “Wait, are you working on Christmas day?” – for once I could answer that last one with a honest no! Hahaha. Smile

I learned a bunch:

  • I had to solder! It has been a loooong time since I’ve done that. My first graduate student, whom I made learn how to solder before I let him graduate, would have laughed at how rusty my skills were!
  • I was surprised to learn, at the start, that the Pi has no analog to digital converter. I stole a quick and dirty trick that lots of people have used to get around this problem: time how long it takes to charge a capacitor up with a photoresistor. This is probably the biggest source of noise in my system, but does for crude measurements.
  • I got to write all my code in Python. Even interrupt handling (ok, no call backs, but still!)
  • The Pi, by default, runs a full build of Linux. Also, python 3! I made full use of this – all my code is in python, and a bit in bash to help it get going. I used things like cron and pip – they were either there, or trivial to install. Really, for this project, I was never consious of the Pi being anything less than a full computer.
  • At first I tried to write auto detection code – that would see any changes in the light levels and write them to a file… which was then served on a nginx simple webserver (seriously – that was about 2 lines of code to install). But the noise in the system plus the fact that we’ve not had a fill so I don’t know what my signal looks like yet… So, that code will have to be revised.
  • In the end, I have to write a file with the raw data in it, and analyze that – at least, until I know what an actual signal looks like. So… how to get that data off the Pi – especially given that I can’t access it anymore now that I’ve left New Jersey? In the end I used some Python code to push the files to OneDrive. Other than figuring out how to deal with OAuth2, it was really easy (and I’m still not done fighting the authentication battle). What will happen if/when it fails? Well… I’ve recorded the commands my Dad will have to execute to get the new authentication files down there. Hopefully there isn’t going to be an expiration!
  • imageTo analyze the raw data I’ve used a new tool I’ve recently learned at work: numpy and Jupyter notebooks. They allow me to produce a plot like this one. The dip near the left hand side of the plot is my Dad shining the flashlight at my sensors to see if I could actually see anything. The joker.

Pretty much the only thing I’d used before was Linux, and some very simple things with an older Raspberry Pi 2. If anyone is on the fence about this – I’d definately recommend trying it out. It is very easy and there are 1000’s of web pages with step by step instructions for most things you’ll want to do!


    by gordonwatts at December 28, 2017 06:25 AM

    John Baez - Azimuth

    The 600-Cell (Part 3)

    There are still a few more things I want to say about the 600-cell. Last time I described the ‘compound of five 24-cells’. David Richter built a model of this, projected from 4 dimensions down to 3:

    It’s nearly impossible to tell from this picture, but it’s five 24-cells inscribed in the 600-cell, with each vertex of the 600-cell being the vertex of just one of these five 24-cells. The trick for constructing it is to notice that the vertices of the 600-cell form a group sitting in the sphere of unit quaternions, and to find a 24-cell whose vertices form a subgroup.

    The left cosets of a subgroup H \subset G are the sets

    gH = \{gh : \; h \in H\}

    They look like copies of H ‘translated’, or in our case ‘rotated’, inside G. Every point of G lies in exactly one coset.

    In our example there are five cosets. Each is the set of vertices of a 24-cell inscribed in the 600-cell. Every vertex of the 600-cell lies in exactly one of these cosets. This gives our ‘compound of five 24-cells’.

    It turns out this trick is part of a family of three tricks, each of which gives a nice compound of 4d regular polytopes. While I’ve been avoiding coordinates, I think they’ll help get the idea across now. Here’s a nice description of the 120 vertices of the 600-cell. We take these points:

    \displaystyle{ (\pm \textstyle{\frac{1}{2}}, \pm \textstyle{\frac{1}{2}},\pm \textstyle{\frac{1}{2}},\pm \textstyle{\frac{1}{2}}) }

    \displaystyle{ (\pm 1, 0, 0, 0) }

    \displaystyle{ \textstyle{\frac{1}{2}} (\pm \Phi, \pm 1 , \pm 1/\Phi, 0 )}

    and all those obtained by even permutations of the coordinates. So, we get

    2^4 = 16

    points of the first kind,

    2 \times 4 = 8

    points of the second kind, and

    2^3 \times 4! / 2 = 96

    points of the third kind, for a total of

    16 + 8 + 96 = 120

    points.

    The 16 points of the first kind are the vertices of a 4-dimensional hypercube, the 4d analogue of a cube:

    The 8 points of the second kind are the vertices of a 4-dimensional orthoplex, the 4d analogue of an octahedron:

    The hypercube and orthoplex are dual to each other. Taking both their vertices together we get the 16 + 8 = 24 vertices of the 24-cell, which is self-dual:

    The hypercube, orthoplex and 24-cell are regular polytopes, as is the 600-cell.

    Now let’s think of any point in 4-dimensional space as a quaternion:

    (a,b,c,d) = a + b i + c j + d k

    If we do this, we can check that the 120 vertices of the 600-cell form a group under quaternion multiplication. As mentioned in Part 1, this group is called the binary icosahedral group or 2\mathrm{I}, because it’s a double cover of the rotational symmetry group of an icosahedron (or dodecahedron).

    We can also check that the 24 vertices of the 24-cell form a group under quaternion multiplication. As mentioned in Part 1, this is called the binary tetrahedral group or 2\mathrm{T}, because it’s a double cover of the rotational symmetry group of a tetrahedron.

    All this is old news. But it’s even easier to check that the 8 vertices of the orthoplex form a group under quaternion multiplication: they’re just

    \pm 1, \pm i, \pm i, \pm k

    This group is often called the quaternion group or \mathrm{Q}. It too is a double cover of a group of rotations! The 180° rotations about the x, y and z axes square to 1 and commute with each other; up in the double cover of the rotation group (the unit quaternions, or \mathrm{SU}(2)) they give elements that square to -1 and anticommute with each other.

    Furthermore, the 180° rotations about the x, y and z axes are symmetries of a regular tetrahedron! This is easiest to visualize if you inscribe the tetrahedron in a cube thus:

    So, up in the double cover of the 3d rotation group we get a chain of subgroups

    \mathrm{Q} \subset 2\mathrm{T} \subset 2\mathrm{I}

    which explains why we’re seeing an orthoplex inscribed in a 24-cell inscribed in a 600-cell! This explanation is more satisfying to me than the one involving coordinates.

    Alas, I don’t see how to understand the hypercube inscribed in the 24-cell in quite this way, since the hypercube is not a subgroup of the unit quaternions. It certainly wasn’t in the coordinates I gave before—but worse, there’s no way to rotate the hypercube so that it becomes a subgroup. There must be something interesting to say here, but I don’t know it. So, I’ll forget the hypercube for now.

    Instead, I’ll use group theory to do something nice with the orthoplex.

    First, look at the orthoplexes sitting inside the 24-cell! We’ve got 8-element subgroup of a 24-element group:

    \mathrm{Q} \subset 2\mathrm{T}

    so it has three right cosets, each forming the vertices of an orthoplex inscribed in the 24-cell. So, we get compound of three orthoplexes: a way of partitioning the vertices of the 24-cell into those of three orthoplexes.

    Second, look at the orthoplexes sitting inside the 600-cell! We’ve got 8-element subgroup of a 120-element group:

    \mathrm{Q} \subset 2\mathrm{I}

    so it has 15 right cosets, each forming the vertices of an orthoplex inscribed in the 600-cell. So, we get a compound of 15 orthoplexes: a way of partitioning the vertices of the 600-cell into those of 15 orthoplexes.

    And third, these fit nicely with what we saw last time: the 24-cells sitting inside the 600-cell! We saw a 24-element subgroup of a 120-element group

    2\mathrm{T} \subset 2\mathrm{I}

    so it has 5 right cosets, each forming the vertices of a 24-cell inscribed in the 600-cell. That gave us the compound of five 24-cells: a way of partitioning the vertices of the 600-cell into those of five 24-cells.

    There are some nontrivial counting problems associated with each of these three compounds. David Roberson has already solved most of these.

    1) How many ways are there of inscribing an orthoplex in a 24-cell?

    2) How many ways are there of inscribing a compound of three orthoplexes in a 24-cell?

    3) How many ways are there of inscribing an orthoplex in a 600-cell? David used a computer to show there are 75. Is there a nice human-understandable argument?

    4) How many ways are there of inscribing a compound of 15 orthoplexes in a 600-cell? David used a computer to show there are 280. Is there a nice human-understandable argument?

    5) How many ways are there of inscribing a 24-cell in a 600-cell? David used a computer to show there are 25. Is there a nice human-understandable argument?

    4) How many ways are there of inscribing a compound of five 24-cells in a 600-cell? David used a computer to show there are 10. Is there a nice human-understandable argument? (It’s pretty easy to prove that 10 is a lower bound.)

    For those who prefer visual delights to math puzzles, here is a model of the compound of 15 orthoplexes, cleverly projected from 4 dimensions down to 3, made by David Richter and some friends:

    It took four people 6 hours to make this! Click on the image to learn more about this amazing shape, and explore David Richter’s pages to see more compounds.

    So far my tale has not encompassed the 120-cell, which is the dual of the 600-cell. This has 600 vertices and 120 dodecahedral faces:

    Unfortunately, like the hypercube, the vertices of the 120-cell cannot be made into a subgroup of the unit quaternions. I’ll need some other idea to think about them in a way that I enjoy. But the 120-cell is amazing because every regular polytope in 4 dimensions can be inscribed in the 120-cell.

    For example, we can inscribe the orthoplex in the 120-cell. Since the orthoplex has 8 vertices while the 120-cell has 600, and

    600/8 = 75

    we might hope for a compound of 75 orthoplexes whose vertices, taken together, are those of the 120-cell. And indeed it exists… and David Richter and his friends have built a model!

    Image credits

    You can click on any image to see its source. The photographs of models of the compound of five 24-cells and the compound of 15 orthoplexes are due to David Richter and friends. The shiny ball-and-strut pictures of the tetrahedron in the cube and the 120-cells were made by Tom Ruen using Robert Webb’s Stella software and placed on Wikicommons. The 2d projections of the hypercube, orthoplex and 24-cell were made by Tom Ruen and placed into the public domain on Wikicommons.

    by John Baez at December 28, 2017 01:00 AM

    December 26, 2017

    Tommaso Dorigo - Scientificblogging

    Christmas Homework
    Scared by the void of Christmas vacations? Unable to put just a few more feet between your mouth and the candy tray? Suffocating in the trivialities of the chit-chat with relatives? I have a solution for you. How about trying to solve a few simple high-energy physics quizzes? 

    I offer three questions below, and you are welcome to think any or all of them over today and tomorrow. In two days I will give my answer, explain the underlying physics a bit, and comment your own answers, if you have been capable of typing them despite your skyrocketing glycemic index.

    read more

    by Tommaso Dorigo at December 26, 2017 12:18 PM

    December 15, 2017

    Andrew Jaffe - Leaves on the Line

    WMAP Breaks Through

    It was announced this morning that the WMAP team has won the $3 million Breakthrough Prize. Unlike the Nobel Prize, which infamously is only awarded to three people each year, the Breakthrough Prize was awarded to the whole 27-member WMAP team, led by Chuck Bennett, Gary Hinshaw, Norm Jarosik, Lyman Page, and David Spergel, but including everyone through postdocs and grad students who worked on the project. This is great, and I am happy to send my hearty congratulations to all of them (many of whom I know well and am lucky to count as friends).

    I actually knew about the prize last week as I was interviewed by Nature for an article about it. Luckily I didn’t have to keep the secret for long. Although I admit to a little envy, it’s hard to argue that the prize wasn’t deserved. WMAP was ideally placed to solidify the current standard model of cosmology, a Universe dominated by dark matter and dark energy, with strong indications that there was a period of cosmological inflation at very early times, which had several important observational consequences. First, it made the geometry of the Universe — as described by Einstein’s theory of general relativity, which links the contents of the Universe with its shape — flat. Second, it generated the tiny initial seeds which eventually grew into the galaxies that we observe in the Universe today (and the stars and planets within them, of course).

    By the time WMAP released its first results in 2003, a series of earlier experiments (including MAXIMA and BOOMERanG, which I had the privilege of being part of) had gone much of the way toward this standard model. Indeed, about ten years one of my Imperial colleagues, Carlo Contaldi, and I wanted to make that comparison explicit, so we used what were then considered fancy Bayesian sampling techniques to combine the data from balloons and ground-based telescopes (which are collectively known as “sub-orbital” experiments) and compare the results to WMAP. We got a plot like the following (which we never published), showing the main quantity that these CMB experiments measure, called the power spectrum (which I’ve discussed in a little more detail here). The horizontal axis corresponds to the size of structures in the map (actually, its inverse, so smaller is to the right) and the vertical axis to how large the the signal is on those scales.

    Grand unified spectrum

    As you can see, the suborbital experiments, en masse, had data at least as good as WMAP on most scales except the very largest (leftmost; this is because you really do need a satellite to see the entire sky) and indeed were able to probe smaller scales than WMAP (to the right). Since then, I’ve had the further privilege of being part of the Planck Satellite team, whose work has superseded all of these, giving much more precise measurements over all of these scales: PlanckCl

    Am I jealous? Ok, a little bit.

    But it’s also true, perhaps for entirely sociological reasons, that the community is more apt to trust results from a single, monolithic, very expensive satellite than an ensemble of results from a heterogeneous set of balloons and telescopes, run on (comparative!) shoestrings. On the other hand, the overall agreement amongst those experiments, and between them and WMAP, is remarkable.

    And that agreement remains remarkable, even if much of the effort of the cosmology community is devoted to understanding the small but significant differences that remain, especially between one monolithic and expensive satellite (WMAP) and another (Planck). Indeed, those “real and serious” (to quote myself) differences would be hard to see even if I plotted them on the same graph. But since both are ostensibly measuring exactly the same thing (the CMB sky), any differences — even those much smaller than the error bars — must be accounted for almost certainly boil down to differences in the analyses or misunderstanding of each team’s own data. Somewhat more interesting are differences between CMB results and measurements of cosmology from other, very different, methods, but that’s a story for another day.

    by Andrew at December 15, 2017 08:11 PM

    December 14, 2017

    Robert Helling - atdotde

    What are the odds?
    It's the time of year, you give out special problems in your classes. So this is mine for the blog. It is motivated by this picture of the home secretaries of the German federal states after their annual meeting as well as some recent discussions on Facebook:
    I would like to call it Summers' problem:

    Let's have two real random variables $M$ and $F$ that are drawn according to two probability distributions $\rho_{M/F}(x)$ (for starters you may both assume to be Gaussians but possibly with different mean and variance). Take $N$ draws from each and order the $2N$ results. What is the probability that the $k$ largest ones are all from $M$ rather than $F$? Express your results in terms of the $\rho_{M/F}(x)$. We are also interested in asymptotic results for $N$ large and $k$ fixed as well as $N$ and $k$ large but $k/N$ fixed.

    Last bonus question: How many of the people that say that they hire only based on merit and end up with an all male board realise that by this they say that women are not as good by quite a margin?

    by Robert Helling (noreply@blogger.com) at December 14, 2017 08:58 AM

    November 30, 2017

    Axel Maas - Looking Inside the Standard Model

    Reaching closure – completing a review
    I did not publish anything here within the last few months, as the review I am writingtook up much more time than expected. A lot of interesting project developments happened also during this time. I will write on them as well later, so that nobody will miss out on the insights we gained and the fun we had with them.

    But now, I want to write about how the review comes along. It has now grown into a veritable almost 120 page document. And actually most of it is texts and formulas, and only very few figures. This makes for a lot of content. Right now, it has reached the status of a release candidate 2. This means I have distributed it to many of my colleagues to comment on it. I also used the draft as lecture notes for a lecture on its contents at a winter school in Odense/Denmark (where I actually wrote this blog entry). Why? Because I wanted to have feedback. What can be understood, and what may I have misunderstood? After all, this review not only looks at my own research. Rather, it compiles knowledge from more than a hundred scientists over 45 years. In fact, some of the results I write about have been obtained before I was born. Especially,I could have overlooked results. With by now dozens of new papers per day, this can easily happen. I have collected more than 330 relevant articles, which I refer to in the review.

    And, of course, I could have misunderstood other people’s results or made mistakes. This needs to be avoided in a review as good as possible.

    Indeed, I had many discussions by now on various aspects of the research I review. I got comments and was challenged. In the end, there was always either a conclusion or the insight that some points, believed to be clear, are not as entirely clear as it seemed. There are always more loopholes, more subtleties, than one anticipates. By this, the review became better, and could collect more insights from many brilliant scientists. And likewise I myself learned a lot.

    In the end, I learned two very important lessons about the physics I review.

    The first is that many more things are connected than I expected. Some issues, which looked to my like a parenthetical remark in the beginning became first remarks at more than one place and ultimately became an issue of their on.

    The second is that the standard modelof particle physics is even more special and more balanced than I thought. I was never really thinking that the standard model is so terrible special. Just one theory among many which happen to fit experiments. But really it is an extremely finely adjusted machinery. Every cog in it is important, and even slight changes will make everything fall apart. All the elements are in constant connection with each other, and influence each other.

    Does this mean anything? Good question. Perhaps it is a sign of an underlying ordering principle. But if it is, I cannot see it (yet?). Perhaps this is just an expression of how a law of nature must be – perfectly balanced. At any rate, it gave me a new perspective of what the standard model is.

    So, as I anticipated writing this review gave me a whole new perspectiveand a lot of insights. Partly by formulating questions and answers more precisely. But, and probably more importantly, I had to explain it to others, and to either successfully defend or adapt it or even correct it.

    In addition, two of the most important lessons about understanding physics I learned were the following:

    One: Take your theory seriously. Do not take a shortcut or use some experience. Literally understand what it means and only then start to interpret.

    Two: Pose your questions (and answers) clearly. Every statement should have a well-defined meaning. Never be vague when you want to make a scientific statement. Be always able to back up a question of “what do you mean by this?” by a precise definition. This seems obvious, but is something you tend to be cavalier about. Don’t.

    So, writing a review not only helps in summarizing knowledge. It also helps to understand this knowledge and realize its implications. And, probably fortunately, it poses new questions. What they are, and what we do about, this is something I will write about in the future.

    So, how does it proceed now? In two weeks I have to deliver the review to the journal which mandated it. At the same time (watch my twitteraccount) it will become available on the preprint server arxiv.org, the standard repository of all elementary particle physics knowledge. Then you can see for yourself what I wrote, and wrote about

    by Axel Maas (noreply@blogger.com) at November 30, 2017 05:15 PM

    November 24, 2017

    Sean Carroll - Preposterous Universe

    Thanksgiving

    This year we give thanks for a simple but profound principle of statistical mechanics that extends the famous Second Law of Thermodynamics: the Jarzynski Equality. (We’ve previously given thanks for the Standard Model Lagrangian, Hubble’s Law, the Spin-Statistics Theorem, conservation of momentum, effective field theory, the error bar, gauge symmetry, Landauer’s Principle, the Fourier Transform, Riemannian Geometry, and the speed of light.)

    The Second Law says that entropy increases in closed systems. But really it says that entropy usually increases; thermodynamics is the limit of statistical mechanics, and in the real world there can be rare but inevitable fluctuations around the typical behavior. The Jarzynski Equality is a way of quantifying such fluctuations, which is increasingly important in the modern world of nanoscale science and biophysics.

    Our story begins, as so many thermodynamic tales tend to do, with manipulating a piston containing a certain amount of gas. The gas is of course made of a number of jiggling particles (atoms and molecules). All of those jiggling particles contain energy, and we call the total amount of that energy the internal energy U of the gas. Let’s imagine the whole thing is embedded in an environment (a “heat bath”) at temperature T. That means that the gas inside the piston starts at temperature T, and after we manipulate it a bit and let it settle down, it will relax back to T by exchanging heat with the environment as necessary.

    Finally, let’s divide the internal energy into “useful energy” and “useless energy.” The useful energy, known to the cognoscenti as the (Helmholtz) free energy and denoted by F, is the amount of energy potentially available to do useful work. For example, the pressure in our piston may be quite high, and we could release it to push a lever or something. But there is also useless energy, which is just the entropy S of the system times the temperature T. That expresses the fact that once energy is in a highly-entropic form, there’s nothing useful we can do with it any more. So the total internal energy is the free energy plus the useless energy,

    U = F + TS. \qquad \qquad (1)

    Our piston starts in a boring equilibrium configuration a, but we’re not going to let it just sit there. Instead, we’re going to push in the piston, decreasing the volume inside, ending up in configuration b. This squeezes the gas together, and we expect that the total amount of energy will go up. It will typically cost us energy to do this, of course, and we refer to that energy as the work Wab we do when we push the piston from a to b.

    Remember that when we’re done pushing, the system might have heated up a bit, but we let it exchange heat Q with the environment to return to the temperature T. So three things happen when we do our work on the piston: (1) the free energy of the system changes; (2) the entropy changes, and therefore the useless energy; and (3) heat is exchanged with the environment. In total we have

    W_{ab} = \Delta F_{ab} + T\Delta S_{ab} - Q_{ab}.\qquad \qquad (2)

    (There is no ΔT, because T is the temperature of the environment, which stays fixed.) The Second Law of Thermodynamics says that entropy increases (or stays constant) in closed systems. Our system isn’t closed, since it might leak heat to the environment. But really the Second Law says that the total of the last two terms on the right-hand side of this equation add up to a positive number; in other words, the increase in entropy will more than compensate for the loss of heat. (Alternatively, you can lower the entropy of a bottle of champagne by putting it in a refrigerator and letting it cool down; no laws of physics are violated.) One way of stating the Second Law for situations such as this is therefore

    W_{ab} \geq \Delta F_{ab}. \qquad \qquad (3)

    The work we do on the system is greater than or equal to the change in free energy from beginning to end. We can make this inequality into an equality if we act as efficiently as possible, minimizing the entropy/heat production: that’s an adiabatic process, and in practical terms amounts to moving the piston as gradually as possible, rather than giving it a sudden jolt. That’s the limit in which the process is reversible: we can get the same energy out as we put in, just by going backwards.

    Awesome. But the language we’re speaking here is that of classical thermodynamics, which we all know is the limit of statistical mechanics when we have many particles. Let’s be a little more modern and open-minded, and take seriously the fact that our gas is actually a collection of particles in random motion. Because of that randomness, there will be fluctuations over and above the “typical” behavior we’ve been describing. Maybe, just by chance, all of the gas molecules happen to be moving away from our piston just as we move it, so we don’t have to do any work at all; alternatively, maybe there are more than the usual number of molecules hitting the piston, so we have to do more work than usual. The Jarzynski Equality, derived 20 years ago by Christopher Jarzynski, is a way of saying something about those fluctuations.

    One simple way of taking our thermodynamic version of the Second Law (3) and making it still hold true in a world of fluctuations is simply to say that it holds true on average. To denote an average over all possible things that could be happening in our system, we write angle brackets \langle \cdots \rangle around the quantity in question. So a more precise statement would be that the average work we do is greater than or equal to the change in free energy:

    \displaystyle \left\langle W_{ab}\right\rangle \geq \Delta F_{ab}. \qquad \qquad (4)

    (We don’t need angle brackets around ΔF, because F is determined completely by the equilibrium properties of the initial and final states a and b; it doesn’t fluctuate.) Let me multiply both sides by -1, which means we  need to flip the inequality sign to go the other way around:

    \displaystyle -\left\langle W_{ab}\right\rangle \leq -\Delta F_{ab}. \qquad \qquad (5)

    Next I will exponentiate both sides of the inequality. Note that this keeps the inequality sign going the same way, because the exponential is a monotonically increasing function; if x is less than y, we know that ex is less than ey.

    \displaystyle e^{-\left\langle W_{ab}\right\rangle} \leq e^{-\Delta F_{ab}}. \qquad\qquad (6)

    (More typically we will see the exponents divided by kT, where k is Boltzmann’s constant, but for simplicity I’m using units where kT = 1.)

    Jarzynski’s equality is the following remarkable statement: in equation (6), if we exchange  the exponential of the average work e^{-\langle W\rangle} for the average of the exponential of the work \langle e^{-W}\rangle, we get a precise equality, not merely an inequality:

    \displaystyle \left\langle e^{-W_{ab}}\right\rangle = e^{-\Delta F_{ab}}. \qquad\qquad (7)

    That’s the Jarzynski Equality: the average, over many trials, of the exponential of minus the work done, is equal to the exponential of minus the free energies between the initial and final states. It’s a stronger statement than the Second Law, just because it’s an equality rather than an inequality.

    In fact, we can derive the Second Law from the Jarzynski equality, using a math trick known as Jensen’s inequality. For our purposes, this says that the exponential of an average is less than the average of an exponential, e^{\langle x\rangle} \leq \langle e^x \rangle. Thus we immediately get

    \displaystyle e^{-\left\langle W_{ab}\right\rangle} \leq \left\langle e^{-W_{ab}}\right\rangle = e^{-\Delta F_{ab}}, \qquad\qquad (8)

    as we had before. Then just take the log of both sides to get \langle W_{ab}\rangle \geq \Delta F_{ab}, which is one way of writing the Second Law.

    So what does it mean? As we said, because of fluctuations, the work we needed to do on the piston will sometimes be a bit less than or a bit greater than the average, and the Second Law says that the average will be greater than the difference in free energies from beginning to end. Jarzynski’s Equality says there is a quantity, the exponential of minus the work, that averages out to be exactly the exponential of minus the free-energy difference. The function e^{-W} is convex and decreasing as a function of W. A fluctuation where W is lower than average, therefore, contributes a greater shift to the average of e^{-W} than a corresponding fluctuation where W is higher than average. To satisfy the Jarzynski Equality, we must have more fluctuations upward in W than downward in W, by a precise amount. So on average, we’ll need to do more work than the difference in free energies, as the Second Law implies.

    It’s a remarkable thing, really. Much of conventional thermodynamics deals with inequalities, with equality being achieved only in adiabatic processes happening close to equilibrium. The Jarzynski Equality is fully non-equilibrium, achieving equality no matter how dramatically we push around our piston. It tells us not only about the average behavior of statistical systems, but about the full ensemble of possibilities for individual trajectories around that average.

    The Jarzynski Equality has launched a mini-revolution in nonequilibrium statistical mechanics, the news of which hasn’t quite trickled to the outside world as yet. It’s one of a number of relations, collectively known as “fluctuation theorems,” which also include the Crooks Fluctuation Theorem, not to mention our own Bayesian Second Law of Thermodynamics. As our technological and experimental capabilities reach down to scales where the fluctuations become important, our theoretical toolbox has to keep pace. And that’s happening: the Jarzynski equality isn’t just imagination, it’s been experimentally tested and verified. (Of course, I remain just a poor theorist myself, so if you want to understand this image from the experimental paper, you’ll have to talk to someone who knows more about Raman spectroscopy than I do.)

    by Sean Carroll at November 24, 2017 02:04 AM

    November 09, 2017

    Robert Helling - atdotde

    Why is there a supercontinent cycle?
    One of the most influential books of my early childhood was my "Kinderatlas"
    There were many things to learn about the world (maps were actually only the last third of the book) and for example I blame my fascination for scuba diving on this book. Also last year, when we visited the Mont-Doré in Auvergne and I had to explain how volcanos are formed to my kids to make them forget how many stairs were still ahead of them to the summit, I did that while mentally picturing the pages in that book about plate tectonics.


    But there is one thing I about tectonics that has been bothering me for a long time and I still haven't found a good explanation for (or at least an acknowledgement that there is something to explain): Since the days of Alfred Wegener we know that the jigsaw puzzle pieces of the continents fit in a way that geologists believe that some hundred million years ago they were all connected as a supercontinent Pangea.
    Pangea animation 03.gif
    By Original upload by en:User:Tbower - USGS animation A08, Public Domain, Link

    In fact, that was only the last in a series of supercontinents, that keep forming and breaking up in the "supercontinent cycle".
    Platetechsimple.png
    By SimplisticReps - Own work, CC BY-SA 4.0, Link

    So here is the question: I am happy with the idea of several (say $N$) plates roughly containing a continent each that a floating around on the magma driven by all kinds of convection processes in the liquid part of the earth. They are moving around in a pattern that looks to me to be pretty chaotic (in the non-technical sense) and of course for random motion you would expect that from time to time two of those collide and then maybe stick for a while.

    Then it would be possible that also a third plate collides with the two but that would be a coincidence (like two random lines typically intersect but if you have three lines they would typically intersect in pairs but typically not in a triple intersection). But to form a supercontinent, you need all $N$ plates to miraculously collide at the same time. This order-$N$ process seems to be highly unlikely when random let alone the fact that it seems to repeat. So this motion cannot be random (yes, Sabine, this is a naturalness argument). This needs an explanation.

    So, why, every few hundred million years, do all the land masses of the earth assemble on side of the earth?

    One explanation could for example be that during those tines, the center of mass of the earth is not in the symmetry center so the water of the oceans flow to one side of the earth and reveals the seabed on the opposite side of the earth. Then you would have essentially one big island. But this seems not to be the case as the continents (those parts that are above sea-level) appear to be stable on much longer time scales. It is not that the seabed comes up on one side and the land on the other goes under water but the land masses actually move around to meet on one side.

    I have already asked this question whenever I ran into people with a geosciences education but it is still open (and I have to admit that in a non-zero number of cases I failed to even make the question clear that an $N$-body collision needs an explanation). But I am sure, you my readers know the answer or even better can come up with one.

    by Robert Helling (noreply@blogger.com) at November 09, 2017 09:35 AM

    October 24, 2017

    Andrew Jaffe - Leaves on the Line

    The Chandrasekhar Mass and the Hubble Constant

    The

    first direct detection of gravitational waves was announced in February of 2015 by the LIGO team, after decades of planning, building and refining their beautiful experiment. Since that time, the US-based LIGO has been joined by the European Virgo gravitational wave telescope (and more are planned around the globe).

    The first four events that the teams announced were from the spiralling in and eventual mergers of pairs of black holes, with masses ranging from about seven to about forty times the mass of the sun. These masses are perhaps a bit higher than we expect to by typical, which might raise intriguing questions about how such black holes were formed and evolved, although even comparing the results to the predictions is a hard problem depending on the details of the statistical properties of the detectors and the astrophysical models for the evolution of black holes and the stars from which (we think) they formed.

    Last week, the teams announced the detection of a very different kind of event, the collision of two neutron stars, each about 1.4 times the mass of the sun. Neutron stars are one possible end state of the evolution of a star, when its atoms are no longer able to withstand the pressure of the gravity trying to force them together. This was first understood by S Chandrasekhar in the early years of the 20th Century, who realised that there was a limit to the mass of a star held up simply by the quantum-mechanical repulsion of the electrons at the outskirts of the atoms making up the star. When you surpass this mass, known, appropriately enough, as the Chandrasekhar mass, the star will collapse in upon itself, combining the electrons and protons into neutrons and likely releasing a vast amount of energy in the form of a supernova explosion. After the explosion, the remnant is likely to be a dense ball of neutrons, whose properties are actually determined fairly precisely by similar physics to that of the Chandrasekhar limit (discussed for this case by Oppenheimer, Volkoff and Tolman), giving us the magic 1.4 solar mass number.

    (Last week also coincidentally would have seen Chandrasekhar’s 107th birthday, and Google chose to illustrate their home page with an animation in his honour for the occasion. I was a graduate student at the University of Chicago, where Chandra, as he was known, spent most of his career. Most of us students were far too intimidated to interact with him, although it was always seen as an auspicious occasion when you spotted him around the halls of the Astronomy and Astrophysics Center.)

    This process can therefore make a single 1.4 solar-mass neutron star, and we can imagine that in some rare cases we can end up with two neutron stars orbiting one another. Indeed, the fact that LIGO saw one, but only one, such event during its year-and-a-half run allows the teams to constrain how often that happens, albeit with very large error bars, between 320 and 4740 events per cubic gigaparsec per year; a cubic gigaparsec is about 3 billion light-years on each side, so these are rare events indeed. These results and many other scientific inferences from this single amazing observation are reported in the teams’ overview paper.

    A series of other papers discuss those results in more detail, covering the physics of neutron stars to limits on departures from Einstein’s theory of gravity (for more on some of these other topics, see this blog, or this story from the NY Times). As a cosmologist, the most exciting of the results were the use of the event as a “standard siren”, an object whose gravitational wave properties are well-enough understood that we can deduce the distance to the object from the LIGO results alone. Although the idea came from Bernard Schutz in 1986, the term “Standard siren” was coined somewhat later (by Sean Carroll) in analogy to the (heretofore?) more common cosmological standard candles and standard rulers: objects whose intrinsic brightness and distances are known and so whose distances can be measured by observations of their apparent brightness or size, just as you can roughly deduce how far away a light bulb is by how bright it appears, or how far away a familiar object or person is by how big how it looks.

    Gravitational wave events are standard sirens because our understanding of relativity is good enough that an observation of the shape of gravitational wave pattern as a function of time can tell us the properties of its source. Knowing that, we also then know the amplitude of that pattern when it was released. Over the time since then, as the gravitational waves have travelled across the Universe toward us, the amplitude has gone down (further objects look dimmer sound quieter); the expansion of the Universe also causes the frequency of the waves to decrease — this is the cosmological redshift that we observe in the spectra of distant objects’ light.

    Unlike LIGO’s previous detections of binary-black-hole mergers, this new observation of a binary-neutron-star merger was also seen in photons: first as a gamma-ray burst, and then as a “nova”: a new dot of light in the sky. Indeed, the observation of the afterglow of the merger by teams of literally thousands of astronomers in gamma and x-rays, optical and infrared light, and in the radio, is one of the more amazing pieces of academic teamwork I have seen.

    And these observations allowed the teams to identify the host galaxy of the original neutron stars, and to measure the redshift of its light (the lengthening of the light’s wavelength due to the movement of the galaxy away from us). It is most likely a previously unexceptional galaxy called NGC 4993, with a redshift z=0.009, putting it about 40 megaparsecs away, relatively close on cosmological scales.

    But this means that we can measure all of the factors in one of the most celebrated equations in cosmology, Hubble’s law: cz=Hd, where c is the speed of light, z is the redshift just mentioned, and d is the distance measured from the gravitational wave burst itself. This just leaves H₀, the famous Hubble Constant, giving the current rate of expansion of the Universe, usually measured in kilometres per second per megaparsec. The old-fashioned way to measure this quantity is via the so-called cosmic distance ladder, bootstrapping up from nearby objects of known distance to more distant ones whose properties can only be calibrated by comparison with those more nearby. But errors accumulate in this process and we can be susceptible to the weakest rung on the chain (see recent work by some of my colleagues trying to formalise this process). Alternately, we can use data from cosmic microwave background (CMB) experiments like the Planck Satellite (see here for lots of discussion on this blog); the typical size of the CMB pattern on the sky is something very like a standard ruler. Unfortunately, it, too, needs to calibrated, implicitly by other aspects of the CMB pattern itself, and so ends up being a somewhat indirect measurement. Currently, the best cosmic-distance-ladder measurement gives something like 73.24 ± 1.74 km/sec/Mpc whereas Planck gives 67.81 ± 0.92 km/sec/Mpc; these numbers disagree by “a few sigma”, enough that it is hard to explain as simply a statistical fluctuation.

    Unfortunately, the new LIGO results do not solve the problem. Because we cannot observe the inclination of the neutron-star binary (i.e., the orientation of its orbit), this blows up the error on the distance to the object, due to the Bayesian marginalisation over this unknown parameter (just as the Planck measurement requires marginalization over all of the other cosmological parameters to fully calibrate the results). Because the host galaxy is relatively nearby, the teams must also account for the fact that the redshift includes the effect not only of the cosmological expansion but also the movement of galaxies with respect to one another due to the pull of gravity on relatively large scales; this so-called peculiar velocity has to be modelled which adds further to the errors.

    This procedure gives a final measurement of 70.0+12-8.0, with the full shape of the probability curve shown in the Figure, taken directly from the paper. Both the Planck and distance-ladder results are consistent with these rather large error bars. But this is calculated from a single object; as more of these events are seen these error bars will go down, typically by something like the square root of the number of events, so it might not be too long before this is the best way to measure the Hubble Constant.

    GW H0

    [Apologies: too long, too technical, and written late at night while trying to get my wonderful not-quite-three-week-old daughter to sleep through the night.]

    by Andrew at October 24, 2017 10:44 AM

    October 17, 2017

    Matt Strassler - Of Particular Significance

    The Significance of Yesterday’s Gravitational Wave Announcement: an FAQ

    Yesterday’s post on the results from the LIGO/VIRGO network of gravitational wave detectors was aimed at getting information out, rather than providing the pedagogical backdrop.  Today I’m following up with a post that attempts to answer some of the questions that my readers and my personal friends asked me.  Some wanted to understand better how to visualize what had happened, while others wanted more clarity on why the discovery was so important.  So I’ve put together a post which  (1) explains what neutron stars and black holes are and what their mergers are like, (2) clarifies why yesterday’s announcement was important — and there were many reasons, which is why it’s hard to reduce it all to a single soundbite.  And (3) there are some miscellaneous questions at the end.

    First, a disclaimer: I am *not* an expert in the very complex subject of neutron star mergers and the resulting explosions, called kilonovas.  These are much more complicated than black hole mergers.  I am still learning some of the details.  Hopefully I’ve avoided errors, but you’ll notice a few places where I don’t know the answers … yet.  Perhaps my more expert colleagues will help me fill in the gaps over time.

    Please, if you spot any errors, don’t hesitate to comment!!  And feel free to ask additional questions whose answers I can add to the list.

    BASIC QUESTIONS ABOUT NEUTRON STARS, BLACK HOLES, AND MERGERS

    What are neutron stars and black holes, and how are they related?

    Every atom is made from a tiny atomic nucleus, made of neutrons and protons (which are very similar), and loosely surrounded by electrons. Most of an atom is empty space, so it can, under extreme circumstances, be crushed — but only if every electron and proton convert to a neutron (which remains behind) and a neutrino (which heads off into outer space.) When a giant star runs out of fuel, the pressure from its furnace turns off, and it collapses inward under its own weight, creating just those extraordinary conditions in which the matter can be crushed. Thus: a star’s interior, with a mass one to several times the Sun’s mass, is all turned into a several-mile(kilometer)-wide ball of neutrons — the number of neutrons approaching a 1 with 57 zeroes after it.

    If the star is big but not too big, the neutron ball stiffens and holds its shape, and the star explodes outward, blowing itself to pieces in a what is called a core-collapse supernova. The ball of neutrons remains behind; this is what we call a neutron star. It’s a ball of the densest material that we know can exist in the universe — a pure atomic nucleus many miles(kilometers) across. It has a very hard surface; if you tried to go inside a neutron star, your experience would be a lot worse than running into a closed door at a hundred miles per hour.

    If the star is very big indeed, the neutron ball that forms may immediately (or soon) collapse under its own weight, forming a black hole. A supernova may or may not result in this case; the star might just disappear. A black hole is very, very different from a neutron star. Black holes are what’s left when matter collapses irretrievably upon itself under the pull of gravity, shrinking down endlessly. While a neutron star has a surface that you could smash your head on, a black hole has no surface — it has an edge that is simply a point of no return, called a horizon. In Einstein’s theory, you can just go right through, as if passing through an open door. You won’t even notice the moment you go in. [Note: this is true in Einstein’s theory. But there is a big controversy as to whether the combination of Einstein’s theory with quantum physics changes the horizon into something novel and dangerous to those who enter; this is known as the firewall controversy, and would take us too far afield into speculation.]  But once you pass through that door, you can never return.

    Black holes can form in other ways too, but not those that we’re observing with the LIGO/VIRGO detectors.

    Why are their mergers the best sources for gravitational waves?

    One of the easiest and most obvious ways to make gravitational waves is to have two objects orbiting each other.  If you put your two fists in a pool of water and move them around each other, you’ll get a pattern of water waves spiraling outward; this is in rough (very rough!) analogy to what happens with two orbiting objects, although, since the objects are moving in space, the waves aren’t in a material like water.  They are waves in space itself.

    To get powerful gravitational waves, you want objects each with a very big mass that are orbiting around each other at very high speed. To get the fast motion, you need the force of gravity between the two objects to be strong; and to get gravity to be as strong as possible, you need the two objects to be as close as possible (since, as Isaac Newton already knew, gravity between two objects grows stronger when the distance between them shrinks.) But if the objects are large, they can’t get too close; they will bump into each other and merge long before their orbit can become fast enough. So to get a really fast orbit, you need two relatively small objects, each with a relatively big mass — what scientists refer to as compact objects. Neutron stars and black holes are the most compact objects we know about. Fortunately, they do indeed often travel in orbiting pairs, and do sometimes, for a very brief period before they merge, orbit rapidly enough to produce gravitational waves that LIGO and VIRGO can observe.

    Why do we find these objects in pairs in the first place?

    Stars very often travel in pairs… they are called binary stars. They can start their lives in pairs, forming together in large gas clouds, or even if they begin solitary, they can end up pairing up if they live in large densely packed communities of stars where it is common for multiple stars to pass nearby. Perhaps surprisingly, their pairing can survive the collapse and explosion of either star, leaving two black holes, two neutron stars, or one of each in orbit around one another.

    What happens when these objects merge?

    Not surprisingly, there are three classes of mergers which can be detected: two black holes merging, two neutron stars merging, and a neutron star merging with a black hole. The first class was observed in 2015 (and announced in 2016), the second was announced yesterday, and it’s a matter of time before the third class is observed. The two objects may orbit each other for billions of years, very slowly radiating gravitational waves (an effect observed in the 70’s, leading to a Nobel Prize) and gradually coming closer and closer together. Only in the last day of their lives do their orbits really start to speed up. And just before these objects merge, they begin to orbit each other once per second, then ten times per second, then a hundred times per second. Visualize that if you can: objects a few dozen miles (kilometers) across, a few miles (kilometers) apart, each with the mass of the Sun or greater, orbiting each other 100 times each second. It’s truly mind-boggling — a spinning dumbbell beyond the imagination of even the greatest minds of the 19th century. I don’t know any scientist who isn’t awed by this vision. It all sounds like science fiction. But it’s not.

    How do we know this isn’t science fiction?

    We know, if we believe Einstein’s theory of gravity (and I’ll give you a very good reason to believe in it in just a moment.) Einstein’s theory predicts that such a rapidly spinning, large-mass dumbbell formed by two orbiting compact objects will produce a telltale pattern of ripples in space itself — gravitational waves. That pattern is both complicated and precisely predicted. In the case of black holes, the predictions go right up to and past the moment of merger, to the ringing of the larger black hole that forms in the merger. In the case of neutron stars, the instants just before, during and after the merger are more complex and we can’t yet be confident we understand them, but during tens of seconds before the merger Einstein’s theory is very precise about what to expect. The theory further predicts how those ripples will cross the vast distances from where they were created to the location of the Earth, and how they will appear in the LIGO/VIRGO network of three gravitational wave detectors. The prediction of what to expect at LIGO/VIRGO thus involves not just one prediction but many: the theory is used to predict the existence and properties of black holes and of neutron stars, the detailed features of their mergers, the precise patterns of the resulting gravitational waves, and how those gravitational waves cross space. That LIGO/VIRGO have detected the telltale patterns of these gravitational waves. That these wave patterns agree with Einstein’s theory in every detail is the strongest evidence ever obtained that there is nothing wrong with Einstein’s theory when used in these combined contexts.  That then in turn gives us confidence that our interpretation of the LIGO/VIRGO results is correct, confirming that black holes and neutron stars really exist and really merge. (Notice the reasoning is slightly circular… but that’s how scientific knowledge proceeds, as a set of detailed consistency checks that gradually and eventually become so tightly interconnected as to be almost impossible to unwind.  Scientific reasoning is not deductive; it is inductive.  We do it not because it is logically ironclad but because it works so incredibly well — as witnessed by the computer, and its screen, that I’m using to write this, and the wired and wireless internet and computer disk that will be used to transmit and store it.)

    THE SIGNIFICANCE(S) OF YESTERDAY’S ANNOUNCEMENT OF A NEUTRON STAR MERGER

    What makes it difficult to explain the significance of yesterday’s announcement is that it consists of many important results piled up together, rather than a simple takeaway that can be reduced to a single soundbite. (That was also true of the black hole mergers announcement back in 2016, which is why I wrote a long post about it.)

    So here is a list of important things we learned.  No one of them, by itself, is earth-shattering, but each one is profound, and taken together they form a major event in scientific history.

    First confirmed observation of a merger of two neutron stars: We’ve known these mergers must occur, but there’s nothing like being sure. And since these things are too far away and too small to see in a telescope, the only way to be sure these mergers occur, and to learn more details about them, is with gravitational waves.  We expect to see many more of these mergers in coming years as gravitational wave astronomy increases in its sensitivity, and we will learn more and more about them.

    New information about the properties of neutron stars: Neutron stars were proposed almost a hundred years ago and were confirmed to exist in the 60’s and 70’s.  But their precise details aren’t known; we believe they are like a giant atomic nucleus, but they’re so vastly larger than ordinary atomic nuclei that can’t be sure we understand all of their internal properties, and there are debates in the scientific community that can’t be easily answered… until, perhaps, now.

    From the detailed pattern of the gravitational waves of this one neutron star merger, scientists already learn two things. First, we confirm that Einstein’s theory correctly predicts the basic pattern of gravitational waves from orbiting neutron stars, as it does for orbiting and merging black holes. Unlike black holes, however, there are more questions about what happens to neutron stars when they merge. The question of what happened to this pair after they merged is still out — did the form a neutron star, an unstable neutron star that, slowing its spin, eventually collapsed into a black hole, or a black hole straightaway?

    But something important was already learned about the internal properties of neutron stars. The stresses of being whipped around at such incredible speeds would tear you and I apart, and would even tear the Earth apart. We know neutron stars are much tougher than ordinary rock, but how much more? If they were too flimsy, they’d have broken apart at some point during LIGO/VIRGO’s observations, and the simple pattern of gravitational waves that was expected would have suddenly become much more complicated. That didn’t happen until perhaps just before the merger.   So scientists can use the simplicity of the pattern of gravitational waves to infer some new things about how stiff and strong neutron stars are.  More mergers will improve our understanding.  Again, there is no other simple way to obtain this information.

    First visual observation of an event that produces both immense gravitational waves and bright electromagnetic waves: Black hole mergers aren’t expected to create a brilliant light display, because, as I mentioned above, they’re more like open doors to an invisible playground than they are like rocks, so they merge rather quietly, without a big bright and hot smash-up.  But neutron stars are big balls of stuff, and so the smash-up can indeed create lots of heat and light of all sorts, just as you might naively expect.  By “light” I mean not just visible light but all forms of electromagnetic waves, at all wavelengths (and therefore at all frequencies.)  Scientists divide up the range of electromagnetic waves into categories. These categories are radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays, listed from lowest frequency and largest wavelength to highest frequency and smallest wavelength.  (Note that these categories and the dividing lines between them are completely arbitrary, but the divisions are useful for various scientific purposes.  The only fundamental difference between yellow light, a radio wave, and a gamma ray is the wavelength and frequency; otherwise they’re exactly the same type of thing, a wave in the electric and magnetic fields.)

    So if and when two neutron stars merge, we expect both gravitational waves and electromagnetic waves, the latter of many different frequencies created by many different effects that can arise when two huge balls of neutrons collide.  But just because we expect them doesn’t mean they’re easy to see.  These mergers are pretty rare — perhaps one every hundred thousand years in each big galaxy like our own — so the ones we find using LIGO/VIRGO will generally be very far away.  If the light show is too dim, none of our telescopes will be able to see it.

    But this light show was plenty bright.  Gamma ray detectors out in space detected it instantly, confirming that the gravitational waves from the two neutron stars led to a collision and merger that produced very high frequency light.  Already, that’s a first.  It’s as though one had seen lightning for years but never heard thunder; or as though one had observed the waves from hurricanes for years but never observed one in the sky.  Seeing both allows us a whole new set of perspectives; one plus one is often much more than two.

    Over time — hours and days — effects were seen in visible light, ultraviolet light, infrared light, X-rays and radio waves.  Some were seen earlier than others, which itself is a story, but each one contributes to our understanding of what these mergers are actually like.

    Confirmation of the best guess concerning the origin of “short” gamma ray bursts:  For many years, bursts of gamma rays have been observed in the sky.  Among them, there seems to be a class of bursts that are shorter than most, typically lasting just a couple of seconds.  They come from all across the sky, indicating that they come from distant intergalactic space, presumably from distant galaxies.  Among other explanations, the most popular hypothesis concerning these short gamma-ray bursts has been that they come from merging neutron stars.  The only way to confirm this hypothesis is with the observation of the gravitational waves from such a merger.  That test has now been passed; it appears that the hypothesis is correct.  That in turn means that we have, for the first time, both a good explanation of these short gamma ray bursts and, because we know how often we observe these bursts, a good estimate as to how often neutron stars merge in the universe.

    First distance measurement to a source using both a gravitational wave measure and a redshift in electromagnetic waves, allowing a new calibration of the distance scale of the universe and of its expansion rate:  The pattern over time of the gravitational waves from a merger of two black holes or neutron stars is complex enough to reveal many things about the merging objects, including a rough estimate of their masses and the orientation of the spinning pair relative to the Earth.  The overall strength of the waves, combined with the knowledge of the masses, reveals how far the pair is from the Earth.  That by itself is nice, but the real win comes when the discovery of the object using visible light, or in fact any light with frequency below gamma-rays, can be made.  In this case, the galaxy that contains the neutron stars can be determined.

    Once we know the host galaxy, we can do something really important.  We can, by looking at the starlight, determine how rapidly the galaxy is moving away from us.  For distant galaxies, the speed at which the galaxy recedes should be related to its distance because the universe is expanding.

    How rapidly the universe is expanding has been recently measured with remarkable precision, but the problem is that there are two different methods for making the measurement, and they disagree.   This disagreement is one of the most important problems for our understanding of the universe.  Maybe one of the measurement methods is flawed, or maybe — and this would be much more interesting — the universe simply doesn’t behave the way we think it does.

    What gravitational waves do is give us a third method: the gravitational waves directly provide the distance to the galaxy, and the electromagnetic waves directly provide the speed of recession.  There is no other way to make this type of joint measurement directly for distant galaxies.  The method is not accurate enough to be useful in just one merger, but once dozens of mergers have been observed, the average result will provide important new information about the universe’s expansion.  When combined with the other methods, it may help resolve this all-important puzzle.

    Best test so far of Einstein’s prediction that the speed of light and the speed of gravitational waves are identical: Since gamma rays from the merger and the peak of the gravitational waves arrived within two seconds of one another after traveling 130 million years — that is, about 5 thousand million million seconds — we can say that the speed of light and the speed of gravitational waves are both equal to the cosmic speed limit to within one part in 2 thousand million million.  Such a precise test requires the combination of gravitational wave and gamma ray observations.

    Efficient production of heavy elements confirmed:  It’s long been said that we are star-stuff, or stardust, and it’s been clear for a long time that it’s true.  But there’s been a puzzle when one looks into the details.  While it’s known that all the chemical elements from hydrogen up to iron are formed inside of stars, and can be blasted into space in supernova explosions to drift around and eventually form planets, moons, and humans, it hasn’t been quite as clear how the other elements with heavier atoms — atoms such as iodine, cesium, gold, lead, bismuth, uranium and so on — predominantly formed.  Yes they can be formed in supernovas, but not so easily; and there seem to be more atoms of heavy elements around the universe than supernovas can explain.  There are many supernovas in the history of the universe, but the efficiency for producing heavy chemical elements is just too low.

    It was proposed some time ago that the mergers of neutron stars might be a suitable place to produce these heavy elements.  Even those these mergers are rare, they might be much more efficient, because the nuclei of heavy elements contain lots of neutrons and, not surprisingly, a collision of two neutron stars would produce lots of neutrons in its debris, suitable perhaps for making these nuclei.   A key indication that this is going on would be the following: if a neutron star merger could be identified using gravitational waves, and if its location could be determined using telescopes, then one would observe a pattern of light that would be characteristic of what is now called a “kilonova” explosion.   Warning: I don’t yet know much about kilonovas and I may be leaving out important details. A kilonova is powered by the process of forming heavy elements; most of the nuclei produced are initially radioactive — i.e., unstable — and they break down by emitting high energy particles, including the particles of light (called photons) which are in the gamma ray and X-ray categories.  The resulting characteristic glow would be expected to have a pattern of a certain type: it would be initially bright but would dim rapidly in visible light, with a long afterglow in infrared light.  The reasons for this are complex, so let me set them aside for now.  The important point is that this pattern was observed, confirming that a kilonova of this type occurred, and thus that, in this neutron star merger, enormous amounts of heavy elements were indeed produced.  So we now have a lot of evidence, for the first time, that almost all the heavy chemical elements on and around our planet were formed in neutron star mergers.  Again, we could not know this if we did not know that this was a neutron star merger, and that information comes only from the gravitational wave observation.

    MISCELLANEOUS QUESTIONS

    Did the merger of these two neutron stars result in a new black hole, a larger neutron star, or an unstable rapidly spinning neutron star that later collapsed into a black hole?

    We don’t yet know, and maybe we won’t know.  Some scientists involved appear to be leaning toward the possibility that a black hole was formed, but others seem to say the jury is out.  I’m not sure what additional information can be obtained over time about this.

    If the two neutron stars formed a black hole, why was there a kilonova?  Why wasn’t everything sucked into the black hole?

    Black holes aren’t vacuum cleaners; they pull things in via gravity just the same way that the Earth and Sun do, and don’t suck things in some unusual way.  The only crucial thing about a black hole is that once you go in you can’t come out.  But just as when trying to avoid hitting the Earth or Sun, you can avoid falling in if you orbit fast enough or if you’re flung outward before you reach the edge.

    The point in a neutron star merger is that the forces at the moment of merger are so intense that one or both neutron stars are partially ripped apart.  The material that is thrown outward in all directions, at an immense speed, somehow creates the bright, hot flash of gamma rays and eventually the kilonova glow from the newly formed atomic nuclei.  Those details I don’t yet understand, but I know they have been carefully studied both with approximate equations and in computer simulations such as this one and this one.  However, the accuracy of the simulations can only be confirmed through the detailed studies of a merger, such as the one just announced.  It seems, from the data we’ve seen, that the simulations did a fairly good job.  I’m sure they will be improved once they are compared with the recent data.

     

     

     

    by Matt Strassler at October 17, 2017 04:03 PM

    October 16, 2017

    Sean Carroll - Preposterous Universe

    Standard Sirens

    Everyone is rightly excited about the latest gravitational-wave discovery. The LIGO observatory, recently joined by its European partner VIRGO, had previously seen gravitational waves from coalescing black holes. Which is super-awesome, but also a bit lonely — black holes are black, so we detect the gravitational waves and little else. Since our current gravitational-wave observatories aren’t very good at pinpointing source locations on the sky, we’ve been completely unable to say which galaxy, for example, the events originated in.

    This has changed now, as we’ve launched the era of “multi-messenger astronomy,” detecting both gravitational and electromagnetic radiation from a single source. The event was the merger of two neutron stars, rather than black holes, and all that matter coming together in a giant conflagration lit up the sky in a large number of wavelengths simultaneously.

    Look at all those different observatories, and all those wavelengths of electromagnetic radiation! Radio, infrared, optical, ultraviolet, X-ray, and gamma-ray — soup to nuts, astronomically speaking.

    A lot of cutting-edge science will come out of this, see e.g. this main science paper. Apparently some folks are very excited by the fact that the event produced an amount of gold equal to several times the mass of the Earth. But it’s my blog, so let me highlight the aspect of personal relevance to me: using “standard sirens” to measure the expansion of the universe.

    We’re already pretty good at measuring the expansion of the universe, using something called the cosmic distance ladder. You build up distance measures step by step, determining the distance to nearby stars, then to more distant clusters, and so forth. Works well, but of course is subject to accumulated errors along the way. This new kind of gravitational-wave observation is something else entirely, allowing us to completely jump over the distance ladder and obtain an independent measurement of the distance to cosmological objects. See this LIGO explainer.

    The simultaneous observation of gravitational and electromagnetic waves is crucial to this idea. You’re trying to compare two things: the distance to an object, and the apparent velocity with which it is moving away from us. Usually velocity is the easy part: you measure the redshift of light, which is easy to do when you have an electromagnetic spectrum of an object. But with gravitational waves alone, you can’t do it — there isn’t enough structure in the spectrum to measure a redshift. That’s why the exploding neutron stars were so crucial; in this event, GW170817, we can for the first time determine the precise redshift of a distant gravitational-wave source.

    Measuring the distance is the tricky part, and this is where gravitational waves offer a new technique. The favorite conventional strategy is to identify “standard candles” — objects for which you have a reason to believe you know their intrinsic brightness, so that by comparing to the brightness you actually observe you can figure out the distance. To discover the acceleration of the universe, for example,  astronomers used Type Ia supernovae as standard candles.

    Gravitational waves don’t quite give you standard candles; every one will generally have a different intrinsic gravitational “luminosity” (the amount of energy emitted). But by looking at the precise way in which the source evolves — the characteristic “chirp” waveform in gravitational waves as the two objects rapidly spiral together — we can work out precisely what that total luminosity actually is. Here’s the chirp for GW170817, compared to the other sources we’ve discovered — much more data, almost a full minute!

    So we have both distance and redshift, without using the conventional distance ladder at all! This is important for all sorts of reasons. An independent way of getting at cosmic distances will allow us to measure properties of the dark energy, for example. You might also have heard that there is a discrepancy between different ways of measuring the Hubble constant, which either means someone is making a tiny mistake or there is something dramatically wrong with the way we think about the universe. Having an independent check will be crucial in sorting this out. Just from this one event, we are able to say that the Hubble constant is 70 kilometers per second per megaparsec, albeit with large error bars (+12, -8 km/s/Mpc). That will get much better as we collect more events.

    So here is my (infinitesimally tiny) role in this exciting story. The idea of using gravitational-wave sources as standard sirens was put forward by Bernard Schutz all the way back in 1986. But it’s been developed substantially since then, especially by my friends Daniel Holz and Scott Hughes. Years ago Daniel told me about the idea, as he and Scott were writing one of the early papers. My immediate response was “Well, you have to call these things `standard sirens.'” And so a useful label was born.

    Sadly for my share of the glory, my Caltech colleague Sterl Phinney also suggested the name simultaneously, as the acknowledgments to the paper testify. That’s okay; when one’s contribution is this extremely small, sharing it doesn’t seem so bad.

    By contrast, the glory attaching to the physicists and astronomers who pulled off this observation, and the many others who have contributed to the theoretical understanding behind it, is substantial indeed. Congratulations to all of the hard-working people who have truly opened a new window on how we look at our universe.

    by Sean Carroll at October 16, 2017 03:52 PM

    Matt Strassler - Of Particular Significance

    A Scientific Breakthrough! Combining Gravitational and Electromagnetic Waves

    Gravitational waves are now the most important new tool in the astronomer’s toolbox.  Already they’ve been used to confirm that large black holes — with masses ten or more times that of the Sun — and mergers of these large black holes to form even larger ones, are not uncommon in the universe.   Today it goes a big step further.

    It’s long been known that neutron stars, remnants of collapsed stars that have exploded as supernovas, are common in the universe.  And it’s been known almost as long that sometimes neutron stars travel in pairs.  (In fact that’s how gravitational waves were first discovered, indirectly, back in the 1970s.)  Stars often form in pairs, and sometimes both stars explode as supernovas, leaving their neutron star relics in orbit around one another.  Neutron stars are small — just ten or so kilometers (miles) across.  According to Einstein’s theory of gravity, a pair of stars should gradually lose energy by emitting gravitational waves into space, and slowly but surely the two objects should spiral in on one another.   Eventually, after many millions or even billions of years, they collide and merge into a larger neutron star, or into a black hole.  This collision does two things.

    1. It makes some kind of brilliant flash of light — electromagnetic waves — whose details are only guessed at.  Some of those electromagnetic waves will be in the form of visible light, while much of it will be in invisible forms, such as gamma rays.
    2. It makes gravitational waves, whose details are easier to calculate and which are therefore distinctive, but couldn’t have been detected until LIGO and VIRGO started taking data, LIGO over the last couple of years, VIRGO over the last couple of months.

    It’s possible that we’ve seen the light from neutron star mergers before, but no one could be sure.  Wouldn’t it be great, then, if we could see gravitational waves AND electromagnetic waves from a neutron star merger?  It would be a little like seeing the flash and hearing the sound from fireworks — seeing and hearing is better than either one separately, with each one clarifying the other.  (Caution: scientists are often speaking as if detecting gravitational waves is like “hearing”.  This is only an analogy, and a vague one!  It’s not at all the same as acoustic waves that we can hear with our ears, for many reasons… so please don’t take it too literally.)  If we could do both, we could learn about neutron stars and their properties in an entirely new way.

    Today, we learned that this has happened.  LIGO , with the world’s first two gravitational observatories, detected the waves from two merging neutron stars, 130 million light years from Earth, on August 17th.  (Neutron star mergers last much longer than black hole mergers, so the two are easy to distinguish; and this one was so close, relatively speaking, that it was seen for a long while.)  VIRGO, with the third detector, allows scientists to triangulate and determine roughly where mergers have occurred.  They saw only a very weak signal, but that was extremely important, because it told the scientists that the merger must have occurred in a small region of the sky where VIRGO has a relative blind spot.  That told scientists where to look.

    The merger was detected for more than a full minute… to be compared with black holes whose mergers can be detected for less than a second.  It’s not exactly clear yet what happened at the end, however!  Did the merged neutron stars form a black hole or a neutron star?  The jury is out.

    At almost exactly the moment at which the gravitational waves reached their peak, a blast of gamma rays — electromagnetic waves of very high frequencies — were detected by a different scientific team, the one from FERMI. FERMI detects gamma rays from the distant universe every day, and a two-second gamma-ray-burst is not unusual.  And INTEGRAL, another gamma ray experiment, also detected it.   The teams communicated within minutes.   The FERMI and INTEGRAL gamma ray detectors can only indicate the rough region of the sky from which their gamma rays originate, and LIGO/VIRGO together also only give a rough region.  But the scientists saw those regions overlapped.  The evidence was clear.  And with that, astronomy entered a new, highly anticipated phase.

    Already this was a huge discovery.  Brief gamma-ray bursts have been a mystery for years.  One of the best guesses as to their origin has been neutron star mergers.  Now the mystery is solved; that guess is apparently correct. (Or is it?  Probably, but the gamma ray discovery is surprisingly dim, given how close it is.  So there are still questions to ask.)

    Also confirmed by the fact that these signals arrived within a couple of seconds of one another, after traveling for over 100 million years from the same source, is that, indeed, the speed of light and the speed of gravitational waves are exactly the same — both of them equal to the cosmic speed limit, just as Einstein’s theory of gravity predicts.

    Next, these teams quickly told their astronomer friends to train their telescopes in the general area of the source. Dozens of telescopes, from every continent and from space, and looking for electromagnetic waves at a huge range of frequencies, pointed in that rough direction and scanned for anything unusual.  (A big challenge: the object was near the Sun in the sky, so it could be viewed in darkness only for an hour each night!) Light was detected!  At all frequencies!  The object was very bright, making it easy to find the galaxy in which the merger took place.  The brilliant glow was seen in gamma rays, ultraviolet light, infrared light, X-rays, and radio.  (Neutrinos, particles that can serve as another way to observe distant explosions, were not detected this time.)

    And with so much information, so much can be learned!

    Most important, perhaps, is this: from the pattern of the spectrum of light, the conjecture seems to be confirmed that the mergers of neutron stars are important sources, perhaps the dominant one, for many of the heavy chemical elements — iodine, iridium, cesium, gold, platinum, and so on — that are forged in the intense heat of these collisions.  It used to be thought that the same supernovas that form neutron stars in the first place were the most likely source.  But now it seems that this second stage of neutron star life — merger, rather than birth — is just as important.  That’s fascinating, because neutron star mergers are much more rare than the supernovas that form them.  There’s a supernova in our Milky Way galaxy every century or so, but it’s tens of millenia or more between these “kilonovas”, created in neutron star mergers.

    If there’s anything disappointing about this news, it’s this: almost everything that was observed by all these different experiments was predicted in advance.  Sometimes it’s more important and useful when some of your predictions fail completely, because then you realize how much you have to learn.  Apparently our understanding of gravity, of neutron stars, and of their mergers, and of all sorts of sources of electromagnetic radiation that are produced in those merges, is even better than we might have thought. But fortunately there are a few new puzzles.  The X-rays were late; the gamma rays were dim… we’ll hear more about this shortly, as NASA is holding a second news conference.

    Some highlights from the second news conference:

    • New information about neutron star interiors, which affects how large they are and therefore how exactly they merge, has been obtained
    • The first ever visual-light image of a gravitational wave source, from the Swope telescope, at the outskirts of a distant galaxy; the galaxy’s center is the blob of light, and the arrow points to the explosion.

    • The theoretical calculations for a kilonova explosion suggest that debris from the blast should rather quickly block the visual light, so the explosion dims quickly in visible light — but infrared light lasts much longer.  The observations by the visible and infrared light telescopes confirm this aspect of the theory; and you can see evidence for that in the picture above, where four days later the bright spot is both much dimmer and much redder than when it was discovered.
    • Estimate: the total mass of the gold and platinum produced in this explosion is vastly larger than the mass of the Earth.
    • Estimate: these neutron stars were formed about 10 or so billion years ago.  They’ve been orbiting each other for most of the universe’s history, and ended their lives just 130 million years ago, creating the blast we’ve so recently detected.
    • Big Puzzle: all of the previous gamma-ray bursts seen up to now have always had shone in ultraviolet light and X-rays as well as gamma rays.   But X-rays didn’t show up this time, at least not initially.  This was a big surprise.  It took 9 days for the Chandra telescope to observe X-rays, too faint for any other X-ray telescope.  Does this mean that the two neutron stars created a black hole, which then created a jet of matter that points not quite directly at us but off-axis, and shines by illuminating the matter in interstellar space?  This had been suggested as a possibility twenty years ago, but this is the first time there’s been any evidence for it.
    • One more surprise: it took 16 days for radio waves from the source to be discovered, with the Very Large Array, the most powerful existing radio telescope.  The radio emission has been growing brighter since then!  As with the X-rays, this seems also to support the idea of an off-axis jet.
    • Nothing quite like this gamma-ray burst has been seen — or rather, recognized — before.  When a gamma ray burst doesn’t have an X-ray component showing up right away, it simply looks odd and a bit mysterious.  Its harder to observe than most bursts, because without a jet pointing right at us, its afterglow fades quickly.  Moreover, a jet pointing at us is bright, so it blinds us to the more detailed and subtle features of the kilonova.  But this time, LIGO/VIRGO told scientists that “Yes, this is a neutron star merger”, leading to detailed study from all electromagnetic frequencies, including patient study over many days of the X-rays and radio.  In other cases those observations would have stopped after just a short time, and the whole story couldn’t have been properly interpreted.

     

     

    by Matt Strassler at October 16, 2017 03:10 PM

    October 13, 2017

    Sean Carroll - Preposterous Universe

    Mind-Blowing Quantum Mechanics

    Trying to climb out from underneath a large pile of looming (and missed) deadlines, and in the process I’m hoping to ramp back up the real blogging. In the meantime, here are a couple of videos to tide you over.

    First, an appearance a few weeks ago on Joe Rogan’s podcast. Rogan is a professional comedian and mixed-martial arts commentator, but has built a great audience for his wide-ranging podcast series. One of the things that makes him a good interviewer is his sincere delight in the material, as evidenced here by noting repeatedly that his mind had been blown. We talked for over two and a half hours, covering cosmology and quantum mechanics but also some bits about AI and pop culture.

    And here’s a more straightforward lecture, this time at King’s College in London. The topic was “Extracting the Universe from the Wave Function,” which I’ve used for a few talks that ended up being pretty different in execution. This one was aimed at undergraduate physics students, some of whom hadn’t even had quantum mechanics. So the first half is a gentle introduction to many-worlds theory and why it’s the best version of quantum mechanics, and the second half tries to explain our recent efforts to emerge space itself out of quantum entanglement.

    I was invited to King’s by Eugene Lim, one of my former grad students and now an extremely productive faculty member in his own right. It’s always good to see your kids grow up to do great things!

    by Sean Carroll at October 13, 2017 03:01 PM

    October 09, 2017

    Alexey Petrov - Symmetry factor

    Non-linear teaching

    I wanted to share some ideas about a teaching method I am trying to develop and implement this semester. Please let me know if you’ve heard of someone doing something similar.

    This semester I am teaching our undergraduate mechanics class. This is the first time I am teaching it, so I started looking into a possibility to shake things up and maybe apply some new method of teaching. And there are plenty offered: flipped classroom, peer instruction, Just-in-Time teaching, etc.  They all look to “move away from the inefficient old model” where there the professor is lecturing and students are taking notes. I have things to say about that, but not in this post. It suffices to say that most of those approaches are essentially trying to make students work (both with the lecturer and their peers) in class and outside of it. At the same time those methods attempt to “compartmentalize teaching” i.e. make large classes “smaller” by bringing up each individual student’s contribution to class activities (by using “clickers”, small discussion groups, etc). For several reasons those approaches did not fit my goal this semester.

    Our Classical Mechanics class is a gateway class for our physics majors. It is the first class they take after they are done with general physics lectures. So the students are already familiar with the (simpler version of the) material they are going to be taught. The goal of this class is to start molding physicists out of students: they learn to simplify problems so physics methods can be properly applied (that’s how “a Ford Mustang improperly parked at the top of the icy hill slides down…” turns into “a block slides down the incline…”), learn to always derive the final formula before plugging in numbers, look at the asymptotics of their solutions as a way to see if the solution makes sense, and many other wonderful things.

    So with all that I started doing something I’d like to call non-linear teaching. The gist of it is as follows. I give a lecture (and don’t get me wrong, I do make my students talk and work: I ask questions, we do “duels” (students argue different sides of a question), etc — all of that can be done efficiently in a class of 20 students). But instead of one homework with 3-4 problems per week I have two types of homework assignments for them: short homeworks and projects.

    Short homework assignments are single-problem assignments given after each class that must be done by the next class. They are designed such that a student need to re-derive material that we discussed previously in class with small new twist added. For example, in the block-down-to-incline problem discussed in class I ask them to choose coordinate axes in a different way and prove that the result is independent of the choice of the coordinate system. Or ask them to find at which angle one should throw a stone to get the maximal possible range (including air resistance), etc.  This way, instead of doing an assignment in the last minute at the end of the week, students have to work out what they just learned in class every day! More importantly, I get to change how I teach. Depending on how they did on the previous short homework, I adjust the material (both speed and volume) discussed in class. I also  design examples for the future sections in such a way that I can repeat parts of the topic that was hard for the students previously. Hence, instead of a linear propagation of the course, we are moving along something akin to helical motion, returning and spending more time on topics that students find more difficult. That’t why my teaching is “non-linear”.

    Project homework assignments are designed to develop understanding of how topics in a given chapter relate to each other. There are as many project assignments as there are chapters. Students get two weeks to complete them.

    Overall, students solve exactly the same number of problems they would in a normal lecture class. Yet, those problems are scheduled in a different way. In my way, students are forced to learn by constantly re-working what was just discussed in a lecture. And for me, I can quickly react (by adjusting lecture material and speed) using constant feedback I get from students in the form of short homeworks. Win-win!

    I will do benchmarking at the end of the class by comparing my class performance to aggregate data from previous years. I’ll report on it later. But for now I would be interested to hear your comments!

     


    by apetrov at October 09, 2017 09:45 PM

    October 05, 2017

    Symmetrybreaking - Fermilab/SLAC

    A radio for dark matter

    Instead of searching for dark matter particles, a new device will search for dark matter waves.

    Header: A radio for dark matter

    Researchers are testing a prototype “radio” that could let them listen to the tune of mysterious dark matter particles. 

    Dark matter is an invisible substance thought to be five times more prevalent in the universe than regular matter. According to theory, billions of dark matter particles pass through the Earth each second. We don’t notice them because they interact with regular matter only very weakly, through gravity.

    So far, researchers have mostly been looking for dark matter particles. But with the dark matter radio, they want to look for dark matter waves.

    Direct detection experiments for dark matter particles use large underground detectors. Researchers hope to see signals from dark matter particles colliding with the detector material. However, this only works if dark matter particles are heavy enough to deposit a detectable amount energy in the collision. 

    “If dark matter particles were very light, we might have a better chance of detecting them as waves rather than particles,” says Peter Graham, a theoretical physicist at the Kavli Institute for Particle Astrophysics and Cosmology, a joint institute of Stanford University and the Department of Energy’s SLAC National Accelerator Laboratory. “Our device will take the search in that direction.”

    The dark matter radio makes use of a bizarre concept of quantum mechanics known as wave-particle duality: Every particle can also behave like a wave. 

    Take, for example, the photon: the massless fundamental particle that carries the electromagnetic force. Streams of them make up electromagnetic radiation, or light, which we typically describe as waves—including radio waves. 

    The dark matter radio will search for dark matter waves associated with two particular dark matter candidates.  It could find hidden photons—hypothetical cousins of photons with a small mass. Or it could find axions, which scientists think can be produced out of light and transform back into it in the presence of a magnetic field.

    “The search for hidden photons will be completely unexplored territory,” says Saptarshi Chaudhuri, a Stanford graduate student on the project. “As for axions, the dark matter radio will close gaps in the searches of existing experiments.”

    Intercepting dark matter vibes

    A regular radio intercepts radio waves with an antenna and converts them into sound. What sound depends on the station. A listener chooses a station by adjusting an electric circuit, in which electricity can oscillate with a certain resonant frequency. If the circuit’s resonant frequency matches the station’s frequency, the radio is tuned in and the listener can hear the broadcast.

    The dark matter radio works the same way. At its heart is an electric circuit with an adjustable resonant frequency. If the device were tuned to a frequency that matched the frequency of a dark matter particle wave, the circuit would resonate. Scientists could measure the frequency of the resonance, which would reveal the mass of the dark matter particle. 

    The idea is to do a frequency sweep by slowly moving through the different frequencies, as if tuning a radio from one end of the dial to the other.

    The electric signal from dark matter waves is expected to be very weak. Therefore, Graham has partnered with a team led by another KIPAC researcher, Kent Irwin. Irwin’s group is developing highly sensitive magnetometers known as superconducting quantum interference devices, or SQUIDs, which they’ll pair with extremely low-noise amplifiers to hunt for potential signals.

    In its final design, the dark matter radio will search for particles in a mass range of trillionths to millionths of an electronvolt. (One electronvolt is about a billionth of the mass of a proton.) This is somewhat problematic because this range includes kilohertz to gigahertz frequencies—frequencies used for over-the-air broadcasting. 

    “Shielding the radio from unwanted radiation is very important and also quite challenging,” Irwin says. “In fact, we would need a several-yards-thick layer of copper to do so. Fortunately we can achieve the same effect with a thin layer of superconducting metal.”

    One advantage of the dark matter radio is that it does not need to be shielded from cosmic rays. Whereas direct detection searches for dark matter particles must operate deep underground to block out particles falling from space, the dark matter radio can operate in a university basement.

    The researchers are now testing a small-scale prototype at Stanford that will scan a relatively narrow frequency range. They plan on eventually operating two independent, full-size instruments at Stanford and SLAC.

    “This is exciting new science,” says Arran Phipps, a KIPAC postdoc on the project. “It’s great that we get to try out a new detection concept with a device that is relatively low-budget and low-risk.” 

    The dark matter disc jockeys are taking the first steps now and plan to conduct their dark matter searches over the next few years. Stay tuned for future results.

    by Manuel Gnida at October 05, 2017 01:23 PM

    October 03, 2017

    Symmetrybreaking - Fermilab/SLAC

    Nobel recognizes gravitational wave discovery

    Scientists Rainer Weiss, Kip Thorne and Barry Barish won the 2017 Nobel Prize in Physics for their roles in creating the LIGO experiment.

    Illustration depicting two black holes circling one another and producing gravitational waves

    Three scientists who made essential contributions to the LIGO collaboration have been awarded the 2017 Nobel Prize in Physics.

    Rainer Weiss will share the prize with Kip Thorne and Barry Barish for their roles in the discovery of gravitational waves, ripples in space-time predicted by Albert Einstein. Weiss and Thorne conceived of LIGO, and Barish is credited with reviving the struggling experiment and making it happen.

    “I view this more as a thing that recognizes the work of about 1000 people,” Weiss said during a Q&A after the announcement this morning. “It’s really a dedicated effort that has been going on, I hate to tell you, for as long as 40 years, people trying to make a detection in the early days and then slowly but surely getting the technology together to do it.”

    Another founder of LIGO, scientist Ronald Drever, died in March. Nobel Prizes are not awarded posthumously.

    According to Einstein’s general theory of relativity, powerful cosmic events release energy in the form of waves traveling through the fabric of existence at the speed of light. LIGO detects these disturbances when they disrupt the symmetry between the passages of identical laser beams traveling identical distances.

    The setup for the LIGO experiment looks like a giant L, with each side stretching about 2.5 miles long. Scientists split a laser beam and shine the two halves down the two sides of the L. When each half of the beam reaches the end, it reflects off a mirror and heads back to the place where its journey began.

    Normally, the two halves of the beam return at the same time. When there’s a mismatch, scientists know something is going on. Gravitational waves compress space-time in one direction and stretch it in another, giving one half of the beam a shortcut and sending the other on a longer trip. LIGO is sensitive enough to notice a difference between the arms as small as 1000th the diameter of an atomic nucleus.

    Scientists on LIGO and their partner collaboration, called Virgo, reported the first detection of gravitational waves in February 2016. The waves were generated in the collision of two black holes with 29 and 36 times the mass of the sun 1.3 billion years ago. They reached the LIGO experiment as scientists were conducting an engineering test.

    “It took us a long time, something like two months, to convince ourselves that we had seen something from outside that was truly a gravitational wave,” Weiss said.

    LIGO, which stands for Laser Interferometer Gravitational-Wave Observatory, consists of two of these pieces of equipment, one located in Louisiana and another in Washington state.

    The experiment is operated jointly by Weiss’s home institution, MIT, and Barish and Thorne’s home institution, Caltech. The experiment has collaborators from more than 80 institutions from more than 20 countries. A third interferometer, operated by the Virgo collaboration, recently joined LIGO to make the first joint observation of gravitational waves.

    by Kathryn Jepsen at October 03, 2017 10:42 AM

    September 28, 2017

    Symmetrybreaking - Fermilab/SLAC

    Conjuring ghost trains for safety

    A Fermilab technical specialist recently invented a device that could help alert oncoming trains to large vehicles stuck on the tracks.

    Photo of a train traveling along the tracks

    Browsing YouTube late at night, Fermilab Technical Specialist Derek Plant stumbled on a series of videos that all begin the same way: a large vehicle—a bus, semi or other low-clearance vehicle—is stuck on a railroad crossing. In the end, the train crashes into the stuck vehicle, destroying it and sometimes even derailing the train. According to the Federal Railroad Administration, every year hundreds of vehicles meet this fate by trains, which can take over a mile to stop.

    “I was just surprised at the number of these that I found,” Plant says. “For every accident that’s videotaped, there are probably many more.”

    Inspired by a workplace safety class that preached a principle of minimizing the impact of accidents, Plant set about looking for solutions to the problem of trains hitting stuck vehicles.

    Railroad tracks are elevated for proper drainage, and the humped profile of many crossings can cause a vehicle to bottom out. “Theoretically, we could lower all the crossings so that they’re no longer a hump. But there are 200,000 crossings in the United States,” Plant says. “Railroads and local governments are trying hard to minimize the number of these crossings by creating overpasses, or elevating roadways. That’s cost-prohibitive, and it’s not going to happen soon.”

    Other solutions, such as re-engineering the suspension on vehicles likely to get stuck, seemed equally improbable.

    After studying how railroad signaling systems work, Plant came up with an idea: to fake the presence of a train. His invention was developed in his spare time using techniques and principles he learned over his almost two decades at Fermilab. It is currently in the patent application process and being prosecuted by Fermilab’s Office of Technology Transfer.

    “If you cross over a railroad track and you look down the tracks, you’ll see red or yellow or green lights,” he says. “Trains have traffic signals too.”

    These signals are tied to signal blocks—segments of the tracks that range from a mile to several miles in length. When a train is on the tracks, its metal wheels and axle connect both rails, forming an electric circuit through the tracks to trigger the signals. These signals inform other trains not to proceed while one train occupies a block, avoiding pileups.

    Plant thought, “What if other vehicles could trigger the same signal in an emergency?” By faking the presence of a train, a vehicle stuck on the tracks could give advanced warning for oncoming trains to stop and stall for time. Hence the name of Plant’s invention: the Ghost Train Generator.

    To replicate the train’s presence, Plant knew he had to create a very strong electric current between the rails. The most straightforward way to do this is with massive amounts of metal, as a train does. But for the Ghost Train Generator to be useful in a pinch, it needs to be small, portable and easily applied. The answer to achieving these features lies in strong magnets and special wire.

    “Put one magnet on one rail and one magnet on the other and the device itself mimics—electrically—what a train would look like to the signaling system,” he says. “In theory, this could be carried in vehicles that are at high risk for getting stuck on a crossing: semis, tour buses and first-response vehicles,” Plant says. “Keep it just like you would a fire extinguisher—just behind the seat or in an emergency compartment.”

    Once the device is deployed, the train would receive the signal that the tracks were obstructed and stop. Then the driver of the stuck vehicle could call for emergency help using the hotline posted on all crossings.

    Plant compares the invention to a seatbelt.

    “Is it going to save your life 100 percent of the time? Nope, but smart people wear them,” he says. “It’s designed to prevent a collision when a train is more than two minutes from the crossing.”

    And like a seatbelt, part of what makes Plant’s invention so appealing is its simplicity.

    “The first thing I thought was that this is a clever invention,” says Aaron Sauers from Fermilab’s technology transfer office, who works with lab staff to develop new technologies for market. “It’s an elegant solution to an existing problem. I thought, ‘This technology could have legs.’”

    The organizers of the National Innovation Summit seem to agree.  In May, Fermilab received an Innovation Award from TechConnect for the Ghost Train Generator. The invention will also be featured as a showcase technology in the upcoming Defense Innovation Summit in October.

    The Ghost Train Generator is currently in the pipeline to receive a patent with help from Fermilab, and its prospects are promising, according to Sauers. It is a nonprovisional patent, which has specific claims and can be licensed. After that, if the generator passes muster and is granted a patent, Plant will receive a portion of the royalties that it generates for Fermilab.

    Fermilab encourages a culture of scientific innovation and exploration beyond the field of particle physics, according to Sauers, who noted that Plant’s invention is just one of a number of technology transfer initiatives at the lab.

    Plant agrees—Fermilab’s environment helped motivate his efforts to find a solution for railroad crossing accidents.

    “It’s just a general problem-solving state of mind,” he says. “That’s the philosophy we have here at the lab.”

    Editor's note: A version of this article was originally published by Fermilab.

    by Daniel Garisto at September 28, 2017 05:33 PM

    Symmetrybreaking - Fermilab/SLAC

    Fermilab on display

    The national laboratory opened usually inaccessible areas of its campus to thousands of visitors to celebrate 50 years of discovery.

    Fermilab on display

    Fermi National Accelerator Laboratory’s yearlong 50th anniversary celebration culminated on Saturday with an Open House that drew thousands of visitors despite the unseasonable heat.

    On display were areas of the lab not normally open to guests, including neutrino and muon experiments, a portion of the accelerator complex, lab spaces and magnet and accelerator fabrication and testing areas, to name a few. There were also live links to labs around the world, including CERN, a mountaintop observatory in Chile, and the mile-deep Sanford Underground Research Facility that will house the international neutrino experiment, DUNE.

    But it wasn’t all physics. In addition to hands-on demos and a STEM fair, visitors could also learn about Fermilab’s art and history, walk the prairie trails or hang out with the ever-popular bison. In all, some 10,000 visitors got to go behind-the-scenes at Fermilab, shuttled around on 80 buses and welcomed by 900 Fermilab workers eager to explain their roles at the lab. Below, see a few of the photos captured as Fermilab celebrated 50 years of discovery.

    by Lauren Biron at September 28, 2017 03:47 PM

    Subscriptions

    Feeds

    [RSS 2.0 Feed] [Atom Feed]


    Last updated:
    January 24, 2018 03:21 AM
    All times are UTC.

    Suggest a blog:
    planet@teilchen.at