Particle Physics Planet

December 10, 2013

Peter Coles - In the Dark

The Cosmic Web at Sussex

Yesterday I had the honour of giving an evening lecture for staff and students at the School of Mathematical and Physical Sciences at the University of Sussex. The event was preceded by a bit of impromptu twilight stargazing with the new telescope our students have just purchased:

You can just about see Venus in the second picture, just to the left of the street light.

Anyway, after briefly pretending to be a proper astronomer it was down to my regular business as a cosmologist and my talk entitled The Cosmic Web. Here is the abstract:

The lecture will focus on the large-scale structure of the Universe and the ideas that physicists are weaving together to explain how it came to be the way it is. Over the last few decades, astronomers have revealed that our cosmos is not only vast in scale – at least 14 billion light years in radius – but also exceedingly complex, with galaxies and clusters of galaxies linked together in immense chains and sheets, surrounding giant voids of (apparently) empty space. Cosmologists have developed theoretical explanations for its origin that involve such exotic concepts as ‘dark matter’ and ‘cosmic inflation’, producing a cosmic web of ideas that is, in some ways, as rich and fascinating as the Universe itself.

And for those of you interested, here are the slides I used for your perusal:

It was quite a large (and  very mixed) audience; it’s always difficult to pitch a talk at the right level in those circumstances so that it’s not too boring for the people who know something already but not too challenging for those who don’t know anything at all. A couple of people walked out about five minutes into the talk, which doesn’t exactly inspire a speaker with confidence, but overall it seemed to go down quite well.

Most of all, thank you to the organizers for the very nice reward of a bottle of wine!

Tommaso Dorigo - Scientificblogging

Top Partners Wanted
No, this is not an article about top models. Rather, the subject of discussion are models that predict the existence of heavy partners of the top quark.

Emily Lakdawalla - The Planetary Society Blog

A Protected Class of Programs at NASA?
The House Science Committee is considering giving a select few NASA programs special protected status against cancellation.

Lubos Motl - string vacua and pheno

The first Amplituhedron paper is out
We've been using the word "Amplituhedron" since September 2013 but only now, the first preprint with this word in the title was released:
The Amplituhedron
The authors, Nima Arkani-Hamed and Jaroslav Trnka ["Yuh-raw-sluff Turn-kuh" if you allow me to bastardize a Czech name), are preparing two more papers, "Into the Amplituhedron" and "Scattering Amplitudes from Positive Geometry", as well as a third paper along with Andrew Hodges, "Three Views of the Amplituhedron".

The today's paper has 36 pages of JHEP $$\rm\LaTeX$$.

These pages are divided to 14 short sections and it seems that one should be able to read the whole paper. So I hope that some of the TRF readers will try to look, too.

Emily Lakdawalla - The Planetary Society Blog

Curiosity results at AGU: Gale crater rocks are old, but have been exposed recently
In a Martian first, the Curiosity science team has measured the age of a Martian rock, in two totally different ways. They presented the result at the 2013 meeting of the American Geophysical Union.

The n-Category Cafe

A Technical Innovation

Here’s a new feature of the Café, thanks to our benevolent host Jacques Distler. If you ever want to see how someone has created some mathematical expression on this blog, there’s an easy way to do it.

With Firefox, you simply double-click on the expression. Try it: $A×{B}^{A}\to BA \times B^A \to B$ or ${x}_{mn}x_\left\{m n\right\}$ or

$\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right). \Biggl\left( \begin\left\{matrix\right\} 1 & 2 \\ 3 & 4 \end\left\{matrix\right\} \Biggr\right). $

A window should pop up showing the TeX source.

With other browsers, I’m not so sure. Try double-clicking. If that doesn’t work, then, according to Jacques’s instructions, you “bring up the MathJax context-menu for the formula, and choose Show Math As $\to \to$ Annotation $\to \to$ TeX”. I don’t know how one brings up this menu. Does anyone else know?

Once you’ve made the TeX source appear, you can cut and paste to your heart’s content. Of course, most users here are fluent in LaTeX. But like most math-oriented websites, we use a variant of TeX that’s a little different from standard LaTeX, so this should turn out to be a helpful feature.

December 09, 2013

Christian P. Robert - xi'an's og

invariant conjugate analysis for exponential families

Here is a paper from Bayesian Analysis that I somehow missed and only become aware thanks to a (more) recent paper of the first author: in 2012, Pierre Druilhet and Denis Pommeret published invariant conjugate analysis for exponential families. The authors define a new class of conjugate families, called Jeffreys’ conjugate priors (JCP) by using Jeffreys’ prior as the reference density (rather than the uniform in regular conjugate families). Following from the earlier proposal of Druilhet and Marin (2007, BA). Both families of course coincide in the case of quadratic variance exponential families. The motivation for using those new conjugate priors is that the family is invariant by a change of parametrisation. And to include Jeffreys’ prior as a special case of conjugate prior. In the special case of the inverse Gaussian distribution, this approach leads to the conjugacy of the inverse normal distribution, a feature I noticed in 1991 when working on an astronomy project. There are two obvious drawbacks to those new conjugate families: one is that the priors are not longer always proper. The other one is that the computations associated with those new priors are more involved, which may explain why the authors propose the MAP as their default estimator. Since posterior expectations of the mean (in the natural representation [in x] of the exponential family) are no longer linear in x.

Filed under: Books, Statistics, University life Tagged: conjugate priors, inverse Gaussian distribution, inverse normal distribution

arXiv blog

The Emerging Technologies Shaping Future 5G Networks

The fifth generation of mobile communications technology will see the end of the “cell” as the fundamental building block of communication networks.

Emily Lakdawalla - The Planetary Society Blog

The Sorry State of Planetary Science Funding In One Chart
If you want to know why Cassini might be terminated early, or why NASA pulled out of its joint Mars mission with Europe, or why the new ASRG power source was put on indefinite hold, this chart has your answer.

Andrew Jaffe - Leaves on the Line

Breaking the silence (updated)

My apologies for being far too busy to post. I’ll be much louder in couple of weeks once we release the Planck data — on March 21. Until then, I have to shut up and follow the Planck rules.

OK, back to editing. (I’ll try to update this post with any advance information as it becomes available.)

Update (on timing, not content): the main Planck press conference will be held on the morning of 21 March at 10am CET at ESA HQ in Paris. There will be a simultaneous UK event (9am GMT) held at the Royal Astronomical Society in London, where the Paris event will be streamed, followed by a local Q&A session. (There will also be a more technical afternoon session in Paris.)

Probably more important for my astrophysics colleagues: the Planck papers will be posted on the ESA website at noon on the 21st, after the press event, and will appear on the ArXiV the following day, 22 March. Be sure to set aside some time next weekend!

Andrew Jaffe - Leaves on the Line

Planck 2013: the science

If you’re the kind of person who reads this blog, then you won’t have missed yesterday’s announcement of the first Planck cosmology results.

The most important is our picture of the cosmic microwave background itself:

But it takes a lot of work to go from the data coming off the Planck satellite to this picture. First, we have to make nine different maps, one at each of the frequencies in which Planck observes, from 30 GHz (with a wavelength of 1 cm) up to 850 GHz (0.350 mm) — note that the colour scales here are the same:

At low and high frequencies, these are dominated by the emission of our own galaxy, and there is at least some contamination over the whole range, so it takes hard work to separate the primordial CMB signal from the dirty (but interesting) astrophysics along the way. In fact, it’s sufficiently challenging that the team uses four different methods, each with different assumptions, to do so, and the results agree remarkably well.

In fact, we don’t use the above CMB image directly to do the main cosmological science. Instead, we build a Bayesian model of the data, combining our understanding of the foreground astrophysics and the cosmology, and marginalise over the astrophysical parameters in order to extract as much cosmological information as we can. (The formalism is described in the Planck likelihood paper, and the main results of the analysis are in the Planck cosmological parameters paper.)

The main tool for this is the power spectrum, a plot which shows us how the different hot and cold spots on our CMB map are distributed: In this plot, the left-hand side (low ℓ) corresponds to large angles on the sky and high ℓ to small angles. Planck’s results are remarkable for covering this whole range from ℓ=2 to ℓ=2500: the previous CMB satellite, WMAP, had a high-quality spectrum out to ℓ=750 or so; ground- and balloon-based experiments like SPT and ACT filled in some of the high-ℓ regime.

It’s worth marvelling at this for a moment, a triumph of modern cosmological theory and observation: our theoretical models fit our data from scales of 180° down to 0.1°, each of those bumps and wiggles a further sign of how well we understand the contents, history and evolution of the Universe. Our high-quality data has refined our knowledge of the cosmological parameters that describe the universe, decreasing the error bars by a factor of several on the six parameters that describe the simplest ΛCDM universe. Moreover, and maybe remarkably, the data don’t seem to require any additional parameters beyond those six: for example, despite previous evidence to the contrary, the Universe doesn’t need any additional neutrinos.

The quantity most well-measured by Planck is related to the typical size of spots in the CMB map; it’s about a degree, with an error of less than one part in 1,000. This quantity has changed a bit (by about the width of the error bar) since the previous WMAP results. This, in turn, causes us to revise our estimates of quantities like the expansion rate of the Universe (the Hubble constant), which has gone down, in fact by enough that it’s interestingly different from its best measurements using local (non-CMB) data, from more or less direct observations of galaxies moving away from us. Both methods have disadvantages: for the CMB, it’s a very indirect measurement, requiring imposing a model upon the directly measured spot size (known more technically as the “acoustic scale” since it comes from sound waves in the early Universe). For observations of local galaxies, it requires building up the famous cosmic distance ladder, calibrating our understanding of the distances to further and further objects, few of which we truly understand from first principles. So perhaps this discrepancy is due to messy and difficult astrophysics, or perhaps to interesting cosmological evolution.

This change in the expansion rate is also indirectly responsible for the results that have made the most headlines: it changes our best estimate of the age of the Universe (slower expansion means an older Universe) and of the relative amounts of its constituents (since the expansion rate is related to the geometry of the Universe, which, because of Einstein’s General Relativity, tells us the amount of matter).

But the cosmological parameters measured in this way are just Planck’s headlines: there is plenty more science. We’ve gone beyond the power spectrum above to put limits upon so-called non-Gaussianities which are signatures of the detailed way in which the seeds of large-scale structure in the Universe was initially laid down. We’ve observed clusters of galaxies which give us yet more insight into cosmology (and which seem to show an intriguing tension with some of the cosmological parameters). We’ve measured the deflection of light by gravitational lensing. And in work that I helped lead, we’ve used the CMB maps to put limits on some of the ways in which our simplest models of the Universe could be wrong, possibly having an interesting topology or rotation on the largest scales.

But because we’ve scrutinised our data so carefully, we have found some peculiarities which don’t quite fit the models. From the days of COBE and WMAP, there has been evidence that the largest angular scales in the map, a few degrees and larger, have some “anomalies” — some of the patterns show strange alignments, some show unexpected variation between two different hemispheres of the sky, and there are some areas of the sky that are larger and colder than is expected to occur in our theories. Individually, any of these might be a statistical fluke (and collectively they may still be) but perhaps they are giving us evidence of something exciting going on in the early Universe. Or perhaps, to use a bad analogy, the CMB map is like the Zapruder film: if you scrutinise anything carefully enough, you’ll find things that look a conspiracy, but turn out to have an innocent explanation.

I’ve mentioned eight different Planck papers so far, but in fact we’ve released 28 (and there will be a few more to come over the coming months, and many in the future). There’s an overall introduction to the Planck Mission, and papers on the data processing, observations of relatively nearby galaxies, and plenty more cosmology. The papers have been submitted to the journal A&A, they’re available on the ArXiV, and you can find a list of them at the ESA site.

Even more important for my cosmology colleagues, we’ve released the Planck data, as well, along with the necessary code and other information necessary to understand it: you can get it from the Planck Legacy Archive. I’m sure we’ve only just begun to get exciting and fun science out of the data from Planck. And this is only the beginning of Planck’s data: just the first 15 months of observations, and just the intensity of the CMB: in the coming years we’ll be analysing (and releasing) more than one more year of data, and starting to dig into Planck’s observations of the polarized sky.

Emily Lakdawalla - The Planetary Society Blog

The Mists of Mars
Two grand canyons fill with fog, one on Earth and one on Mars.

Symmetrybreaking - Fermilab/SLAC

Chinese collider expands particle zoo

China’s Beijing Electron-Positron Collider seems to be hosting a reunion; members of a poorly understood family of particles keep popping up in their data, which may help clarify the properties of this reclusive family.

While much of the world’s attention remains transfixed on the Large Hadron Collider and its discovery of the Higgs boson, a continent away, another, smaller particle accelerator is churning out particles—including at least two brand-new and completely unexpected ones.

Matt Strassler - Of Particular Significance

The Guardian’s Level-Headed Article on Fukushima

[Note: If you missed Wednesday evening's discussion of particle physics involving me, Sean Carroll and Alan Boyle, you can listen to it here.]

I still have a lot of work to do before I can myself write intelligently about the Fukushima Daiichi nuclear plant, and the nuclear accident and cleanup that occurred there. (See here and here for a couple of previous posts about it.) But I did want to draw your attention to one of the better newspaper articles that I’ve seen written about it, by Ian Sample at the Guardian. I can’t vouch for everything that Sample says, but given what I’ve read and investigated myself, I think he finds the right balance. He’s neither scaring people unnecessarily, nor reassuring them that everything will surely be just fine and that there’s no reason to be worried about anything. From what I know and understand, the situation is more or less just as serious and worthy of concern as Sample says it is; but conversely, I don’t have any reason to think it is much worse than what he describes.

Meanwhile, just as I don’t particularly trust anything said by TEPCO, the apparently incompetent and corrupt Japanese power company that runs and is trying to clean up the Fukushima plant, I’m also continuing to see lots of scary articles — totally irresponsible — written by people who should know better but seem bent upon frightening the public. The more wild the misstatements and misleading statements, the better, it seems.

One example of this kind of fear-mongering is to be found here: http://truth-out.org/news/item/19547-fukushima-a-global-threat-that-requires-a-global-response, by Kevin Zeese and Margaret Flowers. It’s one piece of junk after the next: the strategy is to take a fact, take another unrelated fact, quote a non-expert (or quote an expert out of context), stick them all together, and wow! frightening!! But here’s the thing: An experienced and attentive reader will know, after a few paragraphs, to ignore this article. Why?

Because it never puts anything in context. “When contact with radioactive cesium occurs, which is highly unlikely, a person can experience cell damage due to radiation of the cesium particles. Due to this, effects such as nausea, vomiting, diarrhea and bleeding may occur. When the exposure lasts a long time, people may even lose consciousness. Coma or even death may then follow. How serious the effects are depends upon the resistance of individual persons and the duration of exposure and the concentration a person is exposed to, experts say.” Well, how much cesium are we talking about here? Lots or a little? Ah, they don’t tell you that. [The answer: enormous amounts. There's no chance of you getting anywhere near that amount of exposure unless you yourself go wandering around on the Fukushima grounds, and go some place you're really not supposed to go. This didn't even happened to the workers who were at the Fukushima plant when everything was at its worst in March 2011. Even if you ate a fish every week from just off Japan that had a small amount of cesium in it, this would not happen to you.]

Because it makes illogical statements. “Since the accident at Fukushima on March 11, 2011, three reactor cores have gone missing.” Really? Gone missing? Does that make sense? Well then, why is so much radioactive cooling water — which is mentioned later in the article — being stored up at the Fukushima site? Isn’t that water being used to keep those cores cool? And how could that happen if the cores were missing? [The cores melted; it's not known precisely what shape they are in or precisely how much of each is inside or outside the original containment vessel, but they're being successfully cooled by water, so it's clear roughly where they are. They're not "missing"; that's a wild over-statement.]

Because the authors quote people without being careful to explain clearly who they are. “Harvey Wasserman, who has been working on nuclear energy issues for over 40 years,…” Is Harvey Wasserman a scientist or engineer? No.  But he gets lots of press in this article (and elsewhere.) [Wikipedia says: "Harvey Franklin Wasserman (born December 31, 1945) is an American journalist, author, democracy activist, and advocate for renewable energy. He has been a strategist and organizer in the anti-nuclear movement in the United States for over 30 years." I have nothing against Mr. Wasserman and I personally support both renewable energy and the elimination of nuclear power. But as far as I know, Wasserman has no scientific training, and is not an expert on cleaning up a nuclear plant and the risks thereof... and he's an anti-nuclear activist, so you do have to worry he's going to make thing sound worse than they are. Always look up the people being quoted!]

Because the article never once provides balance or nuance: absolutely everything is awful, awful, awful. I’m sorry, but things are never that black and white, or rather, black and black. There are shades of gray in the real world, and it’s important to tease them out a little bit. There are eventualities that would be really terrible, others that would be unfortunate, still others that would merely be a little disruptive in the local area — and they’re not equally bad, nor are they equally likely. [I don't get any sense that the authors are trying to help their readers understand; they're just bashing the reader over the head with one terrifying-sounding thing after another. This kind of article just isn't credible.]

The lesson: one has to be a critical, careful reader, and read between the lines! In contrast to Sample’s article in the Guardian, the document by Zeese and Flowers is not intended to inform; it is intended to frighten, period. I urge you to avoid getting your information from sources like that one. Find reliable, sensible people — Ian Sample is in that category, I think — and stick with them. And I would ignore anything Zeese and Flowers have to say in the future; people who’d write an article like theirs have no credibility.

Peter Coles - In the Dark

Quantum Technology – a Sussex Strength

Amid all the doom and gloom in the Chancellor’s Autumn Statement delivered last week there’s a ray of sunshine for research in Physics in the form of an injection of around £270 million in Quantum Technology. According to the Financial Times,

The money will support a national network of five research centres, covering quantum computing, secure communications, sensors, measurement and simulation.

Details of the scheme are yet to be released, but it seems the network will consist of “regional centres” although how evenly it will be spread across the regions remains to be seen. How many will be in the Midlands, for example?

We’re very happy here with this announcement here in the School of Physics & Astronomy at the University of Sussex as we have a well-established and expanding major research activity in Quantum Technology and an MSc Course called Frontiers of Quantum Technology. Moreover, as members of the South East Physics Network (SEPNet) we seem to be in a good position to be for funds as a truly regional centre. Assuming, that is, that the scheme hasn’t already been divvied up behind closed doors before it was even announced!

The investment announced by the government mirrors a growing realization of the potential for economic exploitation of, e.g., quantum computing which is bound to lead to a new range of career opportunities for budding physics graduates.

I’d welcome any comments from people who know any more information about the details of the new investment, as I’m too lazy to search for it myself…

Jaques Distler - Musings

G2 and Spin(8) Triality

Oscar Chacaltana, Yuji Tachikawa and I are deep in the weeds of nilpotent orbits. One of the things we had to study were the nilpotent orbits of ${𝔤}_{2}\mathfrak\left\{g\right\}_2$, and how they sit in $\mathrm{𝔰𝔬}\left(8\right)\mathfrak\left\{so\right\}\left(8\right)$. Understanding the answer involves an explicit description of $\mathrm{Spin}\left(8\right)Spin\left(8\right)$ triality, which I thought was kinda cute. Few people will care about the nilpotent orbits, but the bit about triality and ${G}_{2}G_2$ might be of some independent interest. So here it is.

$\mathrm{Spin}\left(8\right)Spin\left(8\right)$ has a triality symmetry (an outer autmomorphism of the Lie algebra), which permutes the three 8-dimensional irreducible representations: ${8}_{v}8_v$, ${8}_{s}8_s$, and ${8}_{c}8_c$. ${𝔤}_{2}\subset \mathrm{𝔰𝔬}\left(8\right)\mathfrak\left\{g\right\}_2\subset \mathfrak\left\{so\right\}\left(8\right)$ is the invariant subalgebra. (I’ll conveniently pass back and forth between the complex form of the Lie algebra and the compact real form of the group, as both are of interest to us.) What I want to do is describe that triality symmetry very explicitly and, thereby, the realization of ${𝔤}_{2}\mathfrak\left\{g\right\}_2$. Note that $\mathrm{Spin}\left(8\right)Spin\left(8\right)$ contains an $\left({\mathrm{SU}\left(2\right)}^{4}\right)/{ℤ}_{2}\left(\left\{SU\left(2\right)\right\}^4\right)/\mathbb\left\{Z\right\}_2$ subgroup, under which the adjoint decomposes as $28=\left(3,1,1,1\right)+\left(1,3,1,1\right)+\left(1,1,3,1\right)+\left(1,1,1,3\right)+\left(2,2,2,2\right) 28 = \left(3,1,1,1\right)+\left(1,3,1,1\right)+\left(1,1,3,1\right)+\left(1,1,1,3\right)+\left(2,2,2,2\right) $

Under this decomposition, the action of triality is easy to describe: pick one of the $\mathrm{𝔰𝔩}\left(2\right)\mathfrak\left\{sl\right\}\left(2\right)$ subalgebras to hold fixed, and consider all permutations of the other three (supplemented by the obvious action on the $\left(2,2,2,2\right)\left(2,2,2,2\right)$).

That’s triality. Looked at this way, it seems absurdly simple. The above description gives a perfectly concrete action of triality, as permutations of the generators. And we can push a little harder, and really understand ${𝔤}_{2}\mathfrak\left\{g\right\}_2$, this way.

The subalgebra, invariant under the ${S}_{3}S_3$ permutations, is ${𝔤}_{2}\subset \mathrm{𝔰𝔬}\left(8\right)\mathfrak\left\{g\right\}_2\subset \mathfrak\left\{so\right\}\left(8\right)$, under which

$28=14+7\otimes V 28 = 14 + 7 \otimes V $

where $VV$ is the 2-dimensional irreducible representation of ${S}_{3}S_3$. In terms of our previous decomposition,

${G}_{2}\supset \left(\mathrm{SU}\left(2\right)×{\mathrm{SU}\left(2\right)}_{D}\right)/{ℤ}_{2} G_2 \supset \left(SU\left(2\right)\times \left\{SU\left(2\right)\right\}_D\right)/\mathbb\left\{Z\right\}_2 $

where the first $\mathrm{SU}\left(2\right)SU\left(2\right)$ is the one you kept fixed, and ${\mathrm{SU}\left(2\right)}_{D}\left\{SU\left(2\right)\right\}_D$ is the diagonal $\mathrm{SU}\left(2\right)SU\left(2\right)$ of the three which are permuted by triality. Under this embedding,

$\begin{array}{rl}14& =\left(3,1\right)+\left(1,3\right)+\left(2,4\right)\\ 7& =\left(1,3\right)+\left(2,2\right)\end{array} \begin\left\{split\right\} 14 &= \left(3,1\right)+\left(1,3\right)+\left(2,4\right)\\ 7 &= \left(1,3\right) + \left(2,2\right) \end\left\{split\right\} $

An explicit basis of antisymmetric $8×88\times 8$ matrices which give this ${𝔤}_{2}\mathfrak\left\{g\right\}_2$ subalgebra is as follows. First, we embed ${\mathrm{𝔰𝔩}\left(2\right)}^{4}\left\{\mathfrak\left\{sl\right\}\left(2\right)\right\}^4$, by taking the $8×88\times 8$ matrix to be block-diagonal, with $4×44\times 4$ blocks containing ${\mathrm{𝔰𝔩}\left(2\right)}^{2}\left\{\mathfrak\left\{sl\right\}\left(2\right)\right\}^2$, as

$\begin{array}{cc}\begin{array}{rl}{H}_{L}& ={\sigma }_{2}\otimes 𝟙\\ {X}_{L}& =\frac{1}{2}\left({\sigma }_{3}+i{\sigma }_{1}\right)\otimes {\sigma }_{2}\\ {Y}_{L}& =\frac{1}{2}\left({\sigma }_{3}-i{\sigma }_{1}\right)\otimes {\sigma }_{2}={X}_{L}^{†}\end{array}& ,\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\begin{array}{rl}{H}_{R}& =𝟙\otimes {\sigma }_{2}\\ {X}_{R}& =\frac{1}{2}{\sigma }_{2}\otimes \left({\sigma }_{3}+i{\sigma }_{1}\right)\\ {Y}_{R}& =\frac{1}{2}{\sigma }_{2}\otimes \left({\sigma }_{3}-i{\sigma }_{1}\right)={X}_{R}^{†}\end{array}\end{array} \begin\left\{matrix\right\} \begin\left\{split\right\} H_L &= \sigma_2 \otimes \mathbb\left\{1\right\}\\ X_L &= \tfrac\left\{1\right\}\left\{2\right\} \left(\sigma_3+i\sigma_1\right) \otimes \sigma_2\\ Y_L &= \tfrac\left\{1\right\}\left\{2\right\} \left(\sigma_3-i\sigma_1\right) \otimes \sigma_2 = X_L^\dagger \end\left\{split\right\}&,\qquad\qquad \begin\left\{split\right\} H_R &= \mathbb\left\{1\right\} \otimes \sigma_2\\ X_R &= \tfrac\left\{1\right\}\left\{2\right\} \sigma_2 \otimes \left(\sigma_3+i\sigma_1\right)\\ Y_R &= \tfrac\left\{1\right\}\left\{2\right\} \sigma_2 \otimes \left(\sigma_3-i\sigma_1\right) = X_R^\dagger \end\left\{split\right\} \end\left\{matrix\right\} $

where we’ve chosen the normalization conventions

$\begin{array}{rl}\left[X,Y\right]& =H\\ \left[H,X\right]& =2X\\ \left[H,Y\right]& =-2Y\end{array} \begin\left\{split\right\} \left[X,Y\right]&=H\\ \left[H,X\right]&=2X\\ \left[H,Y\right]&=-2Y \end\left\{split\right\} $

We pick one of these (the ${\mathrm{𝔰𝔩}\left(2\right)}_{L}\left\{\mathfrak\left\{sl\right\}\left(2\right)\right\}_L$ in the upper left-hand block) to hold fixed, and embed our second $\mathrm{𝔰𝔩}\left(2\right)\mathfrak\left\{sl\right\}\left(2\right)$ diagonally in the other three:

$\begin{array}{rl}{H}_{1}& =\frac{1}{2}\left(𝟙+{\sigma }_{3}\right)\otimes {\sigma }_{2}\otimes 1\\ {X}_{1}& =\frac{1}{4}\left(𝟙+{\sigma }_{3}\right)\otimes \left({\sigma }_{3}+i{\sigma }_{1}\right)\otimes {\sigma }_{2}\\ {Y}_{1}& ={X}_{1}^{†}\\ {H}_{2}& =\frac{1}{2}\left(𝟙-{\sigma }_{3}\right)\otimes {\sigma }_{2}\otimes 𝟙+𝟙\otimes 𝟙\otimes {\sigma }_{2}\\ {X}_{2}& =\frac{1}{4}\left(𝟙-{\sigma }_{3}\right)\otimes \left({\sigma }_{3}+i{\sigma }_{1}\right)\otimes {\sigma }_{2}+\frac{1}{2}1\otimes {\sigma }_{2}\otimes \left({\sigma }_{3}+i{\sigma }_{1}\right)\\ {Y}_{2}& ={X}_{2}^{†}\end{array} \begin\left\{split\right\} H_1 &= \tfrac\left\{1\right\}\left\{2\right\} \left(\mathbb\left\{1\right\}+\sigma_3\right)\otimes \sigma_2 \otimes 1\\ X_1 &= \tfrac\left\{1\right\}\left\{4\right\} \left(\mathbb\left\{1\right\}+\sigma_3\right)\otimes \left(\sigma_3+i\sigma_1\right)\otimes\sigma_2\\ Y_1 &= X_1^\dagger\\ H_2 &= \tfrac\left\{1\right\}\left\{2\right\} \left(\mathbb\left\{1\right\}-\sigma_3\right)\otimes \sigma_2 \otimes \mathbb\left\{1\right\} + \mathbb\left\{1\right\}\otimes \mathbb\left\{1\right\} \otimes \sigma_2\\ X_2 &= \tfrac\left\{1\right\}\left\{4\right\} \left(\mathbb\left\{1\right\}-\sigma_3\right)\otimes \left(\sigma_3+i\sigma_1\right)\otimes \sigma_2 + \tfrac\left\{1\right\}\left\{2\right\} 1\otimes \sigma_2\otimes \left(\sigma_3+i\sigma_1\right)\\ Y_2 &= X_2^\dagger \end\left\{split\right\} $

The highest weight of the $\left(2,4\right)\left(2,4\right)$ is

${S}_{1,3}=\frac{1}{4}{\sigma }_{2}\otimes \left({\sigma }_{3}+i{\sigma }_{1}\right)\otimes \left({\sigma }_{3}+i{\sigma }_{1}\right) S_\left\{1,3\right\} = \tfrac\left\{1\right\}\left\{4\right\} \sigma_2\otimes \left(\sigma_3+i\sigma_1\right)\otimes \left(\sigma_3+i\sigma_1\right) $

The remaining ones, e.g., ${S}_{-1,3}=\left[{Y}_{1},{S}_{1,3}\right]S_\left\{-1,3\right\} = \left[Y_1, S_\left\{1,3\right\}\right]$, are obtained by acting with the lowering operators, ${Y}_{1,2}Y_\left\{1,2\right\}$. With this choice of Cartan, the simple roots of ${𝔤}_{2}\mathfrak\left\{g\right\}_2$ correspond to ${X}_{2}X_2$ (short root) and ${S}_{1,-3}S_\left\{1,-3\right\}$ (long root).

$ Layer 1 {X}_{2}X_2 {S}_{1,-3}S_\left\{1,-3\right\} \begin\left\{svg\right\}\end\left\{svg\right\}$

This 8-dimensional representation of ${G}_{2}G_2$, as it’s reducible, is not the most convenient one for studying the representation theory of ${G}_{2}G_2$. But it’s tailor-made for our purpose, which is understanding the embedding in $\mathrm{Spin}\left(8\right)Spin\left(8\right)$. With an explicit embedding in hand, we can manufacture a distinguished triple, $\left(H,X,Y\right)\left(H,X,Y\right)$ for each nilpotent orbit of ${𝔤}_{2}\mathfrak\left\{g\right\}_2$, and see how it sits in $\mathrm{𝔰𝔬}\left(8\right)\mathfrak\left\{so\right\}\left(8\right)$. But that, probably, holds little interest for the general reader, so I’ll end here.

Jaques Distler - Musings

Normal Coordinate Expansion

I’ve been spending several weeks at the Simons Center for Geometry and Physics. Towards the end of my stay, I got into a discussion with Tim Nguyen, about Ricci flow and nonlinear $\sigma \sigma$-models. He’d been reading Friedan’s PhD thesis, alongside [Kevin Costello’s book](http://www.ams.org/bookstore-getitem/item=SURV-170). So I pointed him to some old notes of mine on the normal coordinate expansion, a key ingredient in renormalizing nonlinear $\sigma \sigma$-models, using the background-field method.

Then it occurred to me that the internet would be a much more useful place for those notes. So, since I have some time to kill, in JFK, here they are.

Let $\varphi :\Sigma \to M\phi:\Sigma\to M$ be a map from the worldsheet, $\Sigma \Sigma$ into a Riemannian Manifold, $\left(M,g\right)\left(M,g\right)$. The NL$\sigma \sigma$M action is
(1)$S={\int }_{\Sigma }\frac{1}{2}\left({\varphi }^{*}g\right)\left({\partial }_{\mu },{\partial }_{\mu }\right){d}^{2}xS= \int_\Sigma \tfrac\left\{1\right\}\left\{2\right\}\left(\phi^*g\right)\left(\partial_\mu,\partial_\mu\right) d^2x $
The Normal Coordinate Expansion is a particularly nice parametrization of fluctuations about the classical $\sigma \sigma$-model field $\varphi \phi$. I’ll use index-free notation, wherever possible. The Levi-Cevita connection, $\nabla \nabla$, on $MM$ is torsion-free and metric-compatible,
(2)$\begin{array}{rl}{\nabla }_{X}Y-{\nabla }_{Y}X-\left[X,Y\right]& \equiv T\left(X,Y\right)=0\\ X\left(g\left(Y,Z\right)\right)& =g\left({\nabla }_{X}Y,Z\right)+g\left(Y,{\nabla }_{X}Z\right)\end{array} \begin\left\{split\right\} \nabla_X Y-\nabla_Y X-\left[X,Y\right]&\equiv T\left(X,Y\right)=0\\ X\left(g\left(Y,Z\right)\right)&=g\left(\nabla_X Y,Z\right)+g\left(Y,\nabla_X Z\right) \end\left\{split\right\} $
The Riemann curvature tensor is
(3)$R\left(X,Y\right)={\nabla }_{X}{\nabla }_{Y}-{\nabla }_{Y}{\nabla }_{X}-{\nabla }_{\left[X,Y\right]} R\left(X,Y\right)=\nabla_X\nabla_Y-\nabla_Y \nabla_X-\nabla_\left\{\left[X,Y\right]\right\} $
Consider a 1-parameter family of such $\sigma \sigma$-model maps, ${\varphi }_{t}:\Sigma \to M\phi_t:\Sigma\to M$, $t\in \left[0,1\right]t\in\left[0,1\right]$ with ${\varphi }_{0}=\varphi \phi_0=\phi$, our original $\sigma \sigma$-model map. We can equally-well think about this family as a map $\stackrel{^}{\varphi }:\Sigma ×\left[0,1\right]\to M\hat\phi:\Sigma\times\left[0,1\right]\to M$, with
(4)$\stackrel{^}{\varphi }\left(x,t\right)={\varphi }_{t}\left(x\right) \hat\phi\left(x,t\right)=\phi_t\left(x\right) $
Given $\stackrel{^}{\varphi }\hat\phi$, we can pull back the connection, $\nabla \nabla$, on $MM$ to a connection $\stackrel{^}{\nabla }\hat \nabla$ on $\Sigma ×\left[0,1\right]\Sigma\times \left[0,1\right]$. We don’t want to choose any old 1-parameter family, though. Let $\xi \left(x\right)\xi\left(x\right)$ be a section of the pullback tangent bundle, ${\varphi }^{*}\mathrm{TM}\phi^*TM$. We wish to choose $\stackrel{^}{\varphi }\hat\phi$ so that we can extend $\xi \left(x\right)\xi\left(x\right)$ to $\xi \left(x,t\right)\xi\left(x,t\right)$ such that
(5)$\begin{array}{rl}\xi \left(x,t\right)& ={\stackrel{^}{\varphi }}_{*}\frac{\partial }{\partial t}\\ {\stackrel{^}{\nabla }}_{{\partial }_{t}}\xi & =0\end{array} \begin\left\{split\right\} \xi\left(x,t\right)&=\hat\phi_*\tfrac\left\{\partial\right\}\left\{\partial t\right\}\\ \hat\nabla_\left\{\partial_t\right\}\xi&=0 \end\left\{split\right\} $
How do we achieve this? The idea is that, given a point $x\in \Sigma x\in\Sigma$, $\xi \left(x\right)\xi\left(x\right)$ gives a tangent vector to $MM$ at the point $\varphi \left(x\right)\phi\left(x\right)$. For this “initial condition”, we solve the geodesic equation
(6)$\begin{array}{c}\gamma :\phantom{\rule{thinmathspace}{0ex}}\left[0,1\right]\to M\\ {\stackrel{¨}{\gamma }}^{k}+{\Gamma }_{\mathrm{ij}}^{k}{\stackrel{˙}{\gamma }}^{i}{\stackrel{˙}{\gamma }}^{k}=0\\ \gamma \left(0\right)=\varphi \left(x\right),\phantom{\rule{2em}{0ex}}\stackrel{˙}{\gamma }\left(0\right)=\xi \left(x\right)\end{array} \begin\left\{gathered\right\} \gamma:\, \left[0,1\right]\to M\\ \ddot\left\{\gamma\right\}^k + \Gamma_\left\{ij\right\}^k \dot\left\{\gamma\right\}^i\dot\left\{\gamma\right\}^k=0\\ \gamma\left(0\right)=\phi\left(x\right),\qquad \dot\left\{\gamma\right\}\left(0\right)=\xi\left(x\right) \end\left\{gathered\right\} $
and define the point ${\varphi }_{t}\left(x\right)\in M\phi_t\left(x\right)\in M$ to be $\gamma \left(t\right)\gamma\left(t\right)$. This is guaranteed to be well-defined for small enough $tt$. We can extend it to $t=1t=1$ by considering $\xi \xi$ to be sufficiently “small”. The extension of $\xi \left(x\right)\xi\left(x\right)$ to $\xi \left(x,t\right)\xi\left(x,t\right)$ is simply the one given by parallel-transporting $\xi \xi$ along the curve $\gamma \gamma$, ${\stackrel{^}{\nabla }}_{{\partial }_{t}}\xi =0 \hat\nabla_\left\{\partial_t\right\}\xi=0 $ or, with slight abuse of notation,
(7)${\nabla }_{\xi }\xi =0\nabla_\xi\xi=0 $
(This is an abuse of notation because $\xi \xi$ is not really a tensor on $MM$. We typically have points $x,x\prime \in \Sigma x,x&apos\in\Sigma$ with $\varphi \left(x\right)=\varphi \left(x\prime \right)\phi\left(x\right)=\phi\left(x&apos\right)$ but $\xi \left(x\right)\ne \xi \left(x\prime \right)\xi\left(x\right)\neq\xi\left(x&apos\right)$. This “mistake” will correct itself when we pull back to $\Sigma \Sigma$.) Since $\left[\frac{\partial }{\partial t},{\partial }_{\mu }\right]=0\Bigl\left[\tfrac\left\{\partial\right\}\left\{\partial t\right\},\partial_\mu\bigr\right]=0$ and the connection $\nabla \nabla$ is torsion-free, we can always exchange
(8)${\stackrel{^}{\nabla }}_{{\partial }_{t}}v={\stackrel{^}{\nabla }}_{{\partial }_{\mu }}\xi \hat\nabla_\left\{\partial_t\right\}v =\hat\nabla_\left\{\partial_\mu\right\}\xi $
where
(9)$v={\stackrel{^}{\varphi }}_{*}{\partial }_{\mu } v=\hat\phi_* \partial_\mu $
or, with the same abuse of notation,
(10)${\nabla }_{\xi }v={\nabla }_{v}\xi \nabla_\xi v =\nabla_v \xi $
Taking another covariant derivative with respect to $tt$, we get the equation of geodesic deviation [1] ${\stackrel{^}{\nabla }}_{{\partial }_{t}}^{2}v={\stackrel{^}{\nabla }}_{{\partial }_{t}}{\stackrel{^}{\nabla }}_{{\partial }_{\mu }}\xi =\stackrel{^}{R}\left({\partial }_{t},{\partial }_{\mu }\right)\xi \hat\nabla_\left\{\partial_t\right\}^2v =\hat\nabla_\left\{\partial_t\right\}\hat\nabla_\left\{\partial_\mu\right\}\xi =\hat R\left(\partial_t,\partial_\mu\right)\xi $ or, in our abusive notation,
(11)${\nabla }_{\xi }{\nabla }_{v}\xi =R\left(\xi ,v\right)\xi \nabla_\xi\nabla_v \xi= R\left(\xi,v\right)\xi $
Now say we wish to evaluate the $tt$-dependence of the pull-back of some tensor $TT$ on $MM$, ${\varphi }_{t\prime }^{*}T={e}^{t\prime {\stackrel{^}{\nabla }}_{{\partial }_{t}}}\left({\stackrel{^}{\varphi }}^{*}T\right){\mid }_{t=0} \phi_\left\{t&apos\right\}^* T = e^\left\{t&apos\hat\nabla_\left\{\partial_t\right\}\right\}\left(\hat\phi^*T\right)\vert_\left\{t=0\right\} $ which we can, again, write as
(12)${\varphi }^{*}\left({e}^{t\prime {\nabla }_{\xi }}T\right)={\varphi }^{*}\left(T+t\prime {\nabla }_{\xi }T+\frac{t{\prime }^{2}}{2}{\nabla }_{\xi }^{2}T+\dots \right)\phi^*\left(e^\left\{t&apos\nabla_\xi\right\} T\right) = \phi^*\left(T+t&apos\nabla_\xi T +\tfrac\left\{t&apos^2\right\}\left\{2\right\}\nabla^2_\xi T +\dots\right) $
We are all set to apply this to the $\sigma \sigma$-model Lagrangian,
(13)$\frac{1}{2}\left({\varphi }_{t}^{*}g\right)\left({\partial }_{\mu },{\partial }_{\mu }\right)={\varphi }_{t}^{*}\left(\frac{1}{2}g\left(v,v\right)\right) \tfrac\left\{1\right\}\left\{2\right\}\left(\phi_t^*g\right)\left(\partial_\mu,\partial_\mu\right) = \phi_t^*\left(\tfrac\left\{1\right\}\left\{2\right\} g\left(v,v\right)\right) $
We expand this using (12) and use (7),(10) and (11) to simplify the terms that result. Note that the ${n}^{\mathrm{th}}n^\left\{th\right\}$ order term in (12) is given by $\frac{1}{n}{\nabla }_{\xi }\tfrac\left\{ 1\right\}\left\{n\right\}\nabla_\xi$ of the $\left(n-1{\right)}^{\mathrm{st}}\left(n-1\right)^\left\{st\right\}$ order term. The ${0}^{\mathrm{th}}0^\left\{th\right\}$ order term is $\frac{1}{2}g\left(v,v\right) \tfrac\left\{1\right\}\left\{2\right\}g\left(v,v\right) $ At first order, we get $g\left(v,{\nabla }_{\xi }v\right)=g\left(v,{\nabla }_{v}\xi \right) g\left(v,\nabla_\xi v\right)= g\left(v,\nabla_v \xi\right) $ Next comes $\frac{1}{2}g\left({\nabla }_{\xi }v,{\nabla }_{v}\xi \right)+\frac{1}{2}g\left(v,{\nabla }_{\xi }{\nabla }_{v}\xi \right)=\frac{1}{2}g\left({\nabla }_{v}\xi ,{\nabla }_{v}\xi \right)+\frac{1}{2}g\left(v,R\left(\xi ,v\right)\xi \right) \tfrac\left\{1\right\}\left\{2\right\}g\left(\nabla_\xi v,\nabla_v \xi\right)+\tfrac\left\{1\right\}\left\{2\right\}g\left(v,\nabla_\xi \nabla_v \xi\right)= \tfrac\left\{1\right\}\left\{2\right\}g\left(\nabla_v \xi,\nabla_v \xi\right) + \tfrac\left\{1\right\}\left\{2\right\}g\left(v,R\left(\xi,v\right) \xi\right) $ At ${3}^{\mathrm{rd}}3^\left\{rd\right\}$ order, we get $\begin{array}{rl}\frac{1}{6}\left[2g\left({\nabla }_{v}\xi ,R\left(\xi ,v\right)\xi \right)+g\left({\nabla }_{v}\xi ,R\left(\xi ,v\right)\xi \right)& +g\left(v,\left({\nabla }_{\xi }R\right)\left(\xi ,v\right)\xi \right)+g\left(v,R\left(\xi ,{\nabla }_{v}\xi \right)\xi \right)\right]\\ & =\frac{1}{6}g\left(v,\left(\nabla R\right)\left(\xi ,\xi ,v\right)\xi \right)+\frac{2}{3}g\left({\nabla }_{v}\xi ,R\left(\xi ,v\right)\xi \right)\end{array} \begin\left\{split\right\} \tfrac\left\{1\right\}\left\{6\right\}\left[2g\left(\nabla_v \xi,R\left(\xi,v \right)\xi\right)+g\left(\nabla_v \xi,R\left(\xi,v\right)\xi\right)& +g\left(v,\left(\nabla_\xi R\right)\left(\xi,v\right)\xi\right) +g\left(v,R\left(\xi,\nabla_v\xi\right)\xi\right)\right]\\ &=\tfrac\left\{1\right\}\left\{6\right\} g\left(v,\left(\nabla R\right)\left(\xi,\xi,v\right)\xi\right) +\tfrac\left\{2\right\}\left\{3\right\}g\left(\nabla_v\xi,R\left(\xi,v\right)\xi\right) \end\left\{split\right\} $ where we used the symmetry of the Riemann tensor
(14)$g\left(U,R\left(V,W\right)Z\right)=g\left(W,R\left(Z,U\right)V\right) g\left(U,R\left(V,W\right)Z\right)=g\left(W,R\left(Z,U\right)V\right) $
Finally, at ${4}^{\mathrm{th}}4^\left\{th\right\}$ order, we get $\begin{array}{rl}\frac{1}{24}\left[g\left({\nabla }_{v}\xi ,\left(\nabla R\right)\left(\xi ,\xi ,v\right)\xi \right)& +g\left(v,\left(\nabla \nabla R\right)\left(\xi ,\xi ,\xi ,v\right)\xi \right)+g\left(v,\left(\nabla R\right)\left(\xi ,\xi ,{\nabla }_{v}\xi \right)\xi \right)\right]\\ & +\frac{1}{6}\left[g\left(R\left(\xi ,v\right)\xi ,R\left(\xi ,v\right)\xi \right)+g\left({\nabla }_{v}\xi ,\left(\nabla R\right)\left(\xi ,\xi ,v\right)\xi \right)+g\left({\nabla }_{v}\xi ,R\left(\xi ,{\nabla }_{v}\xi \right)\xi \right)\right]\\ =& \frac{1}{4}g\left({\nabla }_{v}\xi ,\left(\nabla R\right)\left(\xi ,\xi ,v\right)\xi \right)+\frac{1}{6}g\left({\nabla }_{v}\xi ,R\left(\xi ,{\nabla }_{v}\xi \right)\xi \right)\\ & +\frac{1}{6}g\left(R\left(\xi ,v\right)\xi ,R\left(\xi ,v\right)\xi \right)+\frac{1}{24}g\left(v,\left(\nabla \nabla R\right)\left(\xi ,\xi ,\xi ,v\right)\xi \right)\end{array} \begin\left\{split\right\} \tfrac\left\{1\right\}\left\{24\right\}\left[g\left(\nabla_v \xi,\left(\nabla R\right)\left(\xi,\xi,v \right)\xi\right) &+g\left(v,\left(\nabla \nabla R\right)\left(\xi,\xi,\xi,v \right)\xi\right) +g\left(v,\left(\nabla R\right)\left(\xi,\xi,\nabla_v\xi\right)\xi\right) \right]\\ &+\tfrac\left\{1\right\}\left\{6\right\}\left[g\left(R\left(\xi,v\right)\xi,R\left(\xi,v\right)\xi\right) +g\left(\nabla_v \xi,\left(\nabla R\right)\left(\xi,\xi,v\right)\xi\right) +g\left(\nabla_v \xi,R\left(\xi,\nabla_v \xi\right)\xi\right)\right]\\ =&\tfrac\left\{1\right\}\left\{4\right\} g\left(\nabla_v \xi,\left(\nabla R\right)\left(\xi,\xi,v\right)\xi\right) +\tfrac\left\{1\right\}\left\{6\right\}g\left(\nabla_v \xi,R\left(\xi,\nabla_v \xi\right)\xi\right)\\ &+\tfrac\left\{1\right\}\left\{6\right\}g\left(R\left(\xi,v\right)\xi,R\left(\xi,v\right)\xi\right) +\tfrac\left\{1\right\}\left\{24\right\}g\left(v,\left(\nabla \nabla R\right)\left(\xi,\xi,\xi,v \right)\xi\right) \end\left\{split\right\} $ and so forth. Assembling all of these, we obtain
(15)$\begin{array}{rl}{\varphi }_{t}^{*}\frac{1}{2}g\left(v,v\right)={\varphi }^{*}\left[\frac{1}{2}g\left(v,v\right)& +g\left(v,{\nabla }_{v}\xi \right)+\frac{1}{2}g\left({\nabla }_{v}\xi ,{\nabla }_{v}\xi \right)+\frac{1}{2}g\left(v,R\left(\xi ,v\right)\xi \right)\\ & +\frac{1}{6}g\left(v,\left(\nabla R\right)\left(\xi ,\xi ,v\right)\xi \right)+\frac{2}{3}g\left({\nabla }_{v}\xi ,R\left(\xi ,v\right)\xi \right)\\ & +\frac{1}{4}g\left({\nabla }_{v}\xi ,\left(\nabla R\right)\left(\xi ,\xi ,v\right)\xi \right)+\frac{1}{6}g\left({\nabla }_{v}\xi ,R\left(\xi ,{\nabla }_{v}\xi \right)\xi \right)\\ & +\frac{1}{6}g\left(R\left(\xi ,v\right)\xi ,R\left(\xi ,v\right)\xi \right)+\frac{1}{24}g\left(v,\left(\nabla \nabla R\right)\left(\xi ,\xi ,\xi ,v\right)\xi \right)\right]\end{array} \begin\left\{split\right\} \phi_t^*\tfrac\left\{1\right\}\left\{2\right\}g\left(v,v\right)=\phi^*\left[\tfrac\left\{1\right\}\left\{2\right\}g\left(v,v\right)& +g\left(v,\nabla_v \xi\right) +\tfrac\left\{1\right\}\left\{2\right\}g\left(\nabla_v \xi,\nabla_v \xi\right) + \tfrac\left\{1\right\}\left\{2\right\}g\left(v,R\left(\xi,v\right) \xi\right)\\ &+\tfrac\left\{1\right\}\left\{6\right\} g\left(v,\left(\nabla R\right)\left(\xi,\xi,v\right)\xi\right) +\tfrac\left\{2\right\}\left\{3\right\}g\left(\nabla_v\xi,R\left(\xi,v\right)\xi\right)\\ &+\tfrac\left\{1\right\}\left\{4\right\} g\left(\nabla_v \xi,\left(\nabla R\right)\left(\xi,\xi,v\right)\xi\right) +\tfrac\left\{1\right\}\left\{6\right\}g\left(\nabla_v \xi,R\left(\xi,\nabla_v \xi\right)\xi\right)\\ & +\tfrac\left\{1\right\}\left\{6\right\}g\left(R\left(\xi,v\right)\xi,R\left(\xi,v\right)\xi\right) +\tfrac\left\{1\right\}\left\{24\right\}g\left(v,\left(\nabla \nabla R\right)\left(\xi,\xi,\xi,v \right)\xi\right)\right] \end\left\{split\right\} $
Pulling back to $\Sigma \Sigma$, we obtain the desired expansion of the $\sigma \sigma$-model lagrangian. In conventional notation, replace ${v}^{i}={\partial }_{\mu }{\varphi }^{i}v^i=\partial_\mu\phi^i$, $\left({\nabla }_{v}\xi {\right)}^{i}={D}_{\mu }{\xi }^{i}\left(\nabla_v\xi\right)^i=D_\mu \xi^i$ and $g\left(U,R\left(W,Z\right)V\right)={R}_{\mathrm{ijkl}}{U}^{i}{V}^{j}{W}^{k}{Z}^{l}g\left(U,R\left(W,Z\right)V\right)=R_\left\{ijkl\right\}U^i V^j W^k Z^l$ to obtain the expressions found in Friedan [2] or Freedman et al [3].

Exercise 1: Compute the next term in the expansion, $\begin{array}{r}\frac{1}{12}g\left({\nabla }_{v}\xi ,\left(\nabla R\right)\left(\xi ,\xi ,{\nabla }_{v}\xi \right)\xi \right)+\frac{2}{15}g\left(R\left(\xi ,v\right)\xi ,R\left(\xi ,{\nabla }_{v}\xi \right)\xi \right)+\frac{1}{15}g\left({\nabla }_{v}\xi ,\left(\nabla \nabla R\right)\left(\xi ,\xi ,\xi ,v\right)\xi \right)\\ +\frac{7}{60}g\left(R\left(\xi ,v\right)\xi ,\left(\nabla R\right)\left(\xi ,\xi ,v\right)\xi \right)+\frac{1}{120}g\left(v,\left(\nabla \nabla \nabla R\right)\left(\xi ,\xi ,\xi ,\xi ,v\right)\xi \right)\end{array} \begin\left\{split\right\} \tfrac\left\{1\right\}\left\{12\right\}g\left(\nabla_v\xi,\left(\nabla R\right)\left(\xi,\xi,\nabla_v\xi\right)\xi\right) +\tfrac\left\{2\right\}\left\{15\right\}g\left(R\left(\xi,v\right)\xi,R\left(\xi,\nabla_v\xi\right)\xi\right) +\tfrac\left\{1\right\}\left\{15\right\}g\left(\nabla_v\xi,\left(\nabla\nabla R\right)\left(\xi,\xi,\xi,v\right)\xi\right)\\ +\tfrac\left\{7\right\}\left\{60\right\}g\left(R\left(\xi,v\right)\xi,\left(\nabla R\right)\left(\xi,\xi,v\right)\xi\right) +\tfrac\left\{1\right\}\left\{120\right\}g\left(v,\left(\nabla\nabla\nabla R\right)\left(\xi,\xi,\xi,\xi,v\right)\xi\right) \end\left\{split\right\} $

Exercise 2: Let $MM$ be the $nn$-sphere, ${S}^{n}S^n$, with the round metric, ${\mathrm{ds}}^{2}=\frac{4{r}^{2}\left(\mu \right){\mathrm{dx}}^{i}{\mathrm{dx}}^{i}}{\left(1+\mid x{\mid }^{2}{\right)}^{2}} ds^2= \frac\left\{4 r^2\left(\mu\right) dx^i dx^i\right\}\left\{\left(1+|\mathbf\left\{x\right\}|^2\right)^2\right\} $ Show that the solution to the one-loop $\beta \beta$-function equation is ${r}^{2}\left(\mu \right)={r}^{2}\left({\mu }_{0}\right)+\frac{n-1}{4\pi }\mathrm{log}\left(\mu /{\mu }_{0}\right) r^2\left(\mu\right) = r^2\left(\mu_0\right) + \frac\left\{n-1\right\}\left\{4\pi\right\}\log\left(\mu/\mu_0\right) $
[1] S. Weinberg, Gravitation and Cosmology, (Wiley, 1972) p. 148.
[2] D. H. Friedan, “Nonlinear Models in $2+ϵ2+\epsilon$ Dimensions,” Ann. Phys. 163 (1985) 318.
[3] L. Alvarez-Gaume, D. Z. Freedman and S. Mukhi, “The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model,” Ann. Phys. 134 (1981) 85.

Jaques Distler - Musings

Uncertainty

Update (10/18/2012) — Mea Culpa:

Sonia pointed out to me that my (mis)interpretation of Ozawa was too charitable. We ended up (largely due to Steve Weinberg’s encouragement) writing a paper. So… where does one publish simple-minded (but, apparently, hitherto unappreciated) remarks about elementary Quantum Mechanics?

Sonia was chatting with me about this PRL (arXiv version), which seems to have made a splash in the news media and in the blogosphere. She couldn’t make heads or tails of it and (as you will see), I didn’t do much better. But I thought that I would take the opportunity to lay out a few relevant remarks.

Since we’re going to be talking about the Uncertainty Principle, and measurements, it behoves us to formulate our discussion in terms of density matrices.

A quantum system is described in terms of a density matrix, $\rho \rho$, which is a self-adjoint, positive-semidefinite trace-class operator, satisfying

$\mathrm{Tr}\left(\rho \right)=1,\phantom{\rule{2em}{0ex}}\mathrm{Tr}\left({\rho }^{2}\right)\le 1 Tr\left(\rho\right)=1,\qquad Tr\left(\rho^2\right) \leq 1 $

In the Schrödinger picture (which we will use), it evolves unitarily in time

(1)$\rho \left({t}_{2}\right)=U\left({t}_{2},{t}_{1}\right)\rho \left({t}_{1}\right){U\left({t}_{2},{t}_{1}\right)}^{-1}\rho\left(t_2\right) = U\left(t_2,t_1\right) \rho\left(t_1\right) \left\{U\left(t_2,t_1\right)\right\}^\left\{-1\right\} $

except when a measurement is made.

Consider a self-adjoint operator $AA$ (an “observable”). We will assume that $AA$ has a pure point spectrum, and let ${P}_{i}^{\left(A\right)}P^\left\{\left(A\right)\right\}_i$ be the projection onto the ${i}^{\text{th}}i^\left\{\text\left\{th\right\}\right\}$ eigenspace of $AA$.

When we measure $AA$, quantum mechanics computes for us

1. A classical probability distribution for the values on the readout panel of the measuring apparatus. The moments of this probability distribution are computed by taking traces. The ${n}^{\text{th}}n^\left\{\text\left\{th\right\}\right\}$ moment is $⟨{A}^{n}⟩=\mathrm{Tr}\left({A}^{n}\rho \right) \left\{\langle A^n\rangle\right\} = Tr\left(A^n\rho\right) $ In particular, the variance is ${\left(\Delta A\right)}^{2}=\mathrm{Tr}\left({A}^{2}\rho \right)-{\left(\mathrm{Tr}\left(A\rho \right)\right)}^{2} \left\{\left(\Delta A\right)\right\}^2 = Tr\left(A^2\rho\right) - \left\{\left\left(Tr\left(A\rho\right)\right\right)\right\}^2 $
2. A change (which, under the assumptions stated, can be approximated as occurring instantaneously) in the density matrix,
(2)${\rho }_{\text{after}}\equiv \stackrel{^}{\rho }\left(\rho ,A\right)=\sum _{i}{P}_{i}^{\left(A\right)}\rho {P}_{i}^{\left(A\right)}\rho_\left\{\text\left\{after\right\}\right\}\equiv \hat\left\{\rho\right\}\left(\rho,A\right) = \sum_i P^\left\{\left(A\right)\right\}_i \rho P^\left\{\left(A\right)\right\}_i $

Thereafter, the system, described by the new density matrix, $\stackrel{^}{\rho }\hat\left\{\rho\right\}$, again evolves unitarily, according to (1).

The new density matrix, $\stackrel{^}{\rho }\hat\left\{\rho\right\}$, after the measurement1, can be completely characterized by two properties

1. All of the moments of $AA$ are the same as before $⟨{A}^{n}⟩=\mathrm{Tr}\left({A}^{n}\stackrel{^}{\rho }\left(\rho ,A\right)\right)=\mathrm{Tr}\left({A}^{n}\rho \right) \langle A^n\rangle = Tr\left(A^n \hat\left\{\rho\right\}\left(\rho,A\right)\right)= Tr\left(A^n \rho\right) $ (In particular, $\Delta A\Delta A$ is unchanged.) Moreover, for any observable, $CC$, which commutes with $AA$ ( $\left[C,A\right]=0\left[C,A\right]=0$ ), $⟨{C}^{n}⟩=\mathrm{Tr}\left({C}^{n}\stackrel{^}{\rho }\left(\rho ,A\right)\right)=\mathrm{Tr}\left({C}^{n}\rho \right) \langle C^n\rangle = Tr\left(C^n \hat\left\{\rho\right\}\left(\rho,A\right)\right)= Tr\left(C^n \rho\right) $
2. However, the measurement has destroyed all interference between the different eigenspaces of $AA$ $\mathrm{Tr}\left(\left[A,B\right]\stackrel{^}{\rho }\left(\rho ,A\right)\right)=0,\phantom{\rule{2em}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}B Tr\left(\left[A,B\right] \hat\left\{\rho\right\}\left(\rho,A\right)\right) = 0, \qquad \forall\, B $

Note that it is really important that I have assumed a pure point spectrum. If $AA$ has a continuous spectrum, then you have to deal with complications both physical and mathematical. Mathematically, you need to deal with the complications of the Spectral Theorem; physically, you have to put in finite detector resolutions, in order to make proper sense of what a “measurement” does. I’ll explain, later, how to deal with those complications

Now consider two such observables, $AA$ and $BB$. The Uncertainty Principle gives a lower bound on the product

(3)${\left(\Delta A\right)}_{\rho }{\left(\Delta B\right)}_{\rho }\ge \frac{1}{2}\mid \mathrm{Tr}\left(-i\left[A,B\right]\rho \right)\mid \left\{\left(\Delta A\right)\right\}_\rho \left\{\left(\Delta B\right)\right\}_\rho \geq \tfrac\left\{1\right\}\left\{2\right\}\left|Tr\left(-i\left[A,B\right]\rho\right)\right| $

in any state, $\rho \rho$. (Exercise: Generalize the usual proof, presented for “pure states” to the case of density matrices.)

As stated, (3) is not a statement about the uncertainties in any actual sequence of measurements. After all, once you measure $AA$, in state $\rho \rho$, the density matrix changes, according to (2), to

(4)$\stackrel{^}{\rho }\left(\rho ,A\right)=\sum _{i}{P}_{i}^{\left(A\right)}\rho {P}_{i}^{\left(A\right)}\hat\left\{\rho\right\}\left(\rho,A\right) = \sum_i P^\left\{\left(A\right)\right\}_i \rho P^\left\{\left(A\right)\right\}_i $

so a subsequent measurement of $BB$ is made in a different state from the initial one.

The obvious next thing to try is to note that, since the uncertainty of $AA$ in the state $\stackrel{^}{\rho }\left(\rho ,A\right)\hat\left\{\rho\right\}\left(\rho,A\right)$ is the *same* as in the state $\rho \rho$, and since we are measuring $BB$ in the state $\stackrel{^}{\rho }\left(\rho ,A\right)\hat\left\{\rho\right\}\left(\rho,A\right)$, we can apply the Uncertainty Relation, (3) in the state $\stackrel{^}{\rho }\hat\left\{\rho\right\}$, instead of in the state, $\rho \rho$. Unfortunately, $\mathrm{Tr}\left(\left[A,B\right]\stackrel{^}{\rho }\left(\rho ,A\right)\right)=0Tr\left(\left[A,B\right]\hat\left\{\rho\right\}\left(\rho,A\right)\right)=0$, so this leads to an uninteresting lower bound on the product the uncertainties

(5)$\left(\Delta A\right)\left(\Delta B\right)={\left(\Delta A\right)}_{\stackrel{^}{\rho }\left(\rho ,A\right)}{\left(\Delta B\right)}_{\stackrel{^}{\rho }\left(\rho ,A\right)}\ge 0\left(\Delta A\right) \left(\Delta B\right) = \left\{\left(\Delta A\right)\right\}_\left\{\hat\left\{\rho\right\}\left(\rho,A\right)\right\} \left\{\left(\Delta B\right)\right\}_\left\{\hat\left\{\rho\right\}\left(\rho,A\right)\right\} \geq 0 $

for a measurement of $AA$ immediately followed by a measurement of $BB$.

It is, apparently, possible to derive a better lower bound on the product of the uncertainties of successive measurements (which is still, of course, weaker than the “naïve” $\frac{1}{2}\mid \mathrm{Tr}\left(-i\left[A,B\right]\rho \right)\mid \tfrac\left\{1\right\}\left\{2\right\}\left|Tr\left(-i\left[A,B\right]\rho\right)\right|$, which is what you might have guessed for the lower bound, had you not thought about what (3) means). But I don’t know how to even state that result at the level of generality of the above discussion

Instead, I’d like to discuss how one treats measurements, when $AA$ doesn’t have a pure point spectrum. When it’s discussed at all, it’s treated very poorly in the textbooks.

Measuring Unbounded Operators

Let’s go straight to the worst-case, of an unbounded operator, with $\mathrm{Spec}\left(A\right)=ℝSpec\left(A\right)=\mathbb\left\{R\right\}$. Such an operator has no eigenvectors at all. What happens when we measure such an observable? Clearly, the two conditions which characterized the change in the density matrix, in the case of a pure point spectrum,

1. $\mathrm{Tr}\left({A}^{n}\stackrel{^}{\rho }\left(\rho ,A\right)\right)=\mathrm{Tr}\left({A}^{n}\rho \right)Tr\left(A^n \hat\left\{\rho\right\}\left(\rho,A\right)\right)= Tr\left(A^n \rho\right)$
2. $\mathrm{Tr}\left(\left[A,B\right]\stackrel{^}{\rho }\left(\rho ,A\right)\right)=0,\phantom{\rule{2em}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}BTr\left(\left[A,B\right] \hat\left\{\rho\right\}\left(\rho,A\right)\right) = 0, \qquad \forall\, B$

are going to have to be modified. The second condition clearly can’t hold for all choice of $BB$, in the unbounded case (think $A=xA=\mathbf\left\{x\right\}$ and $B=pB=\mathbf\left\{p\right\}$). As to the first condition, we might *hope* that the moments of the classical probability distribution for the observed measurements of $AA$ would be calculated by taking traces with the density matrix, $\stackrel{^}{\rho }\hat\left\{\rho\right\}$. But that probability distribution *depends* on the resolution of the detector, something which the density matrix, $\rho \rho$, knows nothing about.

To keep things simple, let’s specialize to $ℋ={ℒ}^{2}\left(ℝ\right)\mathcal\left\{H\right\}=\mathcal\left\{L\right\}^2\left(\mathbb\left\{R\right\}\right)$ and $A=xA=\mathbf\left\{x\right\}$. Let’s imagine a detector which can measure the particle’s position with a resolution, $\sigma \sigma$. Let’s define a projection operator

${P}_{{x}_{0}}:\phantom{\rule{1em}{0ex}}\psi \left(x\right)↦\int \frac{dy}{\sigma \sqrt{\pi }}{e}^{-\left(\left(x-{x}_{0}{\right)}^{2}+\left(y-{x}_{0}{\right)}^{2}\right)/2{\sigma }^{2}}\psi \left(y\right) P_\left\{x_0\right\}:\quad \psi\left(x\right) \mapsto \int\frac\left\{d y\right\}\left\{\sigma\sqrt\left\{\pi\right\}\right\} e^\left\{-\left\left( \left(x-x_0\right)^2+\left(y-x_0\right)^2\right\right)/2\sigma^2\right\}\psi\left(y\right) $

which reflects the notion that our detector has measured the position to be ${x}_{0}x_0$, to within an accuracy $\sigma \sigma$. Here, I’ve chosen a Gaussian; but really any acceptance function peaked at $x={x}_{0}x=x_0$, and dying away sufficiently fast away from ${x}_{0}x_0$ will do (and may more-accurately reflect the properties of your actual detector).

But I only know how to do Gaussian integrals, so this one is a convenient choice.

This is, indeed, a projection operator: ${P}_{{x}_{0}}{P}_{{x}_{0}}={P}_{{x}_{0}}P_\left\{x_0\right\}P_\left\{x_0\right\} = P_\left\{x_0\right\}$. But integrating over ${x}_{0}x_0$ doesn’t quite give the completeness relation one would want

$\int \frac{d{x}_{0}}{2\sigma \sqrt{\pi }}{P}_{{x}_{0}}\ne 𝟙 \int \frac\left\{d x_0\right\}\left\{2\sigma\sqrt\left\{\pi\right\}\right\} P_\left\{x_0\right\} \neq \mathbb\left\{1\right\} $

$\int \frac{d{x}_{0}}{2\sigma \sqrt{\pi }}{P}_{{x}_{0}}:\phantom{\rule{1em}{0ex}}\psi \left(x\right)↦\int \frac{du}{2\sigma \sqrt{\pi }}{e}^{-{u}^{2}/4{\sigma }^{2}}\psi \left(x+u\right) \int \frac\left\{d x_0\right\}\left\{2\sigma\sqrt\left\{\pi\right\}\right\} P_\left\{x_0\right\}:\quad \psi\left(x\right) \mapsto \int \frac\left\{d u\right\}\left\{2\sigma\sqrt\left\{\pi\right\}\right\} e^\left\{-u^2/4\sigma^2\right\} \psi\left(x+u\right) $

Rather than getting $\psi \left(x\right)\psi\left(x\right)$ back, we get $\psi \left(x\right)\psi\left(x\right)$ smeared against a Gaussian.

To fix this, we need to consider a more general class of projection operators (here, again, the Gaussian acceptance function proves very convenient):

(6)${P}_{{x}_{0},{k}_{0}}:\phantom{\rule{1em}{0ex}}\psi \left(x\right)↦\int \frac{dy}{\sigma \sqrt{\pi }}\mathrm{exp}\left[-\left(\left(x-{x}_{0}{\right)}^{2}+\left(y-{x}_{0}{\right)}^{2}\right)/2{\sigma }^{2}+i{k}_{0}\left(x-y\right)\right]\psi \left(y\right)P_\left\{x_0,k_0\right\}:\quad \psi\left(x\right)\mapsto \int \frac\left\{d y\right\}\left\{\sigma \sqrt\left\{\pi\right\}\right\}\exp\left\left[-\left\left(\left(x-x_0\right)^2 + \left(y-x_0\right)^2\right\right)/2\sigma^2 +i k_0\left(x-y\right)\right\right] \psi\left(y\right) $

These are still projection operators,

${P}_{{x}_{0},{k}_{0}}{P}_{{x}_{0},{k}_{0}}={P}_{{x}_{0},{k}_{0}} P_\left\{x_0,k_0\right\} P_\left\{x_0,k_0\right\} = P_\left\{x_0,k_0\right\} $

But now they obey the completeness relation

(7)$\int \frac{d{x}_{0}d{k}_{0}}{2\pi }{P}_{{x}_{0},{k}_{0}}=𝟙\int \frac\left\{d x_0 d k_0\right\}\left\{2\pi\right\} P_\left\{x_0,k_0\right\} =\mathbb\left\{1\right\} $

so we can now assert that the density matrix after measuring $x\mathbf\left\{x\right\}$ is

(8)$\stackrel{^}{\rho }=\int \frac{d{x}_{0}d{k}_{0}}{2\pi }{P}_{{x}_{0},{k}_{0}}\rho {P}_{{x}_{0},{k}_{0}}\hat\left\{\rho\right\} = \int \frac\left\{d x_0 d k_0\right\}\left\{2\pi\right\} P_\left\{x_0,k_0\right\} \rho P_\left\{x_0,k_0\right\} $

If $\rho \rho$ is represented by the integral kernel, $K\left(x,y\right)K\left(x,y\right)$:

$\rho :\phantom{\rule{1em}{0ex}}\psi \left(x\right)↦\int dy\phantom{\rule{thinmathspace}{0ex}}K\left(x,y\right)\psi \left(y\right) \rho:\quad \psi\left(x\right) \mapsto \int d y\, K\left(x,y\right) \psi\left(y\right) $

then the new density matrix, $\stackrel{^}{\rho }\hat\left\{\rho\right\}$ is represented by the integral kernel

$\stackrel{^}{K}\left(x,y\right)={e}^{-\left(x-y{\right)}^{2}/2{\sigma }^{2}}\int \frac{du}{\sigma \sqrt{2\pi }}{e}^{-{u}^{2}/2{\sigma }^{2}}K\left(x+u,y+u\right) \hat\left\{K\right\}\left(x,y\right) = e^\left\{-\left(x-y\right)^2/2\sigma^2\right\}\int \frac\left\{d u\right\}\left\{\sigma \sqrt\left\{2\pi\right\}\right\} e^\left\{-u^2/2\sigma^2\right\} K\left(x+u,y+u\right) $

Here we see clearly that it has the desired properties:

1. The off-diagonal terms are suppressed; $\stackrel{^}{K}\left(x,y\right)\to 0\hat\left\{K\right\}\left(x,y\right)\to 0$ for $\mid x-y\mid \gg \sigma |x-y|\gg \sigma$.
2. The near-diagonal terms are smeared by a Gaussian, representing the finite resolution of the detector.

Moreover, the moments of the probability distribution for the measured value of $xx$ are given by taking traces with $\stackrel{^}{\rho }\hat\left\{\rho\right\}$:

$⟨{x}^{n}⟩=\mathrm{Tr}\left({x}^{n}\stackrel{^}{\rho }\right) \langle x^n\rangle = Tr\left(\mathbf\left\{x\right\}^n \hat\left\{\rho\right\}\right) $

One easily computes

$\begin{array}{rl}\mathrm{Tr}\left(x\stackrel{^}{\rho }\right)& =\mathrm{Tr}\left(x\rho \right)\\ \mathrm{Tr}\left({x}^{2}\stackrel{^}{\rho }\right)& ={\sigma }^{2}+\mathrm{Tr}\left({x}^{2}\rho \right)\\ \end{array} \begin\left\{split\right\} Tr\left(\mathbf\left\{x\right\} \hat\left\{\rho\right\}\right)&=Tr\left(\mathbf\left\{x\right\} \rho\right)\\ Tr\left(\mathbf\left\{x\right\}^2 \hat\left\{\rho\right\}\right)&=\sigma^2+ Tr\left(\mathbf\left\{x\right\}^2 \rho\right)\\ \end\left\{split\right\} $

So the intrinsic quantum-mechanical uncertainty of the position, $xx$, in the state, $\rho \rho$, adds in quadrature with the systematic uncertainty of the measuring apparatus to produce the measured uncertainty

$\left(\Delta x{\right)}_{\text{measured}}^{2}=\left(\Delta x{\right)}_{\stackrel{^}{\rho }}^{2}=\left(\Delta x{\right)}_{\rho }^{2}+{\sigma }^{2} \left(\Delta x\right)^2_\left\{\text\left\{measured\right\}\right\} = \left(\Delta x\right)^2_\left\{\hat\left\{\rho\right\}\right\} = \left(\Delta x\right)^2_\left\{\rho\right\} + \sigma^2 $

exactly as we expect.

There’s one feature of this Gaussian measuring apparatus which is a little special. Of course, we expect that measuring $xx$ should change the distribution for values of $pp$. Here, the effect (at least on the first few moments) is quite simple

$\begin{array}{rl}\mathrm{Tr}\left(p\stackrel{^}{\rho }\right)& =\mathrm{Tr}\left(p\rho \right)\\ \mathrm{Tr}\left({p}^{2}\stackrel{^}{\rho }\right)& =\frac{1}{{\sigma }^{2}}+\mathrm{Tr}\left({p}^{2}\rho \right)\\ \end{array} \begin\left\{split\right\} Tr\left(\mathbf\left\{p\right\} \hat\left\{\rho\right\}\right)&=Tr\left(\mathbf\left\{p\right\} \rho\right)\\ Tr\left(\mathbf\left\{p\right\}^2 \hat\left\{\rho\right\}\right)&=\frac\left\{1\right\}\left\{\sigma^2\right\}+ Tr\left(\mathbf\left\{p\right\}^2 \rho\right)\\ \end\left\{split\right\} $

If we wanted to compute the effect of measuring $pp$, using a Gaussian detector with systematic uncertainty ${\sigma }_{p}=1/\sigma \sigma_p =1/\sigma$, we would use the same projectors (6) and obtain the same density matrix (8) after the measurement. This leads to very simple formulæ for the uncertainties resulting from successive measurements. Say we start with an initial state, $\rho \rho$, measure $xx$ with a Gaussian detector with systematic uncertainty ${\sigma }_{x}\sigma_x$, and then measure $pp$ with another Gaussian detector with systematic uncertainty ${\sigma }_{p}\sigma_p$. The measured uncertainties are

(9)$\begin{array}{rl}{\left(\Delta x\right)}^{2}& ={\left(\Delta x\right)}_{\rho }^{2}+{\sigma }_{x}^{2}\\ {\left(\Delta p\right)}^{2}& ={\left(\Delta p\right)}_{\stackrel{^}{\rho }}^{2}+{\sigma }_{p}^{2}={\left(\Delta p\right)}_{\rho }^{2}+\frac{1}{{\sigma }_{x}^{2}}+{\sigma }_{p}^{2}\end{array}\begin\left\{split\right\} \left\{\left(\Delta x\right)\right\}^2&= \left\{\left(\Delta x\right)\right\}^2_\rho +\sigma_x^2\\ \left\{\left(\Delta p\right)\right\}^2&= \left\{\left(\Delta p\right)\right\}^2_\left\{\hat\left\{\rho\right\}\right\} +\sigma_p^2=\left\{\left(\Delta p\right)\right\}^2_\rho +\frac\left\{1\right\}\left\{\sigma_x^2\right\}+\sigma_p^2 \end\left\{split\right\} $

You can play around with other, non-Gaussian, acceptance functions to replace (6). You’re limited only by your ability to find a complete set of projectors, satisfying the analogue of (7) and, of course, by your ability to do the requisite integrals.

What you’ll discover is that the Gaussian acceptance function provides the best tradeoff (when, say, you measure $xx$) between the systematic uncertainty in $xx$ and the contribution to the quantum-mechanical uncertainty in $pp$, resulting from the measurement.

Update (9/20/2012):

I looked some more at the Ozawa paper whose “Universally valid reformulation” of the uncertainty principle this PRL proposes to test. Unfortunately, it doesn’t seem nearly as interesting as it did at first glance.

• For observables, $A,BA,B$, with pure point spectra, we can assume “ideal” measuring apparati (whose measured uncertainty equals the inherent quantum-mechanical uncertainty of the observable in the quantum state in which the measurement is made). In that case, his uncertainty relation (see (17) of his paper) reduces to the “uninteresting” (5). Of course that’s trivially satisfied. I believe that a stronger bound can be derived, in this case. But doing so requires more sophisticated techniques than Ozawa uses.
• For unbounded observables, like $x,px,p$, we can see from what I’ve said above that the actual lower bound is *stronger* than the one Ozawa derives. Consider a measurement of $xx$, followed by a measurement of $pp$. From (9), the product of the measured uncertainties2 satisfies
(10)$\begin{array}{rl}{\left(\Delta x\right)}^{2}{\left(\Delta p\right)}^{2}& \ge \frac{5}{4}+{\left(\Delta x\right)}_{\rho }^{2}\left(\frac{1}{{\sigma }_{x}^{2}}+{\sigma }_{p}^{2}\right)+{\left(\Delta p\right)}_{\rho }^{2}{\sigma }_{x}^{2}+{\sigma }_{x}^{2}{\sigma }_{p}^{2}\\ & \ge {\left(\frac{1}{2}+\sqrt{1+{\sigma }_{x}^{2}{\sigma }_{p}^{2}}\right)}^{2}\end{array}\begin\left\{split\right\} \left\{\left(\Delta x\right)\right\}^2\left\{\left(\Delta p\right)\right\}^2 &\geq \frac\left\{5\right\}\left\{4\right\} + \left\{\left(\Delta x\right)\right\}^2_\rho \left\left(\frac\left\{1\right\}\left\{\sigma_x^2\right\}+\sigma_p^2\right\right) + \left\{\left(\Delta p\right)\right\}^2_\rho \sigma_x^2 +\sigma_x^2\sigma_p^2\\ &\geq \left\{\left\left(\frac\left\{1\right\}\left\{2\right\} +\sqrt\left\{1+\sigma_x^2\sigma_p^2\right\}\right\right)\right\}^2 \end\left\{split\right\} $

where the last inequality is saturated by an initial state, $\rho \rho$, which is a pure state consisting of a Gaussian wave packet with carefully-chosen width, ${\left(\Delta x\right)}_{\rho }^{2}=\frac{{\sigma }_{x}^{2}}{2\sqrt{1+{\sigma }_{x}^{2}{\sigma }_{p}^{2}}},\phantom{\rule{2em}{0ex}}{\left(\Delta p\right)}_{\rho }^{2}=\frac{\sqrt{1+{\sigma }_{x}^{2}{\sigma }_{p}^{2}}}{2{\sigma }_{x}^{2}} \left\{\left(\Delta x\right)\right\}^2_\rho = \frac\left\{\sigma_x^2\right\}\left\{2\sqrt\left\{1+\sigma_x^2\sigma_p^2\right\}\right\},\qquad \left\{\left(\Delta p\right)\right\}^2_\rho = \frac\left\{\sqrt\left\{1+\sigma_x^2\sigma_p^2\right\}\right\}\left\{2\sigma_x^2\right\} $

Update (9/28/2012):

Here is, at least, one lower bound (stronger than Ozawa’s stupid bound) for the product of measured uncertainties when $A,BA,B$ have pure point spectra. Let

$\stackrel{^}{B}=\sum _{i}{P}_{i}^{\left(A\right)}B{P}_{i}^{\left(A\right)} \hat\left\{B\right\} = \sum_i P^\left\{\left(A\right)\right\}_i B P^\left\{\left(A\right)\right\}_i $

and

${M}_{i}={P}_{i}^{\left(A\right)}B\left(𝟙-{P}_{i}^{\left(A\right)}\right) M_i = P^\left\{\left(A\right)\right\}_i B \left\left(\mathbb\left\{1\right\} - P^\left\{\left(A\right)\right\}_i\right\right) $

We easily compute

${\left(\Delta B\right)}_{\stackrel{^}{\rho }}^{2}={\left(\Delta \stackrel{^}{B}\right)}_{\rho }^{2}+\sum _{i}\mathrm{Tr}\left({M}_{i}{M}_{i}^{†}\rho \right) \left\{\left(\Delta B\right)\right\}^2_\left\{\hat\left\{\rho\right\}\right\} = \left\{\left(\Delta \hat\left\{B\right\}\right)\right\}^2_\left\{\rho\right\} + \sum_i Tr\left(M_i \M_i^\dagger \rho\right) $

and hence

(11)${\left(\Delta A\right)}_{\rho }^{2}{\left(\Delta B\right)}_{\stackrel{^}{\rho }}^{2}={\left(\Delta A\right)}_{\rho }^{2}{\left(\Delta \stackrel{^}{B}\right)}_{\rho }^{2}+{\left(\Delta A\right)}_{\rho }^{2}\sum _{i}\mathrm{Tr}\left({M}_{i}{M}_{i}^{†}\rho \right)\left\{\left(\Delta A\right)\right\}^2_\rho \left\{\left(\Delta B\right)\right\}^2_\left\{\hat\left\{\rho\right\}\right\} = \left\{\left(\Delta A\right)\right\}^2_\rho \left\{\left(\Delta \hat\left\{B\right\}\right)\right\}^2_\left\{\rho\right\} + \left\{\left(\Delta A\right)\right\}^2_\rho \sum_i Tr\left(M_i \M_i^\dagger \rho\right) $

Since $\left[\stackrel{^}{B},A\right]=0\left[\hat\left\{B\right\},A\right]=0$, the first term is bounded below3 by

(12)${\left(\Delta A\right)}_{\rho }^{2}{\left(\Delta \stackrel{^}{B}\right)}_{\rho }^{2}\ge {\left(\frac{1}{2}\mathrm{Tr}\left(\left\{A,\stackrel{^}{B}\right\}\rho \right)-\mathrm{Tr}\left(A\rho \right)\mathrm{Tr}\left(\stackrel{^}{B}\rho \right)\right)}^{2}\left\{\left(\Delta A\right)\right\}^2_\rho \left\{\left(\Delta \hat\left\{B\right\}\right)\right\}^2_\left\{\rho\right\} \geq \left\{\left\left(\frac\left\{1\right\}\left\{2\right\} Tr\left(\\left\{A,\hat\left\{B\right\}\\right\}\rho\right)-Tr\left(A\rho\right)Tr\left(\hat\left\{B\right\}\rho\right)\right\right)\right\}^2 $

The second term is also positive-semidefinite.

For the classic case of a 2-state system, with $A={J}_{x}A=J_x$ and $B={J}_{y}B=J_y$ (the system considered by the aforementioned PRL), we see that $\stackrel{^}{B}\equiv 0\hat\left\{B\right\}\equiv 0$, and the product of uncertainties is entirely given by the second term of (11).

The most general density matrix for the 2-state system is parametrized by the unit 3-ball

$\rho =\frac{1}{2}\left(𝟙+\stackrel{⇀}{a}\cdot \stackrel{⇀}{\sigma }\right),\phantom{\rule{2em}{0ex}}\stackrel{⇀}{a}\cdot \stackrel{⇀}{a}\le 1 \rho = \frac\left\{1\right\}\left\{2\right\}\left\left(\mathbb\left\{1\right\}+ \vec\left\{a\right\}\cdot\vec\left\{\sigma\right\}\right\right), \qquad \vec\left\{a\right\}\cdot\vec\left\{a\right\}\leq 1 $

The points on the boundary ${S}^{2}=\left\{\stackrel{⇀}{a}\cdot \stackrel{⇀}{a}=1\right\}S^2=\\left\{\vec\left\{a\right\}\cdot\vec\left\{a\right\}=1\\right\}$ correspond to pure states.

Upon measuring $A={J}_{x}\equiv \frac{1}{2}{\sigma }_{x}A=J_x\equiv\tfrac\left\{1\right\}\left\{2\right\}\sigma_x$, the density matrix after the measurement is

$\stackrel{^}{\rho }=\frac{1}{2}\left(𝟙+{a}_{x}{\sigma }_{x}\right) \hat\left\{\rho\right\} = \frac\left\{1\right\}\left\{2\right\}\left\left(\mathbb\left\{1\right\}+ \a_x\sigma_x\right\right) $

and, for a subsequent measurement of ${J}_{y}J_y$,

${\left(\Delta {J}_{x}\right)}_{\rho }^{2}{\left(\Delta {J}_{y}\right)}_{\stackrel{^}{\rho }}^{2}=\frac{1}{16}\left(1-{a}_{x}^{2}\right) \left\{\left(\Delta J_x\right)\right\}^2_\rho\left\{\left(\Delta J_y\right)\right\}^2_\left\{\hat\left\{\rho\right\}\right\} = \frac\left\{1\right\}\left\{16\right\}\left(1-a_x^2\right) $

as “predicted” by (11).

1 Frequently, one wants to ask questions about conditional probabilites: “Given that a measurement of $AA$ yields the value ${\lambda }_{1}\lambda_1$, what is the probability distribution for a subsequent measurement of …”. To answer such questions, one typically works with a new (“projected”) density matrix, $\rho \prime =\frac{1}{Z}{P}_{1}^{\left(A\right)}\rho {P}_{1}^{\left(A\right)}\rho&apos = \frac\left\{1\right\}\left\{Z\right\} P^\left\{\left(A\right)\right\}_1 \rho P^\left\{\left(A\right)\right\}_1$, where the normalization factor $Z=\mathrm{Tr}\left({P}_{1}^{\left(A\right)}\rho \right)Z=Tr\left(P^\left\{\left(A\right)\right\}_1 \rho\right)$ is required to make $\mathrm{Tr}\left(\rho \prime \right)=1Tr\left(\rho&apos\right)=1$. The formalism in the main text of this post is geared, instead, to computing joint probability distributions.

2 Ozawa’s inequality isn’t for the product of the measured uncertainties, $\left(\Delta x\right)\left(\Delta p\right)\left(\Delta x\right)\left(\Delta p\right)$, but rather for the product, $\left(\Delta x\right){\left(\Delta p\right)}_{\stackrel{^}{\rho }}\left(\Delta x\right)\left\{\left(\Delta p\right)\right\}_\left\{\hat\left\{\rho\right\}\right\}$, of the measured uncertainty in $xx$ with the quantum-mechanical uncertainty in $pp$ in the state $\stackrel{^}{\rho }\hat\left\{\rho\right\}$ which results from the $xx$-measurement. To obtain this, just mentally set ${\sigma }_{p}=0\sigma_p=0$ in the above formulæ.

3 Let $S=\left(A-{⟨A⟩}_{\rho }𝟙\right)+{e}^{i\theta }\alpha \left(B-{⟨B⟩}_{\rho }𝟙\right)S= \left\left(A-\left\{\langle A\rangle\right\}_\rho\mathbb\left\{1\right\}\right\right)+e^\left\{i\theta\right\}\alpha \left\left(B-\left\{\langle B\rangle\right\}_\rho\mathbb\left\{1\right\}\right\right)$ for $\alpha \in ℝ\alpha\in\mathbb\left\{R\right\}$. Consider $Q\left(\alpha \right)=\mathrm{Tr}\left(S{S}^{†}\rho \right) Q\left(\alpha\right) = Tr\left(S S^\dagger \rho\right) $ This is a quadratic expression in $\alpha \alpha$, which is positive-semidefinite, $Q\left(\alpha \right)\ge 0Q\left(\alpha\right)\geq 0$. Thus, the discriminant must be negative-semidefinite, $D\le 0D\leq 0$. For $\theta =\pi /2\theta=\pi/2$, this yields the conventional uncertainty relation, ${\left(\Delta A\right)}_{\rho }^{2}{\left(\Delta B\right)}_{\rho }^{2}\ge \frac{1}{4}{\left(\mathrm{Tr}\left(\left[A,B\right]\rho \right)\right)}^{2} \left\{\left(\Delta A\right)\right\}^2_\rho\left\{\left(\Delta B\right)\right\}^2_\rho\geq\frac\left\{1\right\}\left\{4\right\}\left\{\left\left(Tr\left(\left[A,B\right]\rho\right)\right\right)\right\}^2 $ For $\theta =0\theta=0$, it yields ${\left(\Delta A\right)}_{\rho }^{2}{\left(\Delta B\right)}_{\rho }^{2}\ge {\left(\frac{1}{2}\mathrm{Tr}\left(\left\{A,B\right\}\rho \right)-\mathrm{Tr}\left(A\rho \right)\mathrm{Tr}\left(B\rho \right)\right)}^{2} \left\{\left(\Delta A\right)\right\}^2_\rho\left\{\left(\Delta B\right)\right\}^2_\rho\geq\left\{\left\left(\frac\left\{1\right\}\left\{2\right\}Tr\left(\\left\{A,B\\right\}\rho\right)-Tr\left(A\rho\right)Tr\left(B\rho\right)\right\right)\right\}^2 $ which is an expression you sometimes see, in the higher-quality textbooks.

Emily Lakdawalla - The Planetary Society Blog

The ASRG Cancellation in Context
ASRGs could have stretched NASA's limited supply of plutonium to potentially enable missions to the perpetually-shadowed polar craters on our moon, to flyby Uranus, or to float for months on a Titan lake.

December 08, 2013

Christian P. Robert - xi'an's og

MCMSki IV, Jan. 6-9, 2014, Chamonix (news #13)

Now, most poster abstracts have been received (or at least 63 of them),  even though newcomers can still send them to my wordpress address (if they realise the message gets posted immediately!, so the format Subject: firstname secondname (affiliation): title and text: abstract must be respected! No personal message or query please!). We have now above 200 registered participants, with all sessions remaining miraculously full (after a few permutations in the program).

S0 it is time to mention a wee bit of the “ski” side of MCMski. Chamonix has two types of ski passes, Chamonix Le Pass, and Mont Blanc Unlimited, the later allowing a wide access to the Mont Blanc area, up to 3800 meters and in France, Italy, and Switzerland, but presumably harder to exploit to the fullest on a 4 hour afternoon break. (You have to arrange renting skis and buying passes on your own! The conference centre may answer moderate queries but not make any booking.)  The temperature in the town of Chamonix is currently between -7 and 0 (centigrades), with ten centimetres of snow in town. All ski areas will be open by Dec. 21. If you plan to ski the Vallée Blanche from Aiguille du Midi, at 3800m, I strongly advise renting a guide for this ultimate skiing experience!

Big big big news: not only the ski race will take place on Wed., Jan. 08, afternoon, organised by ESF Chamonix, but Antonietta Mira managed to secure one or two pairs of skis for the winner(s) of the race! I doubt there will be other opportunities of that magnitude for winning a magnificent pair of skis made in Italy by Blossom skis. Thanks a lot to Anto!!! And to Blossom skis (whose collection includes a series called FreeTibet.)

Filed under: Mountains, Statistics, Travel, University life Tagged: Blossom skis, Chamonix, doodle, France, Geneva, ISBA conference, Italy, MCMC, MCMSki IV, Mont Blanc, Monte Carlo Statistical Methods, posters, shuttle, simulation, ski

Christian P. Robert - xi'an's og

This week, thanks to a lack of clear instructions (from me) to my students in the Reading Classics student seminar, four students showed up with a presentation! Since I had planned for two teaching blocks, three of them managed to fit within the three hours, while the last one nicely accepted to wait till next week to present a paper by David Cox…

The first paper discussed therein was A new look at the statistical model identification, written in 1974 by Hirotugu Akaike. And presenting the AIC criterion. My student Rozan asked to give the presentation in French as he struggled with English, but it was still a challenge for him and he ended up being too close to the paper to provide a proper perspective on why AIC is written the way it is and why it is (potentially) relevant for model selection. And why it is not such a definitive answer to the model selection problem. This is not the simplest paper in the list, to be sure, but some intuition could have been built from the linear model, rather than producing the case of an ARMA(p,q) model without much explanation. (I actually wonder why the penalty for this model is (p+q)/T, rather than (p+q+1)/T for the additional variance parameter.) Or simulation ran on the performances of AIC versus other xIC’s…

The second paper was another classic, the original GLM paper by John Nelder and his coauthor Wedderburn, published in 1972 in Series B. A slightly easier paper, in that the notion of a generalised linear model is presented therein, with mathematical properties linking the (conditional) mean of the observation with the parameters and several examples that could be discussed. Plus having the book as a backup. My student Ysé did a reasonable job in presenting the concepts, but she would have benefited from this extra-week in including properly the computations she ran in R around the glm() function… (The definition of the deviance was somehow deficient, although this led to a small discussion during the class as to how the analysis of deviance was extending the then flourishing analysis of variance.) In the generic definition of the generalised linear models, I was also reminded of the
generality of the nuisance parameter modelling, which made the part of interest appear as an exponential shift on the original (nuisance) density.

The third paper, presented by Bong, was yet another classic, namely the FDR paper, Controlling the false discovery rate, of Benjamini and Hochberg in Series B (which was recently promoted to the should-have-been-a-Read-Paper category by the RSS Research Committee and discussed at the Annual RSS Conference in Edinburgh four years ago, as well as published in Series B). This 2010 discussion would actually have been a good start to discuss the paper in class, but Bong was not aware of it and mentioned earlier papers extending the 1995 classic. She gave a decent presentation of the problem and of the solution of Benjamini and Hochberg but I wonder how much of the novelty of the concept the class grasped. (I presume everyone was getting tired by then as I was the only one asking questions.) The slides somewhat made it look too much like a simulation experiment… (Unsurprisingly, the presentation did not include any Bayesian perspective on the approach, even though they are quite natural and emerged very quickly once the paper was published. I remember for instance the Valencia 7 meeting in Teneriffe where Larry Wasserman discussed about the Bayesian-frequentist agreement in multiple testing.)

Filed under: Books, Kids, Statistics, University life Tagged: AIC, Akaike's criterion, ARMA models, Benjamini, classics, FAR, generalised linear models, GLMs, Hochberg, John Nelder, Master program, multiple comparisons, seminar, Université Paris Dauphine, Valencia conferences

John Baez - Azimuth

Lebesgue’s Universal Covering Problem

I try to focus on serious problems in this blog, mostly environmental issues and the attempt to develop ‘green mathematics’. But I seem unable to resist talking about random fun stuff now and then. For example, the Lebesgue universal covering problem. It’s not important, it’s just strange… but for some reason I feel a desire to talk about it.

For starters, let’s say the diameter of a region in the plane is the maximum distance between two points in this region. You all know what a circle of diameter 1 looks like. But an equilateral triangle with edges of length 1 also has diameter 1:

After all, two points in this triangle are farthest apart when they’re at two corners. And you’ll notice, if you think about it, that this triangle doesn’t fit inside a circle of diameter 1:

In 1914, the famous mathematician Henri Lebesgue sent a letter to a pal named Pál. And in this letter he asked: what’s the smallest possible region that contains every set in the plane with diameter 1?

He was actually more precise. The phrase “smallest possible” is vague, but Lebesgue wanted the set with the least possible area. The phrase “contains” is also vague, at least as I used it. I didn’t mean that our region should literally contain every set with diameter 1. Only the whole plane does that! I meant that we can rotate or translate any set with diameter 1 until it fits in our region. So a more precise statement is:

What is the smallest possible area of a region $S$ in the plane such that every set of diameter 1 in the plane can be rotated and translated to fit inside $S$?

You see why math gets a reputation of being dry: sometimes when you make a simple question precise it gets longer.

Even this second version of the question is a bit wrong, since it’s presuming there exists a region with smallest possible area that does the job. Maybe you can do it with regions whose area gets smaller and smaller, closer and closer to some number, but you can’t do it with a region whose area equals that number! Also, the word ‘region’ is a bit vague. So if I were talking to a nitpicky mathematician, I might pose the puzzle this way:

What is the greatest lower bound of the measures of closed sets $S \subseteq \mathbb{R}^2$ such that every set $T \subseteq \mathbb{R}^2$ of diameter 1 can be rotated and translated to fit inside $S$?

Anyway, the reason this puzzle is famous is not that it gets dry when you state it precisely. It’s that it’s hard to solve!

It’s called Lebesgue’s universal covering problem. A region in the plane is called a universal covering if it does the job: any set in the plane of diameter 1 can fit inside it, in the sense I made precise.

Pál set out to find universal coverings, and in 1920 he wrote a paper on his results. He found a very nice one: a regular hexagon circumscribed around a circle of diameter 1. Do you see why you can fit the equilateral triangle with side length 1 inside this?

This hexagon has area

$\sqrt{3}/2 \approx 0.86602540$

But in the same paper, Pál showed you could safely cut off two corners of this hexagon and get a smaller universal covering! This one has area

$2 - 2/\sqrt{3} \approx 0.84529946$

He guessed this solution was optimal.

However, in 1936, a guy named Sprague sliced two tiny pieces off Pál’s proposed solution and found a universal covering with area

$\sim 0.84413770$

Here’s the idea:

The big hexagon is Pál's original solution. He then inscribed a regular 12-sided polygon inside this, and showed that you can remove two of the corners, say $B_1B_2B$ and $F_1F_2F,$ and get a smaller universal covering. But Sprague noticed that near $D$ you can also remove the part outside the circle with radius 1 centered at $B_1$, as well as the part outside the circle with radius 1 centered at $F_2.$

Sprague conjectured that this is the best you can do.

But in 1975, Hansen showed you could slice off very tiny corners off Sprague’s solution, each of which reduces the area by just $6 \cdot 10^{-18}.$

I think this microscopic advance, more than anything else, convinced people that Lebesgue’s universal covering problem was fiendishly difficult. Viktor Klee, in a parody of the usual optimistic prophecies people like to make, wrote that:

… progress on this question, which has been painfully slow in the past, may be even more painfully slow in the future.

Indeed, my whole interest in this problem is rather morbid. I don’t know any reason that it’s important. I don’t see it as connected to lots of other beautiful math. It just seems astoundingly hard compared to what you might initially think. I admire people who work on it in the same way I admire people who decide to ski across the Antarctic. It’s fun to read about—from the comfort of my living room, sitting by the fire—but I can’t imagine actually doing it!

In 1980, a guy named Duff did a lot better. All the universal coverings so far were convex subsets of the plane. But he considered nonconvex universal coverings and found one with area:

$\sim 0.84413570$

This changed the game a lot. If we only consider convex universal coverings, it’s possible to prove there’s one with the least possible area—though we don’t know what it is. But if we allow nonconvex ones, we don’t even know this. There could be solutions whose area gets smaller and smaller, approaching some nonzero value, but never reaching it.

So at this point, Lebesgue’s puzzle split in two: one for universal coverings, and one for convex universal coverings. I’ll only talk about the second one, since I don’t know of further progress on the first.

Remember how Hansen improved Sprague’s universal covering by chopping off two tiny pieces? In 1992 he went further. He again sliced two corners off Sprague’s solution. One, the same shape as before, reduced the area by just $6 \cdot 10^{-18}.$ But the other was vastly larger: it reduced the area by a whopping $4 \cdot 10^{-11}.$

I believe this is the smallest convex universal covering known so far.

But there’s more to say. In 2005, Peter Brass and Mehrbod Sharifi came up with a nice lower bound. They showed any convex universal covering must have an area of least

$0.832$

They got this result by a rigorous computer-aided search for the convex set with the smallest area that contains a circle, equilateral triangle and pentagon of diameter 1.

If you only want your convex set to contain a circle and equilateral triangle of diameter 1, you can get away with this:

This gives a lower bound of

$0.825$

But the pentagon doesn’t fit in this set! Here is the least-area convex shape that also contains a pentagon of diameter 1, as found by Brass and Sharifi:

You’ll notice the triangle no longer has the same center as the circle!

To find this result, it was enough to keep the circle fixed, translate the triangle, and translate and rotate the pentagon. So, they searched through a 5-dimensional space, repeatedly computing the area of the convex hull of these three shapes to see how small they could make it. They considered 53,118,162 cases. Of these, 655,602 required further care—roughly speaking, they had to move the triangle and pentagon around slightly to see if they could make the area even smaller.

They say they could have done better if they’d also included a fourth shape, but this was computationally infeasible, since it would take them from a 5-dimensional search space to an 8-dimensional one. It’s possible that one could tackle this using a distributed computing project where a lot of people contribute computer time, like they do in the search for huge prime numbers.

If you hear of more progress on this issue, please let me know! I hope that sometime—perhaps tomorrow, perhaps decades hence—someone will report good news.

References

Hansen’s record-holding convex universal cover is here:

• H. Hansen, Small universal covers for sets of unit diameter, Geometriae Dedicata 42 (1992), 205–213.

It’s quite readable. This is where I got the picture of Pál’s solution and Sprague’s improvement.

The paper on the current best lower bound for convex universal coverings is also quite nice:

• Peter Brass and Mehrbod Sharifi, A lower bound for Lebesgue’s universal cover problem, International Journal of Computational Geometry & Applications 15 (2005), 537–544.

The picture of the equilateral triangle in the circle comes from an earlier paper, which got a lower bound of

$0.8271$

by considering the circle and regular $3^n$-gons of diameter 1 for all $n$:

• Gy. Elekes, Generalized breadths, circular Cantor sets, and the least area UCC, Discrete & Computational Geometry 12 (1994), 439–449.

I have not yet managed to get ahold of Duff’s paper on the record-holding nonconvex universal covering:

• G. F. D. Duff, A smaller universal cover for sets of unit diameter, C. R. Math. Acad. Sci. 2 (1980), 37–42.

One interesting thing is that in 1963, Eggleston found a universal covering that’s minimal in the following sense: no subset of it is a universal covering. However, it doesn’t have the least possible area!

• H. G. Eggleston, Minimal universal covers in $\mathrm{E}^n,$ Israel J. Math. 1 (1963), 149–155.

Later, Kovalev showed that any ‘minimal’ universal covering in this sense is a star domain:

• M. D. Kovalev, A minimal Lebesgue covering exists (Russian), Mat. Zametki 40 (1986), 401–406, 430.

This means there’s a point $x_0$ inside the set such that for any other point $x$ in the set, the line segment connecting $x_0$ and $x$ is in the set. Any convex set is a star domain, but not conversely:

Sean Carroll - Preposterous Universe

The Branch We Were Sitting On

In the latest issue of the New York Review, Cathleen Schine reviews Levels of Life, a new book by Julian Barnes. It’s described as a three-part meditation on grief, following the death of Barnes’s wife Pat Kavanagh.

One of the things that is of no solace to Barnes (and there are many) is religion. He writes:

When we killed–or exiled–God, we also killed ourselves…. No God, no afterlife, no us. We were right to kill Him, of course, this long-standing imaginary friend of ours. And we weren’t going to get an afterlife anyway. But we sawed off the branch we were sitting on. And the view from there, from that height–even if it was only an illusion of a view–wasn’t so bad.

I can’t disagree. Atheists often proclaim the death of God in positively gleeful terms, but it’s important to recognize what was lost–a purpose in living, a natural place in the universe. The loss is not irretrievable; there is nothing that stops us from creating our own meaning even if there’s no supernatural overseer to hand one to us. But it’s a daunting task, one to which we haven’t really faced up.

Peter Coles - In the Dark

The Sad Tale of Veronica Lake

A few weeks ago I indulged myself by watching, during the same evening, a couple of class examples of Film Noir, The Glass Key and The Blue Dahlia The first of these is based on the novel of the same name by Dashiell Hammett and the second has an original screenplay by Raymond Chandler. Both feature the same leading actors, Alan Ladd and Veronica Lake, The Glass Key being the first film featuring this pairing.

There’s a pragmatic reason why Paramount Studios chose Veronica Lake to star with Alan Ladd, namely her size. Alan Ladd was quite a small man, standing  just a shade under 5′ 5″ tall, and the casting directors consequently found it difficult to locate a leading lady who didn’t tower over him. Veronica Lake, however, was only 4′ 11″ and fitted the bill nicely:

It wasn’t just her diminutive stature that propelled Veronica Lake to stardom; she was also very beautiful and managed to project a screen image of cool detachment which made her a perfect choice as femme fatale, a quintessential ingredient of any Film Noir. She’s absolutely great in both the movies I watched, and in many more besides. Her looks and screen presence turned her into a true icon -a vera icon in fact- appropriately enough, because the name Veronica derives from that anagram. The cascade of blond hair, often covering one eye, became a trademark that later found its way into, for example, the character of Jessica Rabbit in the animated film Who Framed Roger Rabbit?

However, her success as a movie star was short-lived and Veronica Lake disappeared from Hollywood entirely in the 1950s. She was rediscovered in the 1960s working as a waitress in a downmarket New York bar, and subsequently made a film called Footsteps in the Snow but it disappeared without trace and failed to revitalize her career. She died in 1973.

So why did an actress of such obvious talent experience such a dramatic reversal of fortune? Sadly, the answer is a familiar one: problems with drink and drugs, struggles with mental illness, a succession of disastrous marriages, and a reputation for being very difficult to work with. Her famous screen persona seems largely to have been a result of narcotics abuse. “I wasn’t a Sex Symbol, I was Sex Zombie”, as she wrote in her biography. She appeared to be detached, because she was stoned.

It’s a sad tale that would cast a shadow over even over the darkest Film Noir but though she paid a heavy price she still left a priceless legacy. Forty years after her death, all that remains of her is what you can see on the screen, and that includes some of the greatest movies of all time.

Andrew Jaffe - Leaves on the Line

Nearly a decade ago, blogging was young, and its place in the academic world wasn’t clear. Back in 2005, I wrote about an anonymous article in the Chronicle of Higher Education, a so-called “advice” column admonishing academic job seekers to avoid blogging, mostly because it let the hiring committee find out things that had nothing whatever to do with their academic job, and reject them on those (inappropriate) grounds.

I thought things had changed. Many academics have blogs, and indeed many institutions encourage it (here at Imperial, there’s a College-wide list of blogs written by people at all levels, and I’ve helped teach a course on blogging for young academics). More generally, outreach has become an important component of academic life (that is, it’s at least necessary to pay it lip service when applying for funding or promotions) and blogging is usually seen as a useful way to reach a wide audience outside of one’s field.

So I was distressed to see the lament — from an academic blogger — “Want an academic job? Hold your tongue”. Things haven’t changed as much as I thought:

… [A senior academic said that] the blog, while it was to be commended for its forthright tone, was so informal and laced with profanity that the professor could not help but hold the blog against the potential faculty member…. It was the consensus that aspiring young scientists should steer clear of such activities.

Depending on the content of the blog in question, this seems somewhere between a disregard for academic freedom and a judgment of the candidate on completely irrelevant grounds. Of course, it is natural to want the personalities of our colleagues to mesh well with our own, and almost impossible to completely ignore supposedly extraneous information. But we are hiring for academic jobs, and what should matter are research and teaching ability.

Of course, I’ve been lucky: I already had a permanent job when I started blogging, and I work in the UK system which doesn’t have a tenure review process. And I admit this blog has steered clear of truly controversial topics (depending on what you think of Bayesian probability, at least).

Christian P. Robert - xi'an's og

amazonly associated thanks (& warnin’)

Following a now well-established pattern, let me (re)warn (the few) unwary ‘Og readers that the links to Amazon.com and to Amazon.fr found on this blog are actually susceptible to earn me a monetary gain [from 4% to 8% on the sales] if a purchase is made by the reader in the 24 hours following the entry on Amazon through this link, thanks to the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to amazon.com/fr. Unlike the pattern of last year, and of the year before last, the mostly purchased item through the links happens to be related to a blog post, since it is Andrew’s book, with 318 copies of its third edition sold through the ‘Og last month! Here are some of the most exotic purchases:

As usual the books I actually reviewed along the past months, positively or negatively, were among the top purchases… Like two dozen copies of The BUGS book. And a dozen of R for dummies. And even a few of The Cartoon Introduction to Statistics. (Despite a most critical review.) Thanks to all of you using those links (for feeding further my book addiction, books that now eventually end up in the math common room in Dauphine or Warwick, once I have read them)!

Filed under: Books, Kids, R, Statistics Tagged: Amazon, book reviews, lollipops, mandoline, Og, pretzels, R, R for dummies

Andrew Jaffe - Leaves on the Line

The next generation of large satellites: PRISM and/or eLISA?

Today was the deadline for submitting so-called “White Papers” proposing the next generation of the European Space Agency satellite missions. Because of the long lead times for these sorts of complicated technical achievements, this call is for launches in the faraway years of 2028 or 2034. (These dates would be harder to wrap my head around if I weren’t writing this on the same weekend that I’m attending the 25th reunion of my university graduation, an event about which it’s difficult to avoid the clichéd thought that May, 1988 feels like the day before yesterday.)

At least two of the ideas are particularly close to my scientific heart.

The Polarized Radiation Imaging and Spectroscopy Mission (PRISM) is a cosmic microwave background (CMB) telescope, following on from Planck and the current generation of sub-orbital telescopes like EBEX and PolarBear: whereas Planck has 72 detectors observing the sky over nine frequencies on the sky, PRISM would have more than 7000 detectors working in a similar way to Planck over 32 frequencies, along with another set observing 300 narrow frequency bands, and another instrument dedicated to measuring the spectrum of the CMB in even more detail. Combined, these instruments allow a wide variety of cosmological and astrophysical goals, concentrating on more direct observations of early Universe physics than possible with current instruments, in particular the possible background of gravitational waves from inflation, and the small correlations induced by the physics of inflation and other physical processes in the history of the Universe.

The eLISA mission is the latest attempt to build a gravitational radiation observatory in space, observing astrophysical sources rather than the primordial background affecting the CMB, using giant lasers to measure the distance between three separate free-floating satellites a million kilometres apart from one another. As a gravitational wave passes through the triangle, it bends space and effectively changes the distance between them. The trio would thereby be sensitive to the gravitational waves produced by small, dense objects orbiting one another, objects like white dwarfs, neutron stars and, most excitingly, black holes. This would give us a probe of physics in locations we can’t see with ordinary light, and in regimes that we can’t reproduce on earth or anywhere nearby.

In the selection process, ESA is supposed to take into account the interests of the community. Hence both of these missions are soliciting support, of active and interested scientists and also the more general public: check out the sites for PRISM and eLISA. It’s a tough call. Both cases would be more convincing with a detection of gravitational radiation in their respective regimes, but the process requires putting down a marker early on. In the long term, a CMB mission like PRISM seems inevitable — there are unlikely to be any technical showstoppers — it’s just a big telescope in a slightly unusual range of frequencies. eLISA is more technically challenging: the LISA Pathfinder effort has shown just how hard it is to keep and monitor a free-floating mass in space, and the lack of a detection so far from the ground-based LIGO observatory, although completely consistent with expectations, has kept the community’s enthusiasm lower. (This will likely change with Advanced LIGO, expected to see many hundreds of sources as soon as it comes online in 2015 or thereabouts.)

Full disclosure: although I’ve signed up to support both, I’m directly involved in the PRISM white paper.

December 07, 2013

Clifford V. Johnson - Asymptotia

Lunch Time in Colour
Finished the inks and threw some (digital) paints over the lunch group. I think it's time to move on to another part of The Project. I've spent far too long fiddling with the light in this. -cvj Click to continue reading this post

Marco Frasca - The Gauge Connection

That strange behavior of supersymmetry…

I am a careful reader of scientific literature and an avid searcher for already published material in peer reviewed journals. Of course, arxiv is essential to accomplish this task and to satisfy my needs for reading. In these days, I am working on Dyson-Schwinger equations. I have written on this a paper (see here) a few years ago but this work is in strong need to be revised. Maybe, some of these days I will take the challenge. Googling around and looking for the Dyson-Schwinger equations applied to the well-known supersymmetric model due to Wess and Zumino, I have uncovered a very exciting track of research that uses Dyson-Schwinger equations to produce exact results in quantum field theory. The paper I have got was authored by Marc Bellon, Gustavo Lozano and Fidel Schaposnik and can be found here. These authors get the Dyson-Schwinger equations for the Wess-Zumino model at one loop and manage to compute the self-energies of the involved fields: A scalar, a fermion and an auxiliary bosonic field. Their equations are yielded for three different self-energies, different for each field. Self-energies are essential in quantum field theory as they introduce corrections to masses in a propagator and so enters into the physical part of an object that is not an observable.

Now, if you are in a symmetric theory like the Wess-Zumino model, such a symmetry, if it is not broken, will yield equal masses to all the components of the multiplet entering into the theory. This means that if you start with the assumption that in this case all the self-energies are equal, you are doing a consistent approximation. This is what Bellon, Lozano and Schaposnik just did. They assumed from the start that all the self-energies are equal for the Dyson Schwinger equations they get and go on with their computations. This choice leaves an open question: What if do I choose different self-energies from the start? Will the Dyson-Schwiner equations drive the solution toward the symmetric one?

This question is really interesting as the model considered is not exactly the one that Witten analysed in his famous paper  on 1982 on breaking of a supersymmetry (you can download his paper here). Supersymmetric model generates non-linear terms and could be amenable to spontaneous symmetry breaking, provided the Witten index has the proper values. The question I asked is strongly related to the idea of a supersymmetry breaking at the bootstrap: Supersymmetry is responsible for its breaking.

So, I managed to numerically solve Dyson-Schwinger equations for the Wess-Zumino model as yielded by Bellon, Lozano and Schaposnik and presented the results in a paper (see here). If you solve them assuming from the start all the self-energies are equal you get the following figure for coupling running from 0.25 to 100 (weak to strong):

It does not matter the way you modify your parameters in the Dyson-Schwinger equations. Choosing them all equal from the start makes them equal forever. This is a consistent choice and this solution exists. But now, try to choose all different self-energies. You will get the following figure for the same couplings:

This is really nice. You see that exist also solutions with all different self-energies and supersymmetry may be broken in this model. This kind of solutions has been missed by the authors. What one can see here is that supersymmetry is preserved for small couplings, even if we started with all different self-energies, but is broken as the coupling becomes stronger. This result is really striking and unexpected. It is in agreement with the results presented here.

I hope to extend this analysis to more mundane theories to analyse behaviours that are currently discussed in literature but never checked for. For these aims there are very powerful tools developed for Mathematica by Markus Huber, Jens Braun and Mario Mitter to get and numerically solve Dyson-Schwinger equations: DoFun anc CrasyDSE (thanks to Markus Huber for help). I suggest to play with them for numerical explorations.

Marc Bellon, Gustavo S. Lozano, & Fidel A. Schaposnik (2007). Higher loop renormalization of a supersymmetric field theory Phys.Lett.B650:293-297,2007 arXiv: hep-th/0703185v1

Edward Witten (1982). Constraints on Supersymmetry Breaking Nuclear Physics B, 202, 253-316 DOI: 10.1016/0550-3213(82)90071-2

Marco Frasca (2013). Numerical study of the Dyson-Schwinger equations for the Wess-Zumino
model arXiv arXiv: 1311.7376v1

Marco Frasca (2012). Chiral Wess-Zumino model and breaking of supersymmetry arXiv arXiv: 1211.1039v1

Markus Q. Huber, & Jens Braun (2011). Algorithmic derivation of functional renormalization group equations and
Dyson-Schwinger equations Computer Physics Communications, 183 (6), 1290-1320 arXiv: 1102.5307v2

Markus Q. Huber, & Mario Mitter (2011). CrasyDSE: A framework for solving Dyson-Schwinger equations arXiv arXiv: 1112.5622v2

Filed under: Computer Science, Particle Physics, Physics Tagged: Dyson-Schwinger equations, Supersymmetry, Supersymmetry breaking, Wess-Zumino model, Witten index

Peter Coles - In the Dark

Caught in the Middle

Academics these days are caught between a rock and a hard place.

On one side we have a government which seems not only malevolent but also utterly incompetent. I cite the recent example of the Department of Business Innovation and Skills, which has completely lost control of its budget, meaning that further cuts are likely to a higher education sector already struggling to cope with the instability generated by constant meddling from successive governments.

On the other we have our students, who are definitely getting a very raw deal compared with those of my generation. Most are justifiably  unhappy with the high level of fees they have to pay. Many also feel generally alienated by the way the country is run, for the benefit of the rich  at the expense of the young and the poor. Recent campus protests across the country are clearly a manifestation of this groundswell of resentment, although in some cases they have clearly been hijacked by extremist elements who will protest about anything at the drop of a hat just for the sake of it.

In between we have us academics, the vast majority of whom agree with the students  that UK higher education is in a complete mess and that the UK government is responsible. However, most of us also believe in the importance of universities as places of research, scholarship and teaching and want to carry out those activities as best we can for the benefit not only of our current students but for society as a whole.

So what should we academics who find ourselves caught  in the middle do?

Unsurprisingly, opinions differ and I don’t claim to speak for anyone but myself when I state mine. I think it’s the responsibility of academic staff to recognize the burden placed on our students by government and in the light of that do absolutely everything in our power to give them the best education we can. That means ensuring that as much of the money coming into universities from tuition fees goes directly towards improving the education of students – better teaching facilities, more and better trained staff and a better all-round experience of campus life. That is the reason that I did not participate in the recent strikes over pay: I absolutely refuse to take any action that would be in any way detrimental to the education of students in my School. Call me a scab if you wish. My conscience is clear. For me it’s not a matter of choice, it’s a matter of responsibility.

So what about the recent wave of student protests? Again, all I can do is give my own opinion (not that of my employer or anyone else) which is that I believe in the right to protest – as long as it’s peaceful – but targeting universities is short-sighted and counterproductive.  I’m sure that all the government is delighted that none of the latest protests have been in Whitehall, which is where the focus of complaint should be, but instead dissipated at arms length in a series of futile and divisive campus demonstrations.

And if one of these protests causes enough disruption that it succeeds in closing down a university for good – and don’t tell me that this government won’t allow that to happen – what good will that have done?

Andrew Jaffe - Leaves on the Line

Teaching mistakes

The academic year has begun, and I’m teaching our second-year Quantum Mechanics course again. I was pretty happy with last year’s version, and the students didn’t completely disagree.

This year, there have been a few changes to the structure of the course — although not as much to the content as I might have liked (“if it ain’t broke, don’t fix it”, although I’d still love to use more of the elegant Dirac notation and perhaps discuss quantum information a bit more). We’ve moved some of the material to the first year, so the students should already come into the course with at least some exposure to the famous Schrödinger Equation which describes the evolution of the quantum wave function. But of course all lecturers treat this material slightly differently, so I’ve tried to revisit some of that material in my own language, although perhaps a bit too quickly.

Perhaps more importantly, we’ve also changed the tutorial system. We used to attempt an imperfect rendition of the Oxbridge small-group tutorial system, but we’ve moved to something with larger groups and (we hope) a more consistent presentation of the material. We’re only on the second term with this new system, so the jury is still out, both in terms of the students’ reactions, and our own. Perhaps surprisingly, they do like the fact that there is more assessed (i.e., explicitly graded, counting towards the final mark in the course) material — coming from the US system, I would like to see yet more of this, while those brought up on the UK system prefer the final exam to carry most (ideally all!) the weight.

So far I’ve given three lectures, including a last-minute swap yesterday. The first lecture — mostly content-free — went pretty well, but I’m not too happy with my performance on the last two: I’ve made a mistake in each of the last two lectures. I’ve heard people say that the students don’t mind a few (corrected) mistakes; it humanises the teachers. But I suspect that the students would, on the whole, prefer less-human, more perfect, lecturing…

Yesterday, we were talking about a particle trapped in a finite potential well — that is, a particle confined to be in a box, but (because of the weirdness of quantum mechanics) with some probability of being found outside. That probability depends upon the energy of the particle, and because of the details of the way I defined that energy (starting at a negative number, instead of the more natural value of zero), I got confused about the signs of some of the quantities I was dealing with. I explained the concepts (I think) completely correctly, but with mistakes in the math behind them, the students (and me) got confused about the details. But many, many thanks to the students who kept pressing me on the issue and helped us puzzle out the problems.

Today’s mistake was less conceptual, but no less annoying — I wrote (and said) “cotangent” when I meant “tangent” (and vice versa). In my notes, this was all completely correct, but when you’re standing up in front of 200 or so students, sometimes you miss the detail on the page in front of you. Again, this was in some sense just a mathematical detail, but (as we always stress) without the right math, you can’t really understand the concepts. So, thanks to the students who saw that I was making a mistake, and my apologies to the whole class.

UR #11: Our Galactic Magnetic Field and Stellar Autopsies

The undergrad research series is where we feature the research that you’re doing. If you’ve missed the previous installments, you can find them under the “Undergraduate Research” category here.

Did you do a summer REU? Working on your senior thesis? Getting an early start on a research project? We want to hear from you! Think you’re up to the challenge of describing your research carefully and clearly to a broad audience, in only one paragraph? Then send us a summary of it!

You can share what you’re doing by clicking on the “Your Research” tab above (or by clicking here) and using the form provided to submit a brief (fewer than 200 words) write-up of your work. The target audience is one familiar with astrophysics but not necessarily your specific subfield, so write clearly and try to avoid jargon. Feel free to also include either a visual regarding your research or else a photo of yourself.

We look forward to hearing from you!

************

Anna Ho
Massachusetts Institute of Technology (MIT)

Anna is a senior at MIT, majoring in physics. During the past two summers, she worked with Scott Ransom through the National Radio Astronomy Observatory REU program, and is currently preparing this work for publication.

The line-of-sight component of the galactic magnetic field strength, for 24 of the 35 millisecond pulsars in the globular cluster Terzan 5 (the 25th is not pictured, in order to more clearly show the gradient.) The field strength changes by 15-20% across the cluster, representing 0.1 µG variations across parsec scales.

Rotation Measures for Globular Cluster Pulsars as a Unique Probe of the Galactic Magnetic Field
As it travels through a magnetic field, a linearly-polarized signal rotates through an angle that is linearly proportional to its wavelength. The scaling factor is called the “rotation measure” (RM) and is a function of both the electron density and the magnetic field strength along the line of sight to the source. We have measured highly-precise RMs for 25 of the 35 millisecond pulsars (MSPs) in the globular cluster Terzan 5, using Green Bank Telescope radio observations at 1.5 GHz and 2 GHz. For each MSP, we use the ratio of RM to electron column density (dispersion measure) to extract the average magnetic field strength along the line of sight to the source. We find that the field strength varies by 15-20% across the cluster, indicating 0.1 µG fluctuations on parsec (several light-year) scales. This represents the first use of dense pulsar populations to probe the small-scale structure of the galactic magnetic field.

************

Ashley Villar
Massachusetts Institute of Technology (MIT)

Ashley is a physics major at MIT in Cambridge, MA. She conducted this study with Prof. Alicia Soderberg and her team in the summer of 2013 during the CfA REU program.

Stellar Autopsies
Core-collapse supernovae (SNe) are highly energetic, cosmic explosions caused by the death of massive stars. Long gamma-ray bursts (GRBs) are fast transients comparable to supernovae in electromagnetic energy. Recently, the two have been observationally linked. We currently believe that these simultaneous events are caused by the collapse of Wolfe-Rayet stars in which a fraction of the ejected material is funneled into a relativistic jet in our line-of-sight. This scenario has intrinsic variability which may depend on mass of the progenitor, the angular momentum, and asymmetries within the explosion. To explore the possibility of asymmetry, we study two supernovae associated with GRBs, SN 2006aj and SN 2003dh, which lie on far ends of the GRB-SNe spectrum. These objects were both extensively studied and modeled using data obtained shortly after peak magnitude, when the temperature of the explosion was high and the optical opacity was large. We study these objects when the optical opacity lowers and deeper layers of the explosion become visible. If asymmetry exists, using spherically symmetric models during different time frames should lead to inconsistent solutions. This study finds no discrepancy, so the explosions are likely to be relatively isotropic.

Geraint Lewis - Cosmic Horizons

The large-scale structure of the halo of the Andromeda Galaxy Part I: global stellar density, morphology and metallicity properties
And now for a major paper from the Pan-Andromeda Archaeological Survey (PAndAS), led by astronomer-extraordinare Rodrigo Ibata. I've written a lot about PAndAS over the years (or maybe a year and a bit I've been blogging here) and we've discovered an awful lot, but one of the key things we wanted to do is measure the size and shape of the stellar halo of the Andromeda Galaxy.

The stellar halo is an interesting place. It's basically made up of the first generation of stars that formed in the dark matter halo in which the spiral galaxy of Andromeda was born, and the properties of the halo are a measure of the  formation history of the galaxy, something we can directly compare to our theoretical models.

But there is always a spanner in the works, and in this case it is the fact that Andromeda, like the Milky Way, is a cannibal and has been eating smaller galaxies. These little galaxies get ripped apart by the gravitational pull of Andromeda, and their stars litter the halo in streams and clumps. As we've seen before, Andromeda has a lot of this debris scattered all over the place.

So, we are left with a problem, namely how do we see the stellar halo, which is quite diffuse and faint, underneath the prominent substructure? This is where this paper comes in.

Well, the first thing is to collect the data, and that's where PAndAS comes in. The below picture confirms just how big the PAndAS survey is, and just how long it took us to get data.
It always amazes me how small the spiral disk of Andromeda is compared to the area we surveyed, but that's what we need to see the stellar halo which should be a couple of hundred kiloparsecs in extent.

Taking the data is only the first step. The next step, the calibration of the data, was, well, painful. I won't go into the detail here, but if you are going to look for faint things, you really need to understand your data at the limit, to understand what's a star, what's a galaxy, what's just noise. There are lots of things you need to consider to make sure the data is nice, uniform and calibrated. But that's what we did :)

Once you've done that, we can ask where the stars are. And here they are;
As you can see, chunks and bumps everywhere, all the dinner leftovers of the cannibal Andromeda. And all of that stuff is in the way of finding the halo!

What do we do? We have to mask out the substructure and search for the underlaying halo. We are in luck, however, as we don't have one map of substructure, we have a few of them. Why? Well, I've written about this before, but the stars in the substructure come from different sized objects, and so them chemical history will be different; in little systems, the heavy elements produced in supernova explosions are not held by their gravitational pull, and so they can be relatively "metal poor", but in larger systems the gas can't escape and gets mixed into the next generation of stars, making them more
"metal-rich".

So, here's our masks as a function of the iron abundance compared to hydrogen.
We see that the giant stream is more metal rich, but as we go to metal poor we see the more extensive substructure, including the South West Cloud.

What do we find? Well, we see the halo (horrah!) and it does what it should - it is brightest near the main body of Andromeda, but gets fainter and fainter towards the edge. Here's a picture of the profile:
It's hard to explain just how faint the halo is, but it is big, basically stretching out to the edge of our PAndAS data, and then beyond, and looks like it accounts for roughly 10% of the stellar mass in Andromeda. It is not inconsequential!

But as we started out noting, its properties provide important clues to the very process of galaxy formation. And it appears that it looks like we would expect from our models of structure formation, with large galaxies being built over time through the accretion of smaller systems.

We're working on a few new tests of the halo, and should hopefully have some more results soon. But for now,  well done Rod!

The large-scale structure of the halo of the Andromeda Galaxy Part I: global stellar density, morphology and metallicity properties

We present an analysis of the large-scale structure of the halo of the Andromeda galaxy, based on the Pan-Andromeda Archeological Survey (PAndAS), currently the most complete map of resolved stellar populations in any galactic halo. Despite copious substructure, the global halo populations follow closely power law profiles that become steeper with increasing metallicity. We divide the sample into stream-like populations and a smooth halo component. Fitting a three-dimensional halo model reveals that the most metal-poor populations ([Fe/H]<-1.7) are distributed approximately spherically (slightly prolate with ellipticity c/a=1.09+/-0.03), with only a relatively small fraction (42%) residing in discernible stream-like structures. The sphericity of the ancient smooth component strongly hints that the dark matter halo is also approximately spherical. More metal-rich populations contain higher fractions of stars in streams (86% for [Fe/H]>-0.6). The space density of the smooth metal-poor component has a global power-law slope of -3.08+/-0.07, and a non-parametric fit shows that the slope remains nearly constant from 30kpc to 300kpc. The total stellar mass in the halo at distances beyond 2 degrees is 1.1x10^10 Solar masses, while that of the smooth component is 3x10^9 Solar masses. Extrapolating into the inner galaxy, the total stellar mass of the smooth halo is plausibly 8x10^9 Solar masses. We detect a substantial metallicity gradient, which declines from [Fe/H]=-0.7 at R=30kpc to [Fe/H]=-1.5 at R=150kpc for the full sample, with the smooth halo being 0.2dex more metal poor than the full sample at each radius. While qualitatively in-line with expectations from cosmological simulations, these observations are of great importance as they provide a prototype template that such simulations must now be able to reproduce in quantitative detail.

December 06, 2013

Quantum Diaries

Has there ever been a paradigm shift?

Yes, once!

Paradigm and paradigm shift are so over used and misused that the world would benefit if they were simply banned.  Originally Thomas Kuhn (1922–1996) in his 1962 book, The Structure of Scientific Revolutions, used the word paradigm to refer to the set of practices that define a scientific discipline at any particular period of time. A paradigm shift is when the entire structure of a field changes, not when someone simply uses a different mathematical formulation. Perhaps it is just grandiosity, everyone thinking their latest idea is earth shaking (or paradigm shifting), but the idea has been so debased that almost any change is called a paradigm shift, down to level of changing the color of ones socks.

The archetypal example, and I would suggest the only real example in the natural and physical sciences, is the paradigm shift from Aristotelian to Newtonian physics. This was not just a change in physics from the perfect motion is circular to an object either is at rest or moves at a constant velocity, unless acted upon by an external force but a change in how knowledge is defined and acquired. There is more here than a different description of motion; the very concept of what is important has changed. In Newtonian physics there is no place for perfect motion but only rules to describe how objects actually behave. Newtonian physics was driven by observation. Newton, himself, went further and claimed his results were derived from observation. While Aristotelian physics is broadly consistent with observation it is driven more by abstract concepts like perfection.  Aristotle (384 BCE – 322 BCE) would most likely have considered Galileo Galilei’s (1564 – 1642) careful experiments beneath him.  Socrates (c. 469 BC – 399 BC) certainly would have. Their epistemology was not based on careful observation.

While there have been major changes in the physical sciences since Newton, they do not reach the threshold needed to call them a paradigm shifts since they are all within the paradigm defined by the scientific method. I would suggest Kuhn was misled by the Aristotle-Newton example where, indeed, the two approaches are incommensurate: What constitutes a reasonable explanation is simply different for the two men. But would the same be true with Michael Faraday (1791 – 1867) and Niels Bohr (1885–1962) who were chronologically on opposite sides of the quantum mechanics cataclysm?  One could easily imagine Faraday, transported in time, having a fruitful discussion with Bohr. While the quantum revolution was indeed cataclysmic, changing mankind’s basic understanding of how the universe worked, it was based on the same concept of knowledge as Newtonian physics. You make models based on observations and validate them through testable predictions.  The pre-cataclysmic scientists understood the need for change due to failed predictions, even if, like Albert Einstein (1879 – 1955) or Erwin Schrödinger (1887 – 1961), they found quantum mechanics repugnant. The phenomenology was too powerful to ignore.

Sir Karl Popper (1902 – 1994) provided another ingredients missed by Kuhn, the idea that science advances by the bold new hypothesis, not by deducing models from observation. The Bohr model of the atom was a bold hypothesis not a paradigm shift, a bold hypothesis refined by other scientists and tested in the crucible of careful observation. I would also suggest that Kuhn did not understand the role of simplicity in making scientific models unique. It is true that one can always make an old model agree with past observations by making it more complex[1]. This process frequently has the side effect of reducing the old models ability to make predictions. It is to remedy these problems that a bold new hypothesis is needed. But to be successful, the bold new hypothesis should be simpler than the modified version of the original model and more crucially must make testable predictions that are confirmed by observation. But even then, it is not a paradigm shift; just a verified bold new hypothesis.

Despite the nay-saying, Kuhn’s ideas did advance the understanding of the scientific method. In particular, it was a good antidote to the logical positivists who wanted to eliminate the role of the model or what Kuhn called the paradigm altogether. Kuhn made the point that is the framework that gives meaning to observations. Combined with Popper’s insights, Kuhn’s ideas paved the way for a fairly comprehensive understanding of the scientific method.

But back to the overused word paradigm, it would be nice if we could turn back the clock and restrict the term paradigm shift to those changes where the before and after are truly incommensurate; where there is no common ground to decide which is better. Or if you like, the demarcation criteria for a paradigm shift is that the before and after are incommensurate[2]. That would rule out the change of sock color from being a paradigm shift. However, we cannot turn back the clock so I will go back to my first suggestion that the word be banned.

[1] This is known as the Duhem-Quine thesis.

[2] There are probably paradigm shifts, even in the restricted meaning of the word, if we go outside science. The French revolution could be considered a paradigm shift in the relation between the populace and the state.

Finding Relics of Galaxy Formation

A schematic illustration of galaxy morphological components. This shows the spiral arms and bulge of a spiral galaxy.

Most galaxies generally fall into two morphological types: spirals and ellipticals. Spiral galaxies are flat, thin disks, with several spiral arms, a central stellar bulge, and possibly a central bar. Elliptical galaxies are smooth, featureless, semi-spherical distributions of stars. Both types have a central nucleus which is believed to house a super massive black hole. Why do galaxies separate into these distinct populations? How do they form?

We aren’t completely sure, but the best theory to date is the hierarchical model of galaxy formation. In this model, galaxies formed in the early universe from the gradual accretion of many smaller galaxy-like clumps of stars, the galaxy “building blocks”. In fact, when we look out to the early universe, the galaxies that we see do appear to be smaller and less well-defined than galaxies in the present universe.

Of course, not every galaxy falls into one of these two morphological categories. A good deal are actually “irregular” galaxies that don’t show any of the clear morphological components of either spirals or ellipticals. The contention of this paper is that some of these irregular galaxies might actually be aged versions of those galactic building blocks that have survived to the present day. The authors use the Sloan Digital Sky Survey data set to search for these sorts of galaxies in the nearby universe.

An example of an elliptical galaxy (top) and a spiral galaxy (bottom). Note the spiral galaxy’s central bar and spiral arms.

Identifying “Genuine Irregular Galaxies”

Maybe some protogalaxy building blocks survived to the present day, but the trouble in finding them is that we know that irregular galaxies can be formed when ordinary galaxies have their morphologies disturbed when they crash into one another. A big part of the data analysis in this paper is dedicated to separating interacting irregular galaxies from genuine irregular galaxies (GIGs). Merging galaxies are fairly easy to identify as two galaxies either colliding or passing close to one another, disturbing each other gravitationally. Galaxies that have recently merged are merger remnants and they appear elongated, with wispy tails.

The authors use the criteria that the galaxy should not be interacting or show evidence of recent interaction, should not have a central bulge, bar, spiral arms, or nucleus, have no nearby partner galaxies, and should not have the smooth light curve of an elliptical galaxy. Altogether this narrows down their sample of GIGs to 33 objects from the SDSS data set, excluding about 50% of galaxies that were otherwise categorized as irregular.

These galaxies do appear to be genuinely irregular: they still have a large fraction of gas, meaning that they haven’t converted all their gas to stars yet; they don’t have a large abundance of heavier elements, meaning that their composition is closer to the primordial abundance of the early universe; they also appear to be less massive, less bright, and smaller than typical galaxies at this redshift. The lack of a bulge in these galaxies indicates that they have not formed a supermassive black hole, either.

All of this points to the fact that these galaxies are still at an early evolutionary phase, and they could help us understand something of the initial building blocks that came together to form spirals and ellipticals; this can in turn help us understand how galaxies form. This study also demonstrates that simple irregular morphological classification doesn’t distinguish between galaxies that are irregular due to interaction and those which are genuinely irregular.

Some of the galaxies classified as genuinely irregular. Notice the lack of smooth stellar distributions and absence of spiral arms, bars, or central nuclei. From Figure 6 in the text.

Sean Carroll - Preposterous Universe

The Spark in the Park

A few years ago, not long after we moved to LA, Jennifer and I got a call from some of the writers on the TV series BONES. There’s already a science component to the show, which features brainy forensic anthropologist Brennan (Emily Deschanel) and her team of lab mates working with fiery FBI agent Booth (David Boreanaz) to solve crimes, most of which involve skeletons and physical evidence in some crucial way. This time they needed some physics input, as they wanted the murderer to be a researcher who used their physics expertise to carry out the crime, and were looking for unusual but realistic ideas. We were able to provide some crucial sociological advice (no, professional research scientists probably wouldn’t meet at a Mensa conference) and consulted with experimentalist friends who would know how to use radioactive substances in potentially lethal ways. I won’t say who, exactly, but when the episode aired they ended up calling the research institute the Collar Lab.

Apparently physicists are a suspiciously violent bunch, because tonight’s episode features another scientist suspect, this time played by Richard Schiff of West Wing fame. I got a chance to consult once again, and this time contributed something a bit more tangible to the set: a collection of blackboards in the physicist’s office. (Which, as in all Hollywood conceptions, is a lot more spacious and ornate than any real physicist’s office I’ve ever seen.) You can see the actual work tonight (8pm ET/PT on Fox), but here’s one that I made up that they didn’t end up using.

It does look like our professor is a theoretical cosmologist of some sort, doesn’t it? The equations here will be familiar to anyone who has carefully read “Dynamical Compactification from de Sitter Space.” The boards that actually will appear on the show are taken mostly from “Attractor Solutions in Scalar-Field Cosmology” and “A Consistent Effective Theory of Long-Wavelength Cosmological Perturbations.” Hey, if I’m going to write down a bunch of equations, they might as well be my equations, right?

But I actually got to be a little more than just a technical scribe. (Although that’s not an unimportant role — not only are the equations themselves gibberish to non-experts, it’s difficult for someone who isn’t familiar with the notation to even accurately transcribe the individual symbols.) No spoilers, but the equation-laden blackboards actually play a prominent role in a scene that appears late in the episode, so I was able to provide an infinitesimally tiny amount of creative input. And the scene itself (the overall conception of which belongs to writers Emily Silver and Stephen Nathan) packs quite an emotional wallop, something not typically associated with a series of equations. I haven’t seen the finished episode yet, but it was a great experience to actually be present on set during filming and watch the sausage being made.

Peter Coles - In the Dark

Haikus for the Day

Invited guest of
the Japanese Embassy

“A Symposium”
they call this. Lectures followed
by wine (hopefully)..

Astronomy and
Space Science unite nations.
One cosmos for all!

Offers

Matt Strassler - Of Particular Significance

No Comet, But Two Crescents

I’m sure you’ve all read in books that Venus is a planet that orbits the Sun and is closer to the Sun than is the Earth. But why learn from books what you can check for yourself?!?

[Note: If you missed Wednesday evening's discussion of particle physics involving me, Sean Carroll and Alan Boyle, you can listen to it here.]

As many feared, Comet ISON didn’t survive its close visit to the Sun, so there’s no reason to get up at 6 in the morning to go looking for it. [You might want to look for dim but pretty Comet Lovejoy, however, barely visible to the naked eye from dark skies.] At 6 in the evening, however, there’s good reason to be looking in the western skies — the Moon (for the next few days) and Venus (for the next few weeks) are shining brightly there.  Right now Venus is about as bright as it ever gets during its cycle.

The very best way to look at them is with binoculars, or a small telescope.  Easily with the telescope, and less easily with binoculars (you’ll need steady hands and sharp eyes, so be patient) you should be able to see that it’s not just the Moon that forms a crescent right now: Venus does too!

If you watch Venus in your binoculars or telescope over the next few weeks, you’ll see proof, with your own eyes, that Venus, like the Earth, orbits the Sun, and it does so at a distance smaller than the distance from the Sun to Earth.

The proof is simple enough, and Galileo himself provided it, by pointing his rudimentary telescope at the Sun 400 years ago, and watching Venus carefully, week by week.  What he saw was this: that when Venus was in the evening sky (every few months it moves from the evening sky to the morning sky, and then back again; it’s never in both),

• it was first rather dim, low in the sky at sunset, and nearly a disk, though a rather small one;
• then it would grow bright, larger, high in the sky at sunset, and develop a half-moon and then a crescent shape;
• and finally it would drop lower in the sky again at sunset, still rather bright, but now a thin crescent that was even larger from tip to tip than before.

The reason for this is illustrated in the figure below, taken from this post [which, although specific in some ways to the sky in February 2012, still has a number of general observations about the skies that apply at any time.]

A planet (such as Mercury or Venus) with an orbit that is smaller than Earth’s has phases like the Moon.  The portion of Venus that is lit is a half-sphere (shown in light color); the portion of Venus we can see is a different half-sphere (dashed red lines); the overlap is shaped like the wedge of an orange and looks like a crescent in the sky.  But unlike the Moon, which is at a nearly fixed distance from Earth, such a planet appears to grow and shrink during its orbit round the Sun, due to its changing distance from Earth. It is always largest when a crescent and smallest when full, and is brightest somewhere in between.

So go dig out those binoculars and telescopes, or use Venus as an excuse to buy new ones! Watch Venus, week by week, as it grows larger in the sky and becomes a thinner crescent, moving ever closer to the sunset horizon.  And a month from now the Moon, having made its orbit round the Earth, will return as a new crescent for you to admire.

Of course there’s another proof that Venus is closer to the Sun than Earth is: on very rare occasions Venus passes directly between the Earth and the Sun.  No more of those “transits” for a long time I’m afraid, but you can see pictures of last June’s transit here, and read about the great scientific value of such transits here

Filed under: Astronomy Tagged: astronomy, moon, venus

Quantum Diaries

One giant leap for the Higgs boson

Both the ATLAS and CMS collaborations at CERN have now shown solid evidence that the new particle discovered in July 2012 behaves even more like the Higgs boson, by establishing that it also decays into particles known as tau leptons, a very heavy version of electrons.

Why is this so important? CMS and ATLAS had already established that the new boson was indeed one type of a Higgs boson. In that case, theory predicted it should decay into several types of particles. So far, decays into W and Z bosons as well as photons were well established. Now, for the first time, both experiments have evidence that it also decays into tau leptons.

The decay of a particle is very much like making change for a coin. If the Higgs boson were a one euro coin, there would be several ways to break it up into smaller coins, but, so far, the change machine seemed to only make change in some particular ways. Now, additional evidence for one more way has been shown.

There are two classes of fundamental particles, called fermions and bosons depending on their spin, their value of angular momentum. Particles of matter (like taus, electrons and quarks) belong to the fermion family. On the other hand, the particles associated with the various forces acting upon these fermions are bosons (like the photons and the W and Z bosons.).

The CMS experiment had already shown evidence for Higgs boson decays into fermions last summer with a signal of 3.4 sigma when combining the tau and b quark channels. A sigma corresponds to one standard deviation, the size of potential statistical fluctuations.  Three sigma is needed to claim evidence while five sigma is usually required for a discovery.

For the first time, there is now solid evidence from a single channel – and two experiments have independently produced it. ATLAS collaboration showed evidence for the tau channel alone with a signal of 4.1 sigma, while CMS obtained 3.4 sigma, both bringing strong evidence that this particular type of decays occurs.

Combining their most recent results for taus and b quarks, CMS now has evidence for decays into fermions at the 4.0 sigma level.

The data collected by the ATLAS experiment (black dots) are consistent with coming from the sum of all backgrounds (colour histograms) plus contributions from a Higgs boson going into a pair of tau leptons (red curve). Below, the background is subtracted from the data to reveal the most likely mass of the Higgs boson, namely 125 GeV (red curve).

CMS is also starting to see decays into pairs of b quarks at the 2.0 sigma-level. While this is still not very significant, it is the first indication for this decay so far at the LHC. The Tevatron experiments have reported seeing it at the 2.8 sigma-level. Although the Higgs boson decays into b quarks about 60% of the time, it comes with so much background that it makes it extremely difficult to measure this particular decay at the LHC.

Not only did the experiments report evidence that the Higgs boson decays into tau leptons, but they also measured how often this occurs. The Standard Model, the theory that describes just about everything observed so far in particle physics, states that a Higgs boson should decay into a pair of tau leptons about 8% of the time. CMS measured a value corresponding to 0.87 ± 0.29 times this rate, i.e. a value compatible with 1.0 as expected for the Standard Model Higgs boson. ATLAS obtained 1.4 +0.5 -0.4, which is also consistent within errors with the predicted value of 1.0.

One of the events collected by the CMS collaboration having the characteristics expected from the decay of the Standard Model Higgs boson to a pair of tau leptons. One of the taus decays to a muon (red line) and neutrinos (not visible in the detector), while the other tau decays into a charged hadron (blue towers) and a neutrino. There are also two forward-going particle jets (green towers).

With these new results, the experiments established one more property that was expected for the Standard Model Higgs boson. What remains to be clarified is the exact type of Higgs boson we are dealing with. Is this indeed the simplest one associated with the Standard Model? Or have we uncovered another type of Higgs boson, the lightest one of the five types of Higgs bosons predicted by another theory called supersymmetry.

It is still too early to dismiss the second hypothesis. While the Higgs boson is behaving so far exactly like what is expected for the Standard Model Higgs boson, the measurements lack the precision needed to rule out that it cannot be a supersymmetric type of Higgs boson. Getting a definite answer on this will require more data. This will happen once the Large Hadron Collider (LHC) resumes operation at nearly twice the current energy in 2015 after the current shutdown needed for maintenance and upgrade.

Meanwhile, these new results will be refined and finalised. But already they represent one small step for the experiments, a giant leap for the Higgs boson.

For all the details, see:

Presentation given by the ATLAS Collaboration on 28 November 2013

Presentation given by the CMS Collaboration on 3 December 2013

Pauline Gagnon

Quantum Diaries

Un pas de géant pour le boson de Higgs

Les collaborations ATLAS et CMS du CERN ont maintenant l’évidence que la nouvelle particule découverte en juillet 2012 se comporte de plus en plus comme le boson de Higgs. Les deux expériences viennent en fait de démontrer que le boson de Higgs se désintègre aussi en particules tau, des particules semblables aux électrons mais beaucoup plus lourdes.

Pourquoi est-ce si important? CMS et l’ATLAS avaient déjà établi que ce nouveau boson était bien un type de boson de Higgs. Si tel est le cas, la théorie prévoit qu’il doit se désintégrer en plusieurs types de particules. Jusqu’ici, seules les désintégrations en bosons W et Z de même qu’en photons étaient confirmées. Pour la première fois, les deux expériences ont maintenant la preuve qu’il se désintègre aussi en particules tau.

La désintégration d’une particule s’apparente beaucoup à faire de la monnaie pour une pièce. Si le boson de Higgs était une pièce d’un euro, il pourrait se briser en différentes pièces de monnaie plus petites. Jusqu’à présent, le distributeur de monnaie semblait seulement donner la monnaie en quelques façons particulières. On a maintenant démontré qu‘il existe une façon supplémentaire.

Il y a deux classes de particules fondamentales, appelées fermions et bosons selon la valeur de quantité de mouvement angulaire. Les particules de matière comme les taus, les électrons et les quarks appartiennent tous à la famille des fermions. Par contre, les particules associées aux diverses forces qui agissent sur ces fermions sont des bosons, comme les photons et les bosons W et Z.

L”été dernier, l’expérience CMS avait déjà apporté la preuve avec un signal de 3.4 sigma que le boson de Higgs se désintégrait en fermions en combinant leurs résultats pour deux types de fermions, les taus et les quarks b. Un sigma correspond à un écart-type, la taille des fluctuations statistiques potentielles. Trois sigma sont nécessaires pour revendiquer une évidence tandis que cinq sigma sont nécessaires pour clamer une découverte.

Pour la première fois, il y a maintenant évidence pour un nouveau canal de désintégration (les taus) – et deux expériences l’ont produit indépendamment. La collaboration ATLAS a montré la preuve pour le canal des taus avec un signal de 4.1 sigma, tandis que CMS a obtenu 3.4 sigma, deux résultats forts prouvant que ce type de désintégrations se produit effectivement.

En combinant leurs résultats les plus récents pour les taus et les quarks b, CMS a maintenant une évidence pour des désintégrations en fermions avec 4.0 sigma.

Les données rassemblées par l’expérience ATLAS (les points noirs) sont en accord avec la somme de tous les évènements venant du bruit de fond (histogrammes en couleur) en plus des contributions venant d’un boson de Higgs se désintégrant en une paire de taus (la ligne rouge). En dessous, le bruit de fond est soustrait des données pour révéler la masse la plus probable du boson de Higgs, à savoir 125 GeV (la courbe rouge).

CMS commence aussi à voir des désintégrations en paires de quarks b avec un signal de 2.0 sigma. Bien que ceci ne soit toujours pas très significatif, c’est la première indication pour cette désintégration jusqu’ici au Grand collisionneur de hadrons (LHC). Les expériences du Tevatron avaient rapporté l’observation de telles désintégrations à 2.8 sigma. Bien que le boson de Higgs se désintègre en quarks b environ 60 % du temps, il y a tant de bruit de fond qu’il est extrêmement difficile de mesurer ces désintégrations au LHC.

Non seulement les expériences ont la preuve que le boson de Higgs se désintègre en paires de taus, mais elles mesurent aussi combien de fois ceci arrive. Le Modèle Standard, la théorie qui décrit à peu près tout ce qui a été observé jusqu’à maintenant en physique des particules, stipule qu’un boson de Higgs devrait se désintégrer en une paire de taus environ 8 % du temps. CMS a mesuré une valeur correspondant à 0.87 ± 0.29 fois ce taux, c’est-à-dire une valeur compatible avec 1.0 comme prévu pour le boson de Higgs du Modèle Standard. ATLAS obtient 1.4 +0.5-0.4, ce qui est aussi consistent avec la valeur de 1.0 à l‘intérieur des marges d’erreur.

Un des événements captés par la collaboration CMS ayant les caractéristiques attendues pour les désintégrations du boson de Higgs du Modèle Standard en une paire de taus. Un des taus se désintègre en un muon (ligne rouge) et en neutrinos (non visibles dans le détecteur), tandis que l’autre tau se désintègre en  hadrons (particules composées de quarks) (tours bleues) et un neutrino. Il y a aussi deux jets de particules vers l’avant (tours vertes).

Avec ces nouveaux résultats, les expériences ont établi une propriété de plus prédite pour le boson de Higgs du Modèle Standard. Reste encore à clarifier le type exact de boson de Higgs que nous avons. Est-ce bien le plus simple des bosons, celui associé au Modèle Standard? Ou avons nous découvert un autre type de boson de Higgs, le plus léger des cinq bosons de Higgs prévus par une autre théorie appelée la supersymétrie.

Il est encore trop tôt pour écarter cette deuxième hypothèse. Tandis que le boson de Higgs se comporte jusqu’ici exactement comme ce à quoi on s’attend pour le boson de Higgs du Modèle Standard, les mesures manquent encore de précision pour exclure qu’il soit de type supersymétrique. Une réponse définitive exige plus de données. Ceci arrivera une fois que le LHC reprendra du service à presque deux fois l’énergie actuelle en 2015 après l’arrêt actuel pour maintenance et consolidation.

En attendant, ces nouveaux résultats seront affinés et finalisés. Déjà ils représentent un petit pas pour les expériences et un bond de géant pour le boson de Higgs.

Pour tous les détails (en anglais seulement)

Présentation donnée par la collaboration ATLAS le 28 novembre 2013

Présentation donnée par la collaboration CMS le 3 décembre 2013

Pauline Gagnon

Pour être averti-e lors de la parution de nouveaux blogs, suivez-moi sur Twitter: @GagnonPauline ou par e-mail en ajoutant votre nom à cette liste de distribution.

CERN Bulletin

"Particle Fever": avant-premiere at CERN Main auditorium on 10 December, at 19:30
CERN people will have the chance to see a preview of the film "Particle Fever" in CERN's main auditorium on Tuesday 10 December at 19:30. The director, Mark Levinson, will be in attendance to speak with the audience about the film after the screening.   CERN and its experiments have been the focus of innumerable television documentaries, news reports, and other media productions.  However, until now, no film about the search for the Higgs Boson has been made for theatrical release in the classic documentary tradition.  "Particle Fever" has received numerous awards and travelled to festivals around the world, where it has consistently played to sell-out audiences.  The film will begin a commercial theatrical run in the United States in early 2014, but CERN people have the chance to see a preview in the CERN main auditorium on Tuesday, 10 December at 19:30.   The director, Mark Levinson, has worked on films such as "The English Patient" and "Cold Mountain", but he also has a PhD in physics. Mark will be in attendance to speak with the audience about the film after the screening.

CERN Bulletin

CinéGlobe invites you to participate in a poster design competition
For its 2014 publicity campaign, CinéGlobe invites CERN people to participate in a poster design competition.  The entries are now on display on the Pas Perdus in the main building, and the CERNois are invited to vote for their favourites.    CinéGlobe is the international festival of short films inspired by science that takes place every two years at CERN, in the Globe of Science and Innovation. From 18 to 23 March 2014, CERN will host the fourth edition of the festival. The mission of the CinéGlobe Film Festival is to challenge the commonly perceived divisions between science and art by demonstrating that they are both essential to interpreting our world. Open to short film creators from around the world, the CinéGlobe festival is truly international, the first three editions having attracted more than 4,000 entries from more than 100 countries around the globe.  In addition to screening some 60 short films, CinéGlobe also hosts musical events, special feature film screenings and panel discussions, open to all both inside and outside CERN.   To vote for the best poster, use the ballot box on the Pas Perdus. For further information, please email info@cineglobe.ch.

CERN Bulletin

Interfon
www.interfon.fr   Rendez-vous sur notre site pour toutes les « News » Interfon « News » « Nouveaux partenaires chez Interfon » Sociétaire Interfon ! Profitez des conditions particulières de nos nouveaux fournisseurs.   Les avantages Db Acoustic : laboratoire équipé des outils les plus récents, bilan auditif gratuit, hautes technologies aux meilleurs prix, essai gratuit d’une aide auditif pendant 3 semaines, protection anti-bruit / anti-eau sur mesure, prêt d’un appareil en cas de panne.   Plus de renseignements : Damien  Boch, audioprothésiste DE, 8 ans d’expérience en Suisse. Tél. (+33) 09 50 37 04 40 - email :  damien.boch@db-acoustic.fr Remise aux sociétaires  7 % dB Acoustic Monsieur Boch Damien 34, Avenue de la République 01630 – Saint-Genis-Pouilly   Y.O.C.O Estate SARL 13B chemin du Levant, 01210   Ferney-Voltaire Y.O.C.O Estate / Bassin Lémanique, Pays de Gex et Haute-Savoie L’immobilier sur mesure personnalisé : nous ne vendons pas une maison, un appartement, un terrain… Nous trouvons le bien immobilier qui correspond à vos attentes, nous établissons une recherche sur mesure. Fini le casse tête pour trouver son bien immobilier, ou la perte de temps passé à effectuer des visites non ciblées sur son temps de travail et sur sa vie privée, Finies les inquiétudes pour trouver son bon emplacement dans une région avec son marché de l’immobilier complexe. Confiez votre projet à Mr Estier Ludovic : Ludovic.estier@yocoestate.com - +33 (0) 6 25 78 35 33 5 % de remise aux sociétaires (sur le montant TTC de votre prestation) www.yocoestate.com   Fermeture des bureaux pour les fêtes de fin d’année Nos bureaux seront fermés pour : le bureau du CERN :  du samedi 21 décembre au dimanche 5 janvier 2014 inclus. le bureau de Saint-Genis-Pouilly :  du mardi 24 décembre au dimanche 5 janvier 2014 inclus. Réouverture le lundi 6 janvier 2014 « Bonnes vacances à tous » Permanences du CERN (Bât 504) : Interfon : du lundi au vendredi (12 h 30 à 15 h 30) tél. 73339 e-mail : interfon@cern.ch. Mutuelle : les jeudis « semaines impaires » (13 h 00 à 15 h 30) tél. 73939, e-mail : interfon@adrea-paysdelain.fr. Bureaux du Technoparc à Saint-Genis-Pouilly : Interfon et Mutuelle : du lundi au vendredi (13 h 30 à 17 h 30) Coopérative : 04 50 42 28 93 interfon@cern.ch Mutuelle : 04 50 42 74 57 interfon@adrea-paysdelain.fr.

CERN Bulletin

Do not miss the Announcements and Events sections of the Bulletin!
The infirmary closure, the start of Horizon 2020, information on the CERN car stickers 2014, an exclusive avant-première, and the Globe Christmas lecture... this information is FOR YOU!

Lubos Motl - string vacua and pheno

Doubly protected Higgs is naturally natural
Nathaniel Craig (now Rutgers) and Kiel Howe (Stanford) released an interesting preprint
Doubling down on naturalness with a supersymmetric twin Higgs
which provides a very nice explicit example why one should never be too ambitious when deducing consequences of naturalness – why "small unnaturalness" is never a problem or a problem that may be solved by a better model.

They consider an extension of the Minimal Supersymmetric Standard Model which protects the Higgs boson by two protection mechanisms. One of them is the supersymmetry, in the usual sense, and the other protection mechanism is (in their particular case) the twin Higgs mechanism.

Supersymmetry has been discussed on this blog frequently – although just a small fraction of the 391 TRF blog entries with the word "supersymmetry" say something really nontrivial about SUSY.

However, as far as I know, I haven't discussed the "twin Higgs models". They are an alternative approach that may explain the lightness of the observed Higgs boson – and as these authors argue in their new paper, this approach may be particularly powerful when it is combined with SUSY.

Many people who are told about the $$E_8\times E_8$$ heterotic string theory (or Hořava-Witten heterotic M-theory) models describing the real Universe like to propose that the "other $$E_8$$ group" could have the same fate as "our $$E_8$$" and it could also be broken to the Standard Model group etc. The states charged under the "other Standard Model group" would represent a shadowy dark sector (you may visualize it as the opposite side of the Hořava-Witten desk-shaped world) that interacts with us weakly (gravitationally plus by some weak interactions) and that is otherwise "analogous" (if not "identical") to the particles we know. Just to be sure, the normal assumption is that the fate of the other $$E_8$$ is different and it may remain unbroken while its gauginos ignite the supersymmetry breaking through their condensate.

So they take two copies of the MSSM and assume a complete $$\ZZ_2$$ symmetry for them, at least for the Higgs potential terms. Accidentally, this $$\ZZ_2$$ is sufficient to guarantee the full $$U(4)$$ symmetry – with the usual "additive" embedding of $$U(2)\times U(2)$$ of both "Standard Models" – for all perturbative terms in the potential. Such a situation is known as an "accidental symmetry", in this case $$U(4)$$.

At the end, the symmetry is broken by other phenomena and the Goldstone theorem guarantees massless or – because the symmetry isn't really exact – light scalar bosons. The observed $$125.6\GeV$$ Higgs boson from the LHC is an example of such light scalar bosons. The two worlds are not completely decoupled in their models, however. A chiral superfield $$S$$ interacts with both worlds via the $$SS$$ superpotential.

The degree of fine-tuning is very small in their model even if the top squarks lie above $$3\TeV$$. The Higgsinos are recommended to be around $$1\TeV$$ which – I note – happens to agree with the estimate of the nearly pure Higgsino LSP in an unconstrained MSSM where the mass was computed as a best fit. I guess that the normal MSSM may still be embedded into the doubled one so the two papers aren't quite incompatible and one could say that there's "diverse evidence" to think that the LSP is a Higgsino and near $$1\TeV$$.

Of course, I am not promising you that a model like that has to be right. It's just interesting and intriguing to know about this "spot" which is more likely than some generic points of the parameter space.

I would like to mention one more paper, one by Norma Susana Mankoč Borštnik of Slovenia (that's one female name, not two or four),
Spin-charge-family theory is explaining appearance of families of quarks and leptons, of Higgs and Yukawa couplings,
that almost claims to have "a theory of everything" unifying the spectrum of the Standard Model including several families into a single representation. The main idea of the "spin-charge-family unification" isn't too different from the wrong claims made by Garrett Lisi and all people on the same frequency. Well, Borštnik's picture is less obviously wrong because she doesn't claim to include gravity – this is what is really impossible to get from similar naive models.

At any rate, all the lepton and quark fields of all generations are claimed to arise from a single chiral 64-dimensional spinor in 13+1 dimensions. You embed $$SO(3,1)$$ to $$SO(13,1)$$ in the obvious additive way and the remaining $$SO(10)$$ dimensions are "enough" to give you some additional degeneracy.

While it's not "immediately wrong", I think that much of my criticism against Garrett Lisi's and similar papers still holds. In particular, the $$SO(13,1)$$ symmetry isn't really exact or unbroken because that would require all the 13+1 dimensions to be uncompactified. So the symmetry has to be broken – morally by a compactification – and because she assumes that the representation theory for the spinors works just like in a flat 13+1-dimensional space, it looks like a toroidal compactification. But that would give us a bad, non-chiral theory. So the $$SO(13,1)$$ has to be broken explicitly, in a different way than the normal compactification, but then I don't understand the rules of the game. Why is she trusting this large symmetry to pick the spectrum if the symmetry is broken and cannot be trusted for most other questions?

I would like one of these attempts to be right but as far as I can say, all of them are wrong because they're using "extra dimensions" – something that is imposed upon us by string/M-theory – but without all the careful analyses of subtleties that string/M-theory demands along with the extra dimensions at the same moment.

Symmetrybreaking - Fermilab/SLAC

US particle physicists look to space

A panel met at SLAC National Accelerator Laboratory to look for promising routes to the study of dark matter, dark energy and other phenomena.

Early this week, about 150 particle physicists gathered at SLAC National Accelerator Laboratory to explore the future of particle physics with a special focus on topics connecting particle physics, cosmology and astrophysics.

Prevalence of Planets that Might be “Earth-like”, Maybe

Scientists who hunt exoplanets smell victory. They’re discovering smaller and smaller rocky planets, getting closer to completing the Copernican revolution. Many people are bullish on the chances of an Earth-twin waiting somewhere out there in the cosmos—out of reach, but just for now. Our current telescopes aren’t powerful enough to prove that a promising exoplanet is Earth 2.0, but we can estimate the prevalence of suitable candidates.

As it happens, finding planets that might be Earth 2.0 (Earth-sized with Earth-like orbits) was the sine qua non for NASA’s Kepler mission.

A few weeks ago, a headline-grabbing study became publicly available on the website of the Proceedings of the National Academy of Science. Led by Erik Petigura, a graduate student at the University of California, Berkeley, the authors mined the Kepler data for planets, focusing on >42,000 Sun-like stars (G and K types) with high apparent brightness in Kepler’s sample. Along with re-discovering hundreds of candidates from the extant Kepler list, they found several new planet candidates with long periods. Correcting for Kepler’s sensitivity, they computed the frequency of Earth-sized (~1-2 Earth-radii) planets with long orbital periods (<400 days).

Overall, Petigura et al. find that 11 ± 4% of Sun-like stars have at least one Earth-sized planet orbiting at a distance where it receives ~1-4 times the stellar radiation that Earth receives at the top of its atmosphere. Kepler’s main mission is over, but this estimate will be continually revisited and likely revised for many years.

Methods and Results

Petigura et al. wrote their own data analysis program (a “pipeline”) to identify transit events. Using quantitative criteria and visual inspection of >16,000 (!!) candidate light curves, they discovered or re-discovered 603 planet candidates. Roughly 10% of these detections are likely false positives—systems that masquerade as an Earth-sized planet orbiting a Sun-like star, but are actually quite different (a Neptune-size planet orbiting one star in a binary system, for instance). High-resolution spectroscopy from the Keck Telescope allowed the authors to measure stellar radii, and thus planetary radii, to within ~10% for all of their planet candidates with orbital periods >100 days. Figure 1 shows their planets (red dots) plotted against orbital period and planetary radius. They calculated the percentages of Sun-like stars that host planets within 23 bins (yellow/green boxes) of period/radius space.

Figure 1 (Petigura, et al. 2013, PNAS): The 603 planets discovered in the Kepler data are shown as red dots. The authors calculated the percentages of Sun-like stars that host planets in 23 binned regions of period/radius space (yellow/green boxes). Quoted uncertainties simply represent the formal errors associated with binomial (i.e., counting) statistics.

Extrapolating from the abundance of large planets with short (<200 days) periods, the authors calculated that ~6% of Sun-like stars have Earth-size planets with orbital periods of 200-400 days. Since the stellar irradiation a planet receives is inversely proportional to the square of the planet-star distance, the authors claimed that 22 ± 8% of Sun-like stars have Earth-sized planets that receive 0.25-4 times the amount of stellar radiation that Earth receives. This corresponds to orbital separations of 0.5-2.0 AU, where 1 AU is roughly the Earth-Sun separation.

(Ordinarily, my job here is to translate jargon-laden prose into something a non-specialist can understand. But Petigura et al. wrote an extraordinarily readable paper. I encourage you, dear reader, to peruse the whole thing for additional details.)

Planets in the “Habitable Zone”: Are they like Earth or just “Earth-like”?

Ultimately, everyone wants to characterize the atmospheres of Earth-sized planets—quantify their thermal structures and chemical compositions, look for water and oxygen, all that jazz. We can’t do any of that, at least not yet. But we can measure their orbital separations. Wouldn’t it be nice if orbital separation, which corresponds to the amount of stellar radiation a planet receives, determined atmospheric properties, and thus surface conditions and habitability? This is where the idea of a “habitable zone” comes in.

Petigura et al. define the habitable zone (HZ) first in a big, blue box on their first page as “where conditions permit surface liquid water.” Later, more verbosely, the HZ is “the set of planetary orbits that permit liquid water on the surface.” Many different authors in previous papers used similar definitions. However, no precise definition of a HZ is universally accepted. Petigura et al. mainly use the aforementioned region of 0.5-2.0 AU for simplicity; the original definition for the Sun was 0.95-1.37 AU (i.e., between Venus and Mars), considerably more conservative. Because the boundaries of the HZ are so fluid, Petigura et al. calculate abundance statistics for four different definitions of the HZ. An Earth-sized planet in the habitable zone, in their argot, is an “Earth-like” planet.

These terms are useful shorthand, but, awkwardly, planets in the “habitable zone” aren’t necessarily habitable and “Earth-like” planets certainly aren’t all like Earth—in the sense of oceans, continents, and critters. Astronomers are all too aware of these niggling caveats, but they typically don’t enumerate them in exhaustive detail.

The habitable zone, as originally defined, is only benevolent to planets identical to Earth. For instance, Earth might survive if you nudged it a little closer to Venus (say, to 0.95 AU), but Venus wouldn’t become habitable even at 1 AU. (Venus is the planet of death, with a thick carbon dioxide atmosphere and surface temperatures high enough to melt lead.) Nowadays, it’s even more confusing—people invoke utterly different atmospheric compositions to expand their definitions of the HZ. At 2 AU, for instance, oceans would freeze on Earth, but maybe not on a planet with a thick hydrogen atmosphere and thus a high greenhouse effect.

Fundamentally, planetary habitability depends on chemistry and the stochastic process of planetary formation and evolution. Moreover, virtually all astronomers agree that life may thrive in many exotic environments, like the oceans of icy moons and planets lost in interstellar space, that are outside any contemporary definition of the HZ.

We might learn that nearly all Earth-sized planets in the HZ are actually inhospitable. But maybe, just maybe, some of these candidates are clement, and the golden age of discovery for which we’ve been hankering will have arrived.

December 05, 2013

Clifford V. Johnson - Asymptotia

Goodbye, Nelson Mandela
Rest in peace… but let your legacy and the lessons of your actions and words forever stay alive and working in our societies worldwide. -cvj Click to continue reading this post

Observing the Next Galactic Supernova

Title: Observing the Next Galactic Supernova
Authors: Scott Adams, C.S. Kochanek, John Beacom, Mark Vagins, K.Z. Stanek
First Author’s Institution: The Ohio State University

Kepler’s drawing of SN 1604. The supernova is the star with the letter “N” pointing at it. in Ophiucus’ left foot.

The last time a supernova was observed within the Milky Way was in 1604 by Johannes Kepler, and was only appreciated by the human eye, since optical telescopes and other measurement devices had not yet been invented. Despite a lack of hard observational data, astronomers have a theoretical framework to describe the processes that occur during a supernova, and numerical simulations are always growing more detailed and sophisticated. Still, without observation, neither theory nor numerical result can be put to the test.

While supernovae in our galaxy are relatively rare, extragalactic supernovae are not. That is because there are countless galaxies that have supernova rates similar to that of the Milky Way. But, due to their distance from Earth are not resolvable and offer little insight into the mechanisms at work during the explosion. Although astronomers haven’t observed supernovae in the Milky Way for several hundred years (read on to find out why this may be), the good news here is that astronomers are developing methods to be ready when the next one happens..

The authors of this paper discuss the likelihood of predicting where in the sky the next supernova will appear and how it should be observed. The trick to predicting their appearance lies with neutrinos–the tiny, light-weight, high-speed, neutral particles that pervade the whole of the Universe. We’ve written about neutrinos a few times here at Astrobites, most recently earlier this week. Neutrinos are produced in a variety of contexts, and they happen to play a special part in the supernova process.

During the collapse of a massive star’s core, neutrinos are produced in huge quantities. Because they are so tiny and interact so weakly with their surroundings, they leave the star before the shock wave that produces the explosion reaches the outer layers of the star. That is, a burst of neutrinos leave the star before the explosion becomes visible. A handful of neutrino detectors around the globe are capable of detecting this neutrino burst, and are part of a network called SNEWS (The SuperNova Early Warning System) that will alert astronomers when that burst is observed.

Astronomers will then have minutes or hours or maybe even days to position their telescopes before the light from the explosion reaches Earth. But, even after all of that, what are the odds that we could actually see the explosion? After all, the predicted occurrence rate is 2-3 per century, yet the last observed explosion happened in 1604. What gives? The most likely scenario is that galactic supernovae have occurred since then, but their visibility has been obscured by dust in the galactic plane.

The authors conducted a statistical study of possible galactic supernovae and found that with our modern fleet of optical telescopes and neutrino observatories, the probability of being able to observe the next galactic supernova is approximately 100 percent, even in visible wavelengths thanks to the sensitivity of space telescopes. This is exciting news for any astronomer who’s anxious to have the once-in-a-career opportunity to study a supernova “up close”. Until then, patience.

John Baez - Azimuth

Talk at the SETI Institute

SETI means ‘Search for Extraterrestrial Intelligence’. I’m giving a talk at the SETI Institute on Tuesday December 17th, from noon to 1 pm. You can watch it live, watch it later on their YouTube channel, or actually go there and see it. It’s free, and you can just walk in at 189 San Bernardo Avenue in Mountain View, California, but please register if you can.

Life’s Struggle to Survive

When pondering the number of extraterrestrial civilizations, it is worth noting that even after it got started, the success of life on Earth was not a foregone conclusion. We recount some thrilling episodes from the history of our planet, some well-documented but others merely theorized: our collision with the planet Theia, the oxygen catastrophe, the snowball Earth events, the Permian-Triassic mass extinction event, the asteroid that hit Chicxulub, and more, including the global warming episode we are causing now. All of these hold lessons for what may happen on other planets.

If you know interesting things about these or other ‘close calls’, please tell me! I’m still preparing my talk, and there’s room for more fun facts. I’ll make my slides available when they’re ready.

The SETI Institute looks like an interesting place, and my host, Adrian Brown, is an expert on the poles of Mars. I’ve been fascinated about the water there, and I’ll definitely ask him about this paper:

• Adrian J. Brown, Shane Byrne, Livio L. Tornabene and Ted Roush, Louth crater: Evolution of a layered water ice mound, Icarus 196 (2008), 433–445.

Louth Crater is a fascinating place. Here’s a photo:

By the way, I’ll be in Berkeley from December 14th to 21st, except for a day trip down to Mountain View for this talk. I’ll be at the Machine Intelligence Research Institute talking to Eliezer Yudkowsky, Paul Christiano and others at a Workshop on Probability, Logic and Reflection. This invitation arose from my blog post here:

If you’re in Berkeley and you want to talk, drop me a line. I may be too busy, but I may not.

Symmetrybreaking - Fermilab/SLAC

10 journals to go open-access in 2014

As part of the SCOAP3 publishing initiative, 10 journals in high-energy physics will offer unrestricted access to their peer-reviewed articles, starting January 1.

At the start of the new year, about 60 percent of the scientific articles in the field of high-energy physics will become freely available online as part of the largest-scale global open-access initiative ever built.

Thanks to a CERN-based publishing initiative called the Sponsoring Consortium for Open Access Publishing in Particle Physics, or SCOAP3, articles from 10 peer-reviewed journals will be available online; authors will retain their copyrights; and new licenses will enable wide re-use of content.

arXiv blog

Physicists Discover World's First Naturally Occurring Topological Insulator

The were first predicted in 2005 and first synthesized in the lab in 2008. Now physicists have discovered a naturally occurring topological insulator that can be mined from the earth’s crust.

Topological insulators are one of the more exciting new materials in science. This stuff is odd because is a conductor on the surface but an insulator inside, rather like a block of ice in which melting water flows around the outside but is trapped as a solid in the middle.

Symmetrybreaking - Fermilab/SLAC

First particle-antiparticle collider now historic site

The European Physical Society has declared the construction site of the Anello di Accumulazione collider in Frascati, Italy, a significant site in physics history.

Measuring roughly 4 feet in diameter and claiming an operational life of only a few years, the Anello di Accumulazione—an early 1960s particle collider (pictured above)—is outwardly unassuming. But the modest machine enabled a new chapter of accelerator physics: It was the first particle-antiparticle collider and the first electron-positron storage ring.

Tommaso Dorigo - Scientificblogging

The Quote Of The Week - Higgs On Anderson's Role In The Higgs Mechanism
"During the years 1962 to 1964 a debate developed about whether the Goldstone theorem could be evaded. Anderson pointed out that in a superconductor the Goldstone mode becomes a massive plasmon mode due to its electromagnetic interaction, and that this mode is just the longitudinal partner of transversely polarized electromagnetic modes, which also are massive (the Meissner effect!). Ths was the first description of what has become known as the Higgs mechanism.

The Great Beyond - Nature blog

More science funding for UK universities

Science played only a minor role in today’s key statement on government spending from George Osborne, the United Kingdom’s chancellor of the exchequer. But he did promise more funding for science courses at universities, as the government seeks to expand the number of students in higher education. To this end, 30,000 extra university places will be created next year, and the current cap on numbers will be abolished entirely the year after that.

Osborne also promised that the United Kingdom would “push the boundaries of scientific endeavour, including in controversial areas”, and confirmed yesterday’s announcement that £270 million (US$441 million) will be invested in quantum technology. A road map for how the long-term capital spending announced earlier this year will be spent is to be produced for next year’s autumn statement. Green issues also featured in today’s speech. The government has been criticized recently for apparently rowing back on its promise to be the ‘greenest ever’. Osborne confirmed that some taxes on energy — including some so-called green levies — will be removed, but he said this would be done in a low-carbon way. A planned increase in tax on petrol will also be cancelled. The Great Beyond - Nature blog Researchers push for more funding as dementia cases rise The number of people living with dementia around the world is now estimated at 44 million, or up 22% from three years ago, according to a report released today by Alzheimer’s Disease International (ADI), a federation of Alzheimer’s associations around the world. The increase on the ADI’s previous finding is due at least in part to improved reporting of dementia prevalence in China and sub-Saharan Africa. And as people live longer, cases of dementia — a catch-all term describing the loss of memory, mental agility and understanding owing to neurodegenerative diseases such as Alzheimer’s — will rise to 76 million by 2030 and to 136 million by 2050, the ADI report says. “The current burden and future impact of the dementia epidemic has been underestimated,” it concludes. The report ratchets up the pressure on funders to invest more into tackling dementia ahead of an 11 December summit in London, at which the World Health Organization and ministers from the G8 (Group of Eight) countries will discuss a global action plan on the condition. “This is a once-in-a-generation opportunity to turn the tide on dementia,” Doug Brown, director of research and development at the Alzheimer’s Society, a charity based in London, told reporters at a briefing yesterday. “We need as much investment in dementia research as we have in cancer,” he said. Indeed, despite well-publicized political commitments — the United Kingdom’s prime minister David Cameron launched a ‘dementia challenge’ in March 2012, and the US government set out plans for extra Alzheimer’s funding in May 2012 — levels of funding remain low. In the United Kingdom, for example, dementia costs the economy £23 billion a year (though mostly not in front-line medical expenses), the Alzheimer’s Society estimates — which is twice the burden of cancer. But public research funding only amounts to some £60 million a year, and that is barely one-eighth of what is spent on cancer research. The problem is similar around the world, Brown says. Nick Fox, a neurologist who heads the Dementia Research Centre at University College London, says, more conservatively, that he hopes the G8 will double dementia funding in the next five years. Drugs designed to fight Alzheimer’s disease have proved disappointing in clinical trials so far. But, says Fox, “some of the trials have been like trying chemotherapy for cancer when the patient is already in a care hospice,” given that Alzheimer’s starts to attack the brain up to a decade before symptoms such as memory loss appear. In a new approach, at least four clinical trials are now planning to treat people who have not yet developed Alzheimer’s symptoms. One is a five-year trial of an antibody, crenezumab, which binds to fragments of neuron-damaging amyloid-β. The drug will be tested in people who carry a rare genetic mutation that makes them certain to get the disease. Another, the Dominantly Inherited Alzheimer’s Network study, will enrol patients with a possible familial risk for Alzheimer’s; a third, by companies Takeda (based in Osaka, Japan) and Zinfandel Pharmaceuticals (based on Durham, North Carolina), hopes to test an experimental drug in people whose genetic makeup suggests elevated risk of Alzheimer’s; and a fourth, known as the A4 study, will treat people who show biomarker evidence of amyloid plaques in positron-emission tomography. The ADI report adds that better care and timely diagnoses are important, too. And dementia is not just a disease of the well-off: though cases are concentrated in the richest and most demographically aged countries, 63% of people with dementia live in low- and middle-income countries where there is limited access to social services and support. December 04, 2013 arXiv blog The Future of Photography: Cameras With Wings (or Rotors) The next generation of photographers will think nothing of taking a camera out of their pocket and releasing it to the skies, like a dove to the wind. Lubos Motl - string vacua and pheno Making exceptional symmetries of SUGRA manifest I found at least two hep-th papers interesting today. Nathan Berkovits brings us some field redefinition that maps his pure spinor formalism to the RNS formalism, using a new method of "dynamical twisting". My understanding is that it's not sufficient to understand why the calculated amplitudes agree. But I will only discuss Exceptional Field Theory I: $$E_{6(6)}$$ covariant Form of M-Theory and Type IIB by Olaf Hohm and Henning Samtleben. The names may sound German to you but it's technically a French-American collaboration. ;-) I don't know the authors but I know all 4 people thanked in the acknowledgements, Liu, Nicolai, Taylor, and Zwiebach. Supergravity (or M-theory) compactified on tori produces lower-dimensional theories with non-compact exceptional continuous (or discrete) symmetries (called the U-duality group in the M-theory case). Exceptional groups are sexy and mysterious, too. It has always been plausible that a decent understanding of the origin of these exceptional symmetries could provide us with a new, spectacularly clear view into string theory's deepest inner workings. It could be just a straightforward technical result without far-reaching implications, too. We can't know for sure. Formulations that make duality symmetries of string/M-theory manifest became popular in recent years. In the case of T-duality, the part of the full U-duality group that is visible at every order of the weak coupling expansion of string theory, the paradigm carries the name "Double Field Theory" or "DFT". Both the circular compact coordinates and their T-dual partners are included as fields. It would be wrong to simply double-count so some constraint has to be added. The new paper extends these techniques to M-theory, beyond the perturbative stringy expansions, where the exceptional symmetries occur. The ordinary well-known exceptional symmetries appear on the compactification of M-theory on $$T^k$$ for $$k=6,7,8$$ where the noncompact symmetries are $$E_{k(k)}(\RR)$$ and U-duality groups are $$E_{k(k)}(\ZZ)$$. In this paper, they discuss the $$k=6$$ case. Note that the supergravity has six compact periodic dimensions but the smallest nontrivial representation of the $$E_6$$-like group is $$27$$-dimensional. So to make this group manifest, we clearly need to add $$27$$ compact dimensions (aside from the $$4+1$$ noncompact ones) and not just $$6$$ of them. The would-be SUGRA theory would have too many degrees of freedom. Most of them are removed again by the following clever $$E_6$$-covariant constraint involving a cubic symmetric tensor invariant of the group $$d^{MNP}$$:$d^{MNK} \partial_N\partial_K A = 0, \quad d^{MNK} \partial_N A \partial_K B = 0$ which should hold for all fields $$A$$ or $$A,B$$ in our $$5+27$$-dimensional spacetime. The bosonic SUGRA fields that live in this extended spacetime are the ordinary fünfbein $$e_\mu^a$$ that remembers the metric in the five noncompact dimensions; $${\mathcal M}_{MN}$$ which are the usual scalars in this supergravity living in the $$E_{6(6)}/USp(8)$$ coset; $$A_\mu^M$$ gauge fields which sort of remember the mixed compact-noncompact components of the metric in the Kaluza-Klein way; and $$B_{\mu\nu,M}$$ which are the newest fields of the "EFT", or "exceptional field theory", namely some new tensor gauge fields. The constraint generalizes (and was constructed by analogy from) the "strong constraint" in DFT which is a stronger version of $$L_0=\tilde L_0$$ needed when the T-dual coordinates are added. The action for these bosonic fields in the $$32$$-dimensional spacetime is kind of natural. I said that $$B$$ were tensor gauge fields so there is a new gauge symmetry for them which doesn't use the ordinary Lie brackets like Lie algebras do. Instead, it uses the E-brackets (equation 2.15). The commutator of gauge transformations dependent on parameter fields $$\Lambda_{1},\Lambda_{2}$$ is proportional to $$\Lambda\partial \Lambda$$ structures, perhaps with two copies of the invariant $$E_6$$ tensor $$d^{MNP}$$. This is pretty cool and the added single derivative on the right-hand side (relatively to the Yang-Mills gauge transformations' commutator which has no derivatives) makes the commutator somewhat supersymmetry-like (but all these objects are bosonic; nevertheless, this bosonic gauge symmetry seems as powerful as supersymmetry in determining all the couplings in the action). You should read the paper itself but I just end up this blog post by saying that they explicitly find the two solutions of the constraints above. One of them preserves $$GL(6)$$ and interprets the vacuum instantly as 11D SUGRA (M-theory...) on a 6-torus; the other one preserves $$GL(5)\times SL(2)$$ and interprets the vacuum as type IIB SUGRA (IIB string) on a five-torus. If that works and if fermions with SUSY may be added (it surely looks like a worm of can to consider $$256$$-dimensional chiral spinors that are minimal in $$32$$ dimensions, it looks like too much but maybe it is exactly OK for SUGRA), this is a new formulation of the maximally supersymmetric supergravity theories that makes the noncompact symmetry group manifest! That's cool but some further progress would probably be needed to define the whole UV-complete string/M-theory (in its maximally supersymmetric vacua) with the manifest U-duality groups. For example, we may ask: Is there a variation of this formalism that defines the so far unknown BFSS-like matrix model for M-theory on $$T^6$$? Or higher? The explanations of the origin of exceptional symmetries in string/M-theory have been among five top research topics of mine in the recent decade or so. I've actually reached conditions similar to the constraints above but from a very different perspective – from attempts to generalize the Dirac quantization rule and T-duality lattices to the exceptional nonperturbative case. The field-theory limit of my construction is probably described in this very paper and it hasn't been clear to me. I will have to think about it more. The authors have announced the exceptional field theory in an August 2013 [PRL] paper that I mostly missed and they are preparing the second part of the today's preprint, too. Tommaso Dorigo - Scientificblogging I Am Writing A Book Inspired by my friend Peter Woit's openness in discussing his work in progress (a thick textbook on the foundations of quantum mechanics), I feel compelled to come out here about my own project. read more The Great Beyond - Nature blog Q&A on the forest agreement in Warsaw Building on several years of negotiations, countries inked a major agreement on forest conservation at the United Nations climate talks in Warsaw on 23 November. The deal formally integrates forest conservation into the international climate agenda by enabling wealthy countries to offset their emissions by paying poor tropical countries to protect their forests. The framework is known as REDD+, which stands for Reducing Emissions from Deforestation and Forest Degradation. The plus sign refers to an additional component that would reward countries for enhancing their forests. Nature talked to Doug Boucher, director of the tropical forests and climate initiative at the Union of Concerned Scientists in Washington DC, about the agreement in Warsaw. How would this programme work? Results-based payments will start with countries fixing a ‘reference level’ — their emissions from deforestation before they start working to reduce it — and putting into place a forest-monitoring system to track emissions. Additionally, they would need to establish safeguards to protect the rights of Indigenous Peoples and environmental values such as biodiversity. They would then take actions, of many kinds, to deal with the drivers of deforestation… After several years, if they have been successful…they will be paid financial compensation. This can come from a variety of sources, including bilateral programs and multilateral programmes. The Green Climate Fund, which has been established and is going to start receiving contributions in the coming year, is expected to become one of the major sources of compensation in years to come. How much would it cost? And given the state of climate negotiations more broadly, is there financing to pay for it? The estimates of several years back generally agreed that we would need US$20 billion or so per year, once REDD+ got going globally in the 2020s. The experience of Brazil and Norway has shown that major emissions reductions can made at costs considerably less than such estimates would have predicted, but we’re not at all sure that this can be done in many other countries. So for the time being I think those older estimates are still our best guesses.

Right now, the financing isn’t there at that scale. More worrisome is that there’s no clear plan among donor countries to increase funding over the next several years… The Green Climate Fund has been established, but countries are going to have to fill it if they want REDD+ to succeed globally.

The agreement took longer than expected. Fears arose about ‘carbon cowboys’, social and environmental safeguards as well as the actual structure of the policy. What were some of the outstanding issues, and how were they resolved?

Safeguards were one of the major issues, and important decisions in Cancun, Durban and here in Warsaw put those into place. The technical aspects — especially how to establish reference levels and how to measure, report and verify reductions with respect to them — required a lot of complicated discussions, but ultimately they were resolved by establishing rules that were scientifically sound but not overly complicated, with technical assessment by the global scientific community. Financing took the longest, and although we now have the procedures for payment in place… we still don’t have the necessary money.

To benefit from forest management, countries must be able to monitor their forests. How will science play into the process?

Science will be critical to monitoring, and we’re fortunate that over the last decade or so…the technology necessary to monitor forests has improved significantly… There have also been important advances in our understanding of the drivers of deforestation and the global economic patterns, including trade and changing diets, that underlie them. So the science is now up to the task, if the resources to implement it are made available.

Where do we go from here? And how long will it take to begin implementing the agreement?

This can start for many countries in 2014, but as I mentioned, some have already been doing it, with considerable success. Leaders such as Brazil, Costa Rica, Vietnam and Norway have already been implementing REDD+ without waiting for the agreement in Warsaw. They’ve shown the way; other countries can now follow.

Matt Strassler - Of Particular Significance

Wednesday: Sean Carroll & I Interviewed Again by Alan Boyle

Today, Wednesday December 4th, at 8 pm Eastern/5 pm Pacific time, Sean Carroll and I will be interviewed again by Alan Boyle on “Virtually Speaking Science”.   The link where you can listen in (in real time or at your leisure) is

What is “Virtually Speaking Science“?  It is an online radio program that presents, according to its website:

• Informal conversations hosted by science writers Alan Boyle, Tom Levenson and Jennifer Ouellette, who explore the explore the often-volatile landscape of science, politics and policy, the history and economics of science, science deniers and its relationship to democracy, and the role of women in the sciences.

Sean Carroll is a Caltech physicist, astrophysicist, writer and speaker, blogger at Preposterous Universe, who recently completed an excellent and now prize-winning popular book (which I highly recommend) on the Higgs particle, entitled “The Particle at the End of the Universe“.  Our interviewer Alan Boyle is a noted science writer, author of the book “The Case for Pluto“, winner of many awards, and currently NBC News Digital’s science editor [at the blog  "Cosmic Log"].

Sean and I were interviewed in February by Alan on this program; here’s the link.  I was interviewed on Virtually Speaking Science once before, by Tom Levenson, about the Large Hadron Collider (here’s the link).  Also, my public talk “The Quest for the Higgs Particle” is posted in their website (here’s the link to the audio and to the slides).

Filed under: Astronomy, Higgs, History of Science, LHC Background Info, Particle Physics, Public Outreach, Quantum Gravity, Science News, The Scientific Process Tagged: astronomy, DarkMatter, DoingScience, gravity, Higgs, LHC, particle physics, PublicPerception, PublicTalks

Clifford V. Johnson - Asymptotia

New Tool
Actually, this new tool is pretty old school, and I love it! There are times when I want to have a change of venue while doing rather detailed work for The Project… perhaps go sit in a cafe for a change, instead of at the drawing desk. But when not at the drawing desk, I could not use the lovely large illuminated magnifying glass […] Click to continue reading this post

Clifford V. Johnson - Asymptotia

Fail Lab Episode 12 – Finale
….In which Crystal and many of the crew chill out on the sofa after a long hard season of shows and express some of their gut feelings about the whole business. Well done on an excellent series of shows, Crystal, Patrick, James, and all the other behind the scenes people! (Warning: - This episode may not be for the squeamish!) Embed below: […] Click to continue reading this post

December 03, 2013

Symmetrybreaking - Fermilab/SLAC

Science writer Mike Perricone presents his favorite books on particle physics and a recommended reading list for the LHC/Higgs Era (2008 to the present).

It’s been two decades and thousands of popular science books since particle physicist and former Fermilab director Leon Lederman first served up science on wry in The God Particle, his contemplation on the subatomic particles and forces that make up the fundamental building blocks of nature. Lederman’s devilish moniker for the elusive Higgs boson might have rubbed some scientists the wrong way, but the catchy name for the then-theoretical boson has proved a useful hook for science writers, editors and readers.

Matt Strassler - Of Particular Significance

The Fast and Glamorous Life of a Theoretical Physicist

Ah, the fast-paced life of a theoretical physicist!  I just got done giving a one-hour talk in Rome, given at a workshop for experts on the ATLAS experiment, one of the two general purpose experiments at the Large Hadron Collider [LHC]. Tomorrow morning I’ll be talking with a colleague at the Rutherford Appleton Lab in the U.K., an expert from CMS (the other general purpose experiment at the LHC). Then it’s off to San Francisco, where tomorrow (Wednesday, 5 p.m. Pacific Time, 8 p.m. Eastern), at the Exploratorium, I’ll be joined by Caltech’s Sean Carroll, who is an expert on cosmology and particle physics and whose book on the Higgs boson discovery just won a nice prize, and we’ll be discussing science with science writer Alan Boyle, as we did back in February. [You can click here to listen in to Wednesday's event.]  Next, on Thursday I’ll be at a meeting hosted in Stony Brook, on Long Island in New York State, discussing a Higgs-particle-related scientific project with theoretical physics colleagues as far flung as Hong Kong.  On Friday I shall rest.

“How does he do it?”, you ask. Hey, a private jet is a wonderful thing! Simple, convenient, no waiting at the gate; I highly recommend it! However — I don’t own one. All I have is Skype, and other Skype-like software.  My words will cross the globe, but my body won’t be going anywhere this week.

We should not take this kind of communication for granted! If the speed of light were 186,000 miles (300,000 kilometers) per hour, instead of 186,000 miles (300,000 kilometers) per second, ordinary life wouldn’t obviously change that much, but we simply couldn’t communicate internationally the way we do. It’s 4100 miles (6500 kilometers) across the earth’s surface to Rome; light takes about 0.02 seconds to travel that distance, so that’s the fastest anything can travel to make the trip. But if light traveled 186,000 miles per hour, then it would take over a minute for my words to reach Rome, making conversation completely impossible. A back-and-forth conversation would be difficult even between New York and Boston — for any signal to travel the 200 miles (300 kilometers) would require four seconds, so you’d be waiting for 8 seconds to hear the other person answer your questions. We’d have similar problems — slightly less severe — if the earth were as large as the sun.  And someday, as we populate the solar system, we’ll actually have this problem.

So think about that next time you call or Skype or otherwise contact a distant friend or colleague, and you have a conversation just as though you were next door, despite your being separated half-way round the planet. It’s a small world (and a fast one) after all.

Filed under: The Scientific Process Tagged: DoingScience, relativity

John Baez - Azimuth

Rolling Hypocycloids

The hypocycloid with n cusps is the curve traced out by a point on a circle rolling inside a circle whose radius is n times larger.

The hypocycloid with 2 cusps is sort of strange:

It’s just a line segment! It’s called the Tusi couple.

The hypocycloid with 3 cusps is called the deltoid, because it’s shaped like the Greek letter Δ:

The hypocycloid with 4 cusps is called the astroid, because it reminds people of a star:

I got interested in hypocycloids while writing some articles here:

My goal was to explain a paper John Huerta and I wrote about the truly amazing things that happen when you roll a ball on a ball 3 times as big. But I got into some interesting digressions.

While pondering hypocycloids in the ensuing discussion, the mathematician, physicist, programmer and science fiction author Greg Egan created this intriguing movie:

It’s a deltoid rolling inside an astroid. It fits in a perfectly snug way, with all three corners touching the astroid at all times!

Why does it work? It’s related to some things that physicists like: SU(3) and SU(4).

SU(n) is the group of n × n unitary matrices with determinant 1. Physicists like Pauli and Heisenberg got interested in SU(2) when they realized it describes the rotational symmetries of an electron. You need it to understand the spin of an electron. Later, Gell-Mann got interested in SU(3) because he thought it described the symmetries of quarks. He won a Nobel prize for predicting a new particle based on this theory, which was then discovered.

We now know that SU(3) does describe symmetries of quarks, but not in the way Gell-Mann thought. It turns out that quarks come in 3 ‘colors’—not real colors, but jokingly called red, green and blue. Similarly, electrons come in 2 different spin states, called up and down. Matrices in SU(3) can change the color of a quark, just as matrices in SU(2) can switch an electron’s spin from up to down, or some mix of up and down.

SU(4) would be important in physics if quarks came in 4 colors. In fact there’s a theory saying they do, with electrons and neutrinos being examples of quarks in their 4th color state! It’s called the Pati–Salam theory, and you can see an explanation here:

• John Baez and John Huerta, The algebra of grand unified theories.

It’s a lot of fun, because it unifies leptons (particles like electrons and neutrinos) and quarks. There’s even a chance that it’s true. But it’s not very popular these days, because it has some problems. It predicts that protons decay, which we haven’t seen happen yet.

Anyway, the math of SU(3) and SU(4) is perfectly well understood regardless of their applications to physics. And here’s the cool part:

If you take a matrix in SU(3) and add up its diagonal entries, you can get any number in the complex plane that lies inside a deltoid. If you take a matrix in SU(4) and add up its diagonal entries, you can get any number in the complex plane that lies inside an astroid. And using how SU(3) fits inside SU(4), you can show that a deltoid moves snugly inside an astroid!

The deltoid looks like it’s rolling, hence the title of this article. But Egan pointed out that it’s not truly ‘roll’ in the sense of mechanics—it slides a bit as it rolls.

But the really cool part—the new thing I want to show you today—is that this pattern continues!

For example, an astroid moves snugly inside a 5-pointed shape, thanks to how SU(4) sits inside SU(5). Here’s a movie of that, again made by Greg Egan:

In general, a hypocycloid with n cusps moves snugly inside a hypocycloid with n + 1 cusps. But this implies that you can have a hypocycloid with 2 cusps moves inside one with 3 cusps moving inside one with 4 cusps, etcetera! I’ve been wanting to see this for a while, but yesterday Egan created some movies showing this:

Depending on the details, you can get different patterns:

Egan explains:

In the first movie, every n-cusped hypocycloid moves inside the enclosing (n+1)-cusped hypocycloid at the same angular velocity and in the same direction, relative to the enclosing one … which is itself rotating, except for the very largest one. So the cumulative rotational velocity is highest for the line, lower for the deltoid, lower still for the astroid, in equal steps until you hit zero for the outermost hypocycloid.

In the second movie, the magnitude of the relative angular velocity is always the same between each hypocycloid and the one that encloses it, but the direction alternates as you go up each level.

Here’s one where he went up to a hypocycloid with 10 cusps:

Note how sometimes all the cusps meet!

I wonder, can a deltoid roll in a 5-hypocycloid? I haven’t worked through the math of just how the SU(n) → SU(n+1) embedding guarantees a fit here.

(And also, are there corresponding pictures we could draw to illustrate the embeddings of other Lie groups? This could be a lovely way to illustrate a wide range of relationships if it could be done more generally.)﻿

Egan figured out that yes, a deltoid can move nicely in a hypocycloid with 5 cusps:

He writes:

To make this, I took the border of a deltoid in “standard position” (as per the trace theorem) to be:

$\mathrm{deltoid}(t) = 2 \exp(i t) + \exp(-2 i t)$

Suppose I apply the linear function:

$f(s, z) = \exp(-2 i s / 5) z + 2 \exp(3 i s / 5)$

to the whole deltoid. Then at the point s=t on the deltoid, we have:

$f(s, \mathrm{deltoid}(s)) = 4 \exp(3 i s / 5) + \exp(-4 (3 i s / 5))$

which is a point on a 5-cusped hypocycloid in “standard position”. Of course with this choice of parameters you need to take $s$ from $0$ through to $10 \pi/3,$ not $2 \pi,$ to complete a cycle.

Using the same trick, you can get a hypocyloid with n cusps to move inside one with m cusps whenever n ≤ m. As for the more general question of how different Lie groups give different ‘generalized hypocycloids’, and how fitting one Lie group in another lets you roll one generalized hypocycloid in another, the current state of the art is here:

But there is still more left to do!

On a somewhat related note, check out this fractal obtained by rolling a circle inside a circle 4 times as big that’s rolling in a circle 4 times as big… and so on: astroidae ad infinitum!

arXiv blog

Data Mining Reveals the Secret to Getting Good Answers

If you want a good answer, ask a decent question. That’s the startling conclusion to a study of online Q&As.

If you spend any time programming, you’ll probably have come across the question and answer site Stack Overflow. The site allows anybody to post a question related to programing and receive answers from the community.

December 02, 2013

ZapperZ - Physics and Physicists

Genius Materials on the ISS
Advances in material science on the International Space Station.

Pay attention, kids. These are physics applications that have direct impacts on your lives. You are using, at this very moment, things that were first studied as part of physics/material science.

Zz.

Symmetrybreaking - Fermilab/SLAC

Baryonic acoustic oscillations

Scientists have found a way to study sound waves from the early universe to learn more about its history and contents.

Baryonic acoustic oscillations are sound waves from the early universe. Scientists have found a way to study these sound waves to learn more about the universe’s history and contents. Shortly after the big bang, when the universe was hot and dense, regions with greater concentrations of light and matter had higher pressure than others. Acoustic waves—governed by the same laws of physics that describe how sound travels to our ears—rippled outward from those high-pressure regions.

Tommaso Dorigo - Scientificblogging

Photos of Physicists In Their Environments
Just a quick link to allow you to browse a very nice set of pictures taken at CERN by Andri Pol. The subjects are physicists in their daily activities - brainstorming at the blackboard, cycling around the lab, bitching about the mess in the common coffee room, or working at various pieces of hardware:
you can see them here. Enjoy!

IceCube Detects Extraterrestrial Neutrinos

Figure 1. A diagram showing how the detectors in the IceCube grid measure the signal of a neutrino. The colors represent the time that each detector received the signal. This is an example of a “track” event.

Background: Neutrinos and IceCube

Most of astronomy focuses on measuring light from distant objects in our universe. Photons of light — whether radio, infrared, visible, ultraviolet, x-ray, or gamma ray — are fundamentally the same, just from different parts of the electromagnetic spectrumNeutrinos, on the other hand, are a completely different type of particle, but they also bring us information about the cosmos. Photons are mass-less particles that fly through the universe at the speed of light, while neutrinos do have a very small mass and travel at nearly the speed of light. Photons interact easily with matter, while neutrinos have no electric charge (they are “neutral”, as their name suggests), so they are unaffected by the electromagnetic force and pass through most materials effortlessly. Detecting photons is relatively easy; detecting neutrinos is not.

The IceCube neutrino detector is designed to find the signal of the rare neutrino that does interact with material while passing through the Earth. IceCube consists of 5,160 light-sensing detectors arranged in a giant 3D grid within the ice under the South Pole. When a neutrino collides with an ice molecule, it creates a muon that then radiates a cone of blue light (called Cherenkov radiation) as it propagates through the ice. This light is detected by the grid of sensors, which allows the original neutrino’s incoming trajectory to be reconstructed (see Figure 1). For more detail on how Cherenkov radiation is produced, see this recent astrobite by Justin Vasel.

Detections: Extraterrestrial or Not?

When the IceCube collaboration analyzed the two years of data collected by their sensors from May 2010 to May 2012, they found the signals of many neutrinos. We know that neutrinos are produced by extraterrestrial sources, but neutrinos and muons are also generated in our own atmosphere — which is not very interesting from an astronomical perspective. So was IceCube actually probing the distant universe or just our atmosphere?

Figure 2. The energy distribution of detected neutrinos. The colored curves show the predicted signal from the atmosphere and the black points show what IceCube detected. The high energy detections are very likely extraterrestrial.

The first step in finding the answer was to examine the energy of these neutrinos. The energy distribution of atmospheric neutrinos peaks at low energies (see Figure 2), so the collaboration focused only on neutrinos traveling with energies above 30 TeV (1 TeV is one trillion electron volts). They found 28 of these high-energy neutrinos, whereas they expected to see only 10 atmospheric neutrinos in this energy range. Of these 28, nine had energies above 100 TeV (whereas the atmosphere was only expected to produce one), and two had over 1,000 TeV of energy. Thus, they must have detected several neutrinos of extraterrestrial origin.

Second, they examined the shapes of the light signals produced in the grid of detectors. The shape of the signal is either a spherical “shower” event or a linear “track”, depending on the type of reaction with the ice. They find that the majority of their detections are “showers”, which is expected from extraterrestrial neutrinos but not atmospheric ones.    This also suggests that many of the detected neutrinos were extraterrestrial.

Finally, the collaboration considered the distribution of the directions of incoming neutrinos. They detected more neutrinos coming into the Earth from the south (downwards into the detector from the point of view of someone standing at the south pole) than from the north (upwards into the detector). Considering that the Earth itself will shield some of the neutrinos coming in from the north , their data are actually consistent with extraterrestrial neutrinos coming from all directions.

Where Did These Extraterrestrial Neutrinos Come From?

Figure 3 shows the directions from which each neutrino originated, although the researchers have not yet been able to identify their sources. While there is a hint of clustering near the direction of the center of our galaxy (where the super-massive black hole is a potential neutrino-producer), the authors note that this is not statistically significant.

Neutrinos originate from many astrophysical phenomena such as supernovae and active galactic nuclei (AGN), and they may help us understand the origin of cosmic rays (neutrinos are produced when cosmic ray particles are accelerated). IceCube will need to wait for more data before it can definitively identify the extraterrestrial neutrino sources, but it has, for now, shown that they exist.

Figure 3. The locations in the sky from which the 28 detected neutrinos originated. The gray line is the galactic plane and the gray square (towards the lower left) is the galactic center. There is no evidence for significant clustering of neutrinos from any one source.

Axel Maas - Looking Inside the Standard Model

Knowing the limits
Some time ago, I have presented one of the methods I am using: The so-called perturbation theory. This fancy name signifies the following idea: If we know something, and we add just a little disturbance (a perturbation) to it, then this will not change things too much. If this is the case, then we can systematically give the consequences of the perturbation. Mathematically, this is done by first calculating the direct impact of the perturbation (the leading order). Then we look at the first indirection, which involves not only the direct effect, but also the simplest indirect effect, and so on.

Back then, I already wrote that, nice as the idea sounds, it is not possible to describe everything by it. Although it works in many cases very beautifully. But this leaves us with the question when does it not work. We cannot know this exactly. This would require to know the theory perfectly, and then there would be no need in the first place to do perturbation theory. So how can we then know what we are doing?

The second problem is that in many cases anything but perturbation theory is technically extremely demanding. Thus the first thing one checks is the simplest one: Whether perturbation theory makes itself sense. Indeed, it turns out that usually perturbation theory starts to produce nonsense if we increase the strength of the perturbation too far. This indicates clearly the breakdown of our assumptions, and thus the breakdown of perturbation theory. However, this is a best-case scenario. Hence, one wonders whether this approach could be fooling us. Indeed, it could be that this approximation breaks down long before it gets critical. So that it first produces bad (or even wrong) answers before it produces nonsensical ones.

This seems like serious trouble. What can be done to avoid it? There is no way inside perturbation theory to deal with it. One way is, of course, to compare to experiment. However, this is not always the best choice. On the one hand it is always possible that our underlying theory actually fails. Then we would misinterpret the failure of our ideas of nature as the failure of our methods. One would therefore like to have a more controllable way. In addition, we often reduce complex problems to simpler ones, to make them tractable. But the simpler problems often do not have a direct realization in nature, and thus we have no experimental access to them. Then this way is also not possible.

Currently, I find myself in such a situation. I want to understand, in the context of my Higgs research, to which extent perturbation theory can be used. In this context, the perturbation is usually the mass of the Higgs. The question then becomes: Up to which Higgs mass is perturbation theory still reliable? Perturbation theory itself predicts its failure at not more than eight time the mass of the observed Higgs particle. The question is, whether this is adequate, or whether this is too optimistic.

How can I answer this question? Well, here enters my approach not to rely only on a single method. It is true that we are not able to calculate as much with different methods than perturbation theory, just because anything else is too complicated. But if we concentrate on a few questions, enough resources are available to calculate things otherwise. The important task is then to make a wise choice. I.e. a choice from which one can read off the desired answer, in the present case whether perturbation theory applies or not. And at the same time to do something one can afford to calculate.

My present choice is to look at the relation of the W boson mass and the Higgs mass. If perturbation theory works, there is a close relation between both, if everything else is adjusted in a suitable way. The perturbative result can be found already in textbooks for physic students. To check it, I am using numerical simulations of both particles and their interactions. Even this simple question is an expensive endeavor, and several ten-thousand days of computing time (we always calculate how much time it would take a single computer to do all the work all by itself) have been invested. The results I found so far are intriguing, but not yet conclusive. However, in just a few weeks more time, it seems, that the fog will finally lift, and at least something can be said. I am looking with great anticipation to this date. Since either of two things will happen: Something unexpected, or something reassuring.