# Particle Physics Planet

## July 21, 2018

### Christian P. Robert - xi'an's og

Orfeo ed Euridice

Last weekend, I went to Versailles [despite the local trains best effort to keep me from it] to listen to a choir version of Orpheo ed Eurydice, Gluck’s opera. A magnificent baroque opera, possibly in par with Monteverdi’s. Unfortunately, this version was given in French and with a piano, which simply cancelled all baroque aspects of the piece (and in addition exhibited too clearly the paucity of the text!). Although I found this version rendered beautifully by Maria Callas:

### Lubos Motl - string vacua and pheno

Cohl Furey understands neither field theory nor octonions
Her mathematical masturbations are physically meaningless

Quanta Magazine's Natalie Wolchover wrote a cheesy celebration
The Peculiar Math That Could Underlie the Laws of Nature
of a would-be theory of everything by Ms Cohl Furey that is claimed to be based on the octonions $${\mathbb O}$$. If you read it, the human part of the story as well as the spirit of the mathematics used for physics sounds virtually isomorphic to Garrett Lisi and his would-be theories of everything based on the exceptional group $$E_8$$. So people are told about some ingenious outsider who has some very non-intellectual hobbies but who can still get closer to a theory of everything than all the professional physicists combined by insisting that an exceptional mathematical structure underlies the patterns of particles and fields in Nature.

The main difference is that Lisi is just a "surfer dude" while Furey is a "ski-accordion-yoga-mat-rented-car gal trained in martial arts, as her muscular physique betrays". Cool: the surfer dude is clearly a second-rate genius when compared to Furey now. ;-)

But both of these people misunderstand the meaning of the mathematical structures they claim to love – and they are trying to use them in field theory in ways that destroy the relationships that make the beloved mathematical structure what it should be; or that are impossible in quantum field theory.

Needless to say, both Lisi and Furey are parts of a much larger crackpot movement. When such positive articles about meaningless claims are published e.g. in the Quanta Magazine, you may be sure that there will be positive reactions because lots of other numerologists and crackpots arrive, praise each other, and promote their own twists on the crackpottery.

Recall that Lisi wanted to claim that he had found something important about $$E_8$$ or even discovered it – he's done neither. And the largest exceptional simple Lie group was used in his model almost as a gauge group in the grand unification but not quite. He wants to be more ambitious so the 248-dimensional adjoint representation should include not only gauge bosons but also all the quarks and fermions.

It's not really possible and as a result, his construction is nothing else than a deceitful lipstick masking a wishful thinking that cannot be backed by anything except for pure numerology. The $$E_8$$ adjoint representation simply has a high enough number of components which is why he may place the fields of the Standard Model "somewhere into it". But do the fields feel comfortable (thanks, Andrei)? Is the adjoint of $$E_8$$ the right description of the mutual relationships between the fields? It's not.

In real physics, the gauge groups that allow chiral (left-right-asymmetric) fermions need to have complex representations – which means representations that can be complex conjugated to get unequivalent but analogous ones (the complex conjugation is used for the switch from particles to antiparticles). $$E_8$$ only has real representations (the complex conjugation does nothing to them at all) so it can't possibly describe left-right-asymmetric phenomena such as the left-handed (chiral) neutrinos that differ from the right-handed (chiral) antineutrinos. On top of that, the union of bosons and fermions within the same representation is impossible. He claims to unite the bosons and fermions through a symmetry but the only symmetry that may relate bosons and fermions is a Grassmann-odd i.e. fermionic symmetry i.e. supersymmetry. The $$E_8$$ symmetry is a bosonic one so it simply cannot relate the properties of bosons to properties of fermions.

In heterotic string theory and its duals, one can start with the $$E_8$$ gauge group and get down to the Standard Model but that's because the $$E_8$$ is broken to $$E_6$$ or $$SO(10)$$ etc. – and those do admit complex representations – by the gauge fields on the compactification manifolds, something that is not available in the purely four-dimensional model building such as Lisi's framework. So Lisi's constructions don't pass even the most basic tests. You just can't unify all the fields into an $$E_8$$ representation in this way.

Furey's would-be theory of everything using the octonions $${\mathbb O}$$ is analogously flawed.

She claims the fields of the Standard Model to be something like representations of some algebra related to octonions in some incoherent way. Note that $$\RR,\CC,\HHH,\OO$$ are 1-, 2-, 4-, 8-dimensional division algebras (where you can add and multiply numbers with one or more real components, and where the inverse exists for every nonzero number). She isn't satisfied with the octonions themselves so she also uses the Dixon algebra$\RR \otimes \CC \otimes \HHH \otimes \OO.$ Let me tell you a secret. The Dixon algebra isn't named after a great physicist (Lance Dixon) but after a complete crackpot (Geoffrey Dixon) and it makes absolutely no sense. There is no interesting algebraic structure that could be described in this way. In fact, the sequence of symbols may be seen to be silly very easily. In particular, $$\RR\otimes$$ is a trivial factor because the tensor multiplication of a vector space $$V$$. with the 1-dimensional space of real numbers changes nothing about $$V$$.

It's very clear what Geoffrey Dixon, Cohl Furey, and other crackpots think about these division algebras. They think it must be a great idea to have generalizations of the real numbers whose dimensions are powers of two, and they can do even better by picking higher powers. For example, the tensor product above is supposed to have the dimension $$1\times 2 \times 4 \times 8 = 64$$.

But that's a totally wrong lesson that someone may extract from the division algebras. In fact, the octonions are connected with the "highest power of two" where a division algebra exists and there's nothing truly analogous in higher power-of-two dimensions. And it's also meaningless to "tensor multiply" quaternions with octonions. You know, the basis vectors of $$\HHH\otimes \OO$$ should be proportional to products of imaginary units from $$\HHH$$ and imaginary units from $$\OO$$. But there's no mathematically interesting rule to define the products from two copies of quaternions or octonions (or one quaternion and one octonion). Interesting multiplication tables only exist on one copy of quaternions, or one copy of octonions. And if you only define the tensor product formally, like $$j\otimes C$$, and assume that the quaternions act from one side and octonions from the other, it's no good because quaternions and octonions only deserve the name if they can be multiplied from both sides. On top of that, it's just wrong to talk about octonionic representations of groups and algebras because the action of groups and algebras must be associative and octonions are not! One may construct algebras of octonionic matrices – the $$3\times 3$$ "Hermitian" octonionic matrices with the anticommutator have the $$F_4$$ automorphism group – but because of the non-associativity of $$\OO$$, they don't have representations that are "columns of octonions" in any sense!

The full multiplication table is the cool set of gems that defines the beauty of quaternions and octonions; I discuss the tables at the bottom of this text ("the bonus"). If you don't care what the multiplication table is or if you propose a wrong one, you won't get the beauty of the quaternions and octonions! Crackpots clearly fail to get the trivial point but it's very important, anyway:
Quaternions and octonions are much more than powers of two.
If you claim that you have made a revolution using octonions without using the multiplication table, it's exactly like if you claim that you are a Formula One expert or champion because you've noticed that the number of a formula's wheels is a power of two (and you may also become a super-champion if the car has 64 wheels instead). The Formula One – and the art of its driving – is much more than the number four. It matters how the wheels are connected to the engine, what the driver does in the vehicle, and so on. I think that most laymen understand this trivial statement in the case of Formula One but for some reason, the equally obvious claim about the division algebras seems incomprehensible to many.

Or, as Peter Morgan puts it in his most meaningful comment under Wolchover's article (I added this quote half a day after this blog post was written down):
The paper linked to above, having introduced the non-associative octonions, takes a page to show that the left action of $$\CC\otimes \OO$$ on itself is isomorphic as an associative composition algebra to the Clifford algebra $$C \ell(6)$$, then the non-associativity of the octonions plays no further part.
To me the move from $$\CC\otimes\HHH$$ to $$C \ell(4)$$ seems artificial, more justified by it being possible to use $$C\ell(6)\otimes C\ell(4)$$ to get to $$C \ell(10)$$ and thence to $$SU(5)$$ than by any really principled argument.

With apologies, the whole seems more a marketing effort for $$C\ell(10)$$ and Georgi's $$SU(5)$$ than, for example, a principled introduction of non-associativity.
Incidentally, in a video, she also repeats the widespread view that quaternions underlie special relativity which is really a misconception for the same reason: the spacetime is 4D but there's no natural "product" defined on a pair of spacetime points so it makes no sense to represent the four-vectors as quaternions. When the four-vectors are represented by quaternions, it's another mere bookkeeping device (which is possible because both have four components) and the product – the main mathematical structure that makes quaternions quaternions – remains unused.

Also, just like Lisi's work, her way to assemble the Standard Model fields is pure numerology. She has a high enough number of components, so she throws the quark and lepton fields somewhere in them. All the actual octonionic structure is totally broken by this treatment. You know, when you claim to have a construction based on octonions, you should notice that the octonions have the 14-dimensional $$G_2$$ automorphism group (a symmetry renaming the octonions into others, a subgroup of $$SO(7)$$).
But if your construction has no trace of the $$G_2$$ symmetry, it doesn't have anything to do with octonions.
There's no $$G_2$$ symmetry left in her construction which is why it's deceitful to say it has something to do with octonions. Well, it doesn't even have the $$SO(3)$$ automorphism group of the quaternions. So all the claims about connecting octonions with physics of the Standard Model are totally spurious. (Some of these claims are hers, most of them are copied from earlier crackpots of the same kind plus some semi-legit researchers trying a wrong track, starting in the 1970s with Günaydin – so much of this criticism is primarily directed against these older authors on whose shoulders she is standing.)

Octonions and other exceptional structures are great and I love them – and use them in my research very often. But the question whether the octonions have something to do with the gauge group of the Standard Model and the representations of quarks and leptons is a question. I think that the obvious comparisons of fingerprints, an analysis we can make, makes it almost certain that the answer is No, the Standard Model fields just don't have anything to do with octonions.

Crackpots like Furey don't ever try to answer questions impartially. The relevance of the octonions is treated as a dogma and Furey and others are ready to destroy the inner workings of both the octonions and the Standard Model in their futile efforts to save the dogma. The dogma is almost certainly wrong.

You may look at her papers. Physicists don't read them much, in the small number of followups, most of them are her own while Geoffrey Dixon, Lee Smolin, and a few other crackpots dominate the rest. Physicists don't read them because they don't really make sense.

OK, take the latest one from June 2018, so far with 0 citations. The title combines the Standard Model group, division algebras, and ladder operators. If you don't think about the content, it looks like a perfectly legitimate paper with the correct grammar, right kind of mathematical symbols and jargon, and a professional ratio between words and formulae, among other things.

But if you have the expertise, read it, and think about it, you immediately see that it's just a pile of hogwash. (It has gotten into a journal – similar authors persistently send their nonsensical papers everywhere and it's statistically guaranteed that a referee who doesn't want to bother or fight emerges and such papers get occasionally published. This referee's blunder is clearly the main reason why the paper was hyped in the Quanta Magazine, too.) She basically claims to derive the 12-dimensional Standard Model group as some part of the 24-dimensional $$SU(5)$$ grand unified group of Georgi and Glashow – it is supposed to be the part that respects some incoherent rules mentioning tensor products of division algebras, Clifford algebras, tensor products of division algebras, left ideals, ladder operators, and other things.

Well, one could say she's an actual great example of someone who is "lost in math". She uses lots of these phrases from algebra but her mixture of these words doesn't make any sense. She constantly pretends to have found some new laws of physics but there are none. In fact, one could argue that none of these things ever appears in physics of quantum field theory. If you want to organize fields,
fields just form representations of groups.
To make it even more constraining, generators of Lie groups correspond to gauge fields and these Lie groups should better be compact (because the norms on the Lie algebra are linked to probabilities that should better be positively definite). That's the actual framework we have. Operators (such as fields) form algebras under the multiplication (and the commutator, that from Lie algebras, is even more widespread than the product itself) and that's it. There's nothing else and if you added some different algebraic structure to the organization of fields in quantum field theories, that would be a technical yet far-reaching development, indeed. If you could meaningfully add semigroups, ideals that aren't just representations of groups, algebras that aren't Lie algebras etc., quantum fields that have to transform as (not just under) Clifford algebras by themselves, physicists would care. But it would have to work.

Furey's construction doesn't work and she doesn't care.

By the way, her isolation of the 12 generators out of 24 generators of $$SU(5)$$ – yes, it's been known for a while that it's exactly one-half – isn't a new observation (you can find it on this blog and in other papers, I guess) – but the justification is just nonsense. One may define a parity on the adjoint representation of $$SU(5)$$ and define the generators of the Standard Model group to be positive (even) and the remaining one to be negative (odd). The parity will correctly (multiplicatively) behave under the commutator.

So there's some way to semi-naturally segregate the $$SU(5)$$ generators into the Standard Model (desirable) ones and the unwanted ones. But that's still far from having a physical mechanism that actually breaks $$SU(5)$$ to the Standard Model group. She claims to have and not have $$SU(5)$$ at the same moment. And she claims to break $$SU(5)$$ to the Standard Model without the Higgs fields. The Standard Model group is picked because it's compatible with some of her ideas or constructions. In actual field theory, the field content and its consistency follows its own rules. She should either obey these rules, or justify some replacement for these rules and/or the replacement for the Higgs, symmetry-breaking mechanism etc.

But there must still be a replacement. If it makes any sense to talk about $$SU(5)$$ at any stage of the construction, there has to be something that breaks it to the Standard Model. In quantum field theory-based model building, it's always the Higgs fields. In string theory, one has new tools such as the Wilson lines around cycles of the compactification manifold. Indeed, those are new physical – and characteristically stringy – mechanisms that may break $$SU(5)$$ to the Standard Model. But she has nothing. So either she talks about theories with the $$SU(5)$$ symmetry or she doesn't. Her own answer really makes it clear that she doesn't have any $$SU(5)$$. So why is she mentioning it at all?

Lisi's and Furey's are efforts that belong to a much more widespread subcommunity of the crackpot movement – whose members also write papers about the "graviweak" unification. If you forgot about "graviweak" folks, those claim that they may embed the Lorentz group and the Standard Model group into a larger, simple group. Lisi is basically an example of that, too. However, the Lorentz group acts on the spacetime while the gauge groups don't – they act inside a point. So they are clearly qualitatively different and cannot be related by any symmetry to each other. In practice, the graviweak people misunderstand the difference between the diffeomorphism symmetry of general relativity (which moves points to other places) and the local Lorentz group (that doesn't). But those are completely different things. The diffeomorphism group of general relativity is an example of a generalized "gauge symmetry" but it's simply not an example of the gauge symmetry of the Yang-Mills type.

So Lisi, Furey, and many others just don't understand the mathematical structures (they're using) too well – and they misunderstand their actual and possible relationship with physics completely. But we read many more articles lionizing these crackpots than we read about the actual exciting physics research. John Baez actually and surprisingly gave a rather reasonable feedback for Wolchover's article, including some technicalities. On the other hand, Pierre Ramond and especially Michael Duff made it sound as her 0-citation meaningless article really revolutionizes particle physics.

In particular, Mike Duff said that it could be revolutionary and have other adjectives and so on. Oh, really, Mike? Haven't you noticed that the paper is pure crackpottery? Have you noticed that when you praise such a thing that makes no sense, you're just absolutely full of šit?

I believe that Duff should understand it and he has some political reasons why he praises this crackpottery. But most of the laymen – even those who have dedicated some time to trying to superficially follow theoretical or particle physics – just don't distinguish real physics from crackpottery of Furey's type. So everything that is controlled by the laymen is more or less guaranteed to gradually replace physicists with crackpots. Those who flatter the laymen and those who own windsurfing boards and yoga mats will clearly be preferred (and get all the Nobel prizes when the movement conquers the Scandinavian institutions).

The Quanta Magazine is a worrisome borderline example. It's funded by Jim Simons who has been an excellent mathematician close to theoretical physics – exactly the type of person who used to understand (and maybe still understands?) the criticism I wrote above. But even the Quanta Magazine which is funded by Simons ended up being a medium that – whenever it writes about fundamental physics – lionizes crackpots and attacks actual top physicists most of the time. Crackpottery is so much more attractive for a larger number of readers – and the number of crackpots is vastly higher than the number of physicists.

Much of the serious research still deserves the allocation of time and energy of the researcher and the funding. In a world increasingly controlled by stupid laymen, is the top-tier serious research in pure science sustainable at all? Isn't it guaranteed that crackpots teamed up with other crackpots are going to overtake not only the Quanta Magazines but the universities such as Cambridge as well?

I am grateful to have spent at least a part of my life in an epoch when this wasn't the case, in a world where physicists and crackpots knew their places, the places weren't the same, and where it was possible for a physicist to explain why crackpots' ideas don't work. I am afraid the mankind is going to deteriorate into a bunch of stupid animals again.

Bonus: the actual beauty of the division algebras

$$\RR,\CC,\HHH,\OO$$ have dimensions 1,2,4,8, respectively. But if you just know the dimensions, you're extremely far from understanding why these are the only four division algebras. I won't prove that they are the only ones here but I will sketch it so that you understand some of the beauty if you focus.

$$\RR$$ are the real numbers such as $$-20.18$$. You may add them, subtract them, multiply them, and divide them – unless the denominator is zero. Everyone should learn how to do it from the elementary school or elsewhere as a kid.

$$\CC$$ are complex numbers of the form $$x+iy$$. The multiplication inverse of that number is$\frac{1}{x+iy} = \frac{x-iy}{x^2+y^2}.$ If you use $$i^2=-1$$, you may check that the right hand side times $$x+iy$$ is equal to one. The complex numbers are great e.g. because the $$n$$-th degree algebraic equation has exactly $$n$$ roots $$x_i\in\CC$$ – some of them may coincide. You may prove it e.g. by studying the phase of the polynomial for $$|x|\to\infty$$ in the complex plane $$x=r\cdot \exp(i\phi)\in\CC$$. The phase $$\phi_p$$ of the polynomial winds along a circle around the origin $$n$$ times. Because $$\phi_p$$ is ill-defined when the polynomial is zero, the points $$x\in\CC$$ where the polynomial vanishes allow you to change the winding number by one, so there must be $$n$$ of them.

Sorry if I were too concise.

Complex numbers are really more natural than the real numbers. They are absolutely needed as probability amplitudes in quantum mechanics. You know, the total probability has to be fixed but the amplitudes must oscillate. The only way how they (think about the energy eigenstates to make it simple) can oscillate yet preserve an invariant is for them to be complex so that the phase oscillates while the absolute value stays the same.

Also, in representation theory of groups, all the representations are complex by default. The real or quaternionic/pseudoreal representations may be defined as initially complex representations with some extra structure – a complex conjugation defined by the antilinear "structure map" $$j$$ that doesn't spoil the remaining operations. In this sense, real and quaternionic representations are equally far from the most fundamental and simplest representations – those are the complex ones. This "centrality" of the complex numbers is misunderstood by all the members of that movement – they either think that $$\RR$$ are the most fundamental among the three, or the highest-dimensional algebras such as $$\HHH,\OO$$ are the fundamental building blocks.

The quaternion $$z\in\HHH$$ is a number of the form$a+ib+jc+kd$ where $$i,j,k$$ are three imaginary units obeying$i^2=j^2=k^2=ijk=jki=kij=-1$ So the unit $$i$$ may be identified with the complex imaginary unit, but so can $$j$$ or $$k$$. And $$ij=k=-ji$$ and cyclic permutations define the totally associative but maximally non-commutative multiplication table of the three imaginary units. The three-dimensional space generated by $$i,j,k$$ may be rotated by $$SO(3)$$ transformations – the quaternions may be $$SO(3)$$-renamed – so that the multiplicative relationships between them remain the same. We say that $$SO(3)$$ is the automorphism group of the quaternions.

Why is it a division algebra? It's because the inverse is defined analogously as for the complex numbers$\frac{1}{a+ib+jc+kd} = \frac{a-ib-jc-kd}{a^2+b^2+c^2+d^2}.$ I just changed the signs of the imaginary "coordinates" $$b,c,d$$ and divided it by the squared Euclidean length of the four-dimensional vector. Why are the numbers inverse to each other? Well, it's because$(a\!+\!ib\!+\!jc\!+\!kd)(a\!-\!ib\!-\!jc\!-\!kd) = a^2+b^2+c^2+d^2.$ Why is it so? The terms $$a^2,b^2,c^2,d^2$$ are obviously there if you use the distributive law – because $$i^2=-1$$ and the minus sign cancels because there are opposite signs in front of $$b$$ in the two terms etc.

And all the mixed terms cancel because they are either of the form$a\cdot (-ib) + (+ib) \cdot a = 0$ or of the form$(ib)\cdot (-jc) + (+jc)\cdot (-jb) = 0.$ The $$ab$$-like terms cancelled because there were explicitly opposite signs in the two terms. The $$bc$$-like terms cancelled because $$ij=-ji$$ – because the imaginary units anticommute. There are three imaginary units because they may be used as the two factors and one result in a multiplication table. You won't find a larger associative division algebra.

If you defined a "simpler" but "uglier" multiplication table, e.g. if all the products of the imaginary units were $$\pm k$$, the "complex conjugate over the squared norm" would still be inverse but such an inverse wouldn't be unique or the multiplication wouldn't be associative. An even simpler sick example: if you defined all the products of imaginary units to be $$\pm 1$$, while preserving the anticommutativity, whole families of such "broken quaternions" would behave as the same number under multiplication.

So the existence of the quaternions is really linked to the fact that a three-dimensional vector is also a two-form, something that determines the rotation of the remaining two, orthogonal vectors.

Now, the octonions are also a division algebra. They have seven imaginary units, let me call them $$i,j,k,A,B,C,D$$. These seven units may be written using products of three basic ones, $$i,j,A$$, and parentheses. The multiplication table of the seven units is such that the squares such as $$B^2$$ are equal to minus one; and the product of two different units is anti-commutative, e.g. $$CD=-DC$$, just like for the quaternions.

On top of that, the product of three units such as $$(ij)A$$ is maximally non-associative whenever it differs from $$\pm 1$$. It means that e.g. $(ij)A = -i(jA).$ The permutation of two different units in a product flips the sign; and the rearrangement of the parentheses in products such as one above flips the sign, too (whenever the product differs from $$\pm 1$$). In some binary counting, the basic imaginary units $$i,j,A$$ may be associated with binary numbers $$100,010,001$$ and the 7 units, given by 3 bits, are multiplied by adding the three bits modulo two. However, the precise sign matters and it's such that you impose the "maximal non-commutativity" and "maximum non-associativity".

That's a great algebraic structure where the inverse of a nonzero number exists and is unique. The automorphism group isn't the whole 21-dimensional $$SO(7)$$ in this case. It's just a group that remembers some relationships between the 7 imaginary units that are encoded in the 3 bits – bits that may be added separately. So the symmetry group ends up being the 14-dimensional $$G_2$$ only. Only 2/3 of the generators are still symmetries of the multiplication table. Why 2/3?

Pick one of the 7 units, it doesn't matter which one, for example $$i$$. The remaining 6 units $$j,k,A,B,C,D$$, may be divided to pairs that, along with the $$i$$, may build the quaternionic triplets:$ijk, iAB, iCD$ Now, $$SO(7)$$ is generated by 21 generators of rotations of 2-planes, such as the rotation of $$jk$$ into each other, $$AB$$ into each other, and $$CD$$ into each other. In $$SO(7)$$, there would be three parameters (angles) for these three generators. But only if $$\phi_1+\phi_2+\phi_3=0$$, the octonionic structure – including the right multiplicative relationship with $$i$$ – is preserved. You may see it e.g. by realizing that $$i,j,A$$ are "fundamental" but $$C,D$$ may already be written in terms of products of $$i,j,A$$ with parentheses – so whether $$C,D$$ have to rotate to preserve the structure of the octonions is already determined by thee rotations involving $$i,j,k=ij,A,B=iA$$.

The octonionic multiplication table may be reconstructed from remembering 7 "triplets that embed quaternions into octonions". These 7 triplets contain $$7\times 3=21$$ pairs of octonionic imaginary units which is all unordered pairs of octonionic units. That's why the number of imaginary units in a construction that works must be $$1+2\times 3$$ because the combination number "7 choose 2" is $$7\times 6 / 2\times 1$$ and it must be the same as $$7\times 3$$. That's why there is no 10-dimensional, 16-dimensional, 32-dimensional, or 64-dimensional extension of the octonions.

Even if you don't fully understand what I just wrote, I want to convey a more general point: the exceptional traits of the quaternions and especially octonions require that you analyze the multiplication table of the imaginary units, including the precise signs of the products (which remember whether the structures are commutative and/or associative). If you don't analyze the multiplication tables of your 4- or 8- or 16- or 32- or 64-component "numbers", then you are not working with anything like quaternions or octonions!

You are just talking about some generic vector spaces whose dimensions are powers of two and you are doing so because you have heard that it's deep. But without the multiplication table (and/or without the anticommutators of the SUSY generators – powers of two also appear in supersymmetry) – and without the additional constraints that the table imposes on the usage of these division algebras – there is absolutely nothing deep about the vector spaces with these dimensions! And that's the case of Furey et al. She just doesn't get what makes the octonions deep. She only uses the word "octonion" to make her claims sound sexy – but her observation is nothing more than "the number of fields in the Standard Model is smaller than a power of two".

### ZapperZ - Physics and Physicists

University Research Made Your Smartphone
A lot of people are ignorant of the fact that a smartphone, or any device, for that matter, is a result of research work done by many people and organization and over a very long time. The iPhone was not solely the work of Apple. Apple benefited from all the scientific and technological progress and accumulation of knowledge to be able to produce such a device. These knowledge and progress are often done many years ago by researchers who work on a particular topic that eventually found an application in a smartphone.

I found this interesting website that highlights how research that originated out of universities under various funding agencies, resulted in the smartphone that we currently have. It lists one aspect of each of the major component of a smartphone that had it initial incubation in university research. A lot of these research work is physics-related. It is why I continue to say that physics isn't just the LHC or the Higgs or the blackhole. It is also your MRI, your iPhone, your GPS, etc...

If you need more background info on this, check out this page.

Zz.

## July 20, 2018

### Christian P. Robert - xi'an's og

the naming of the Dead [book review]

When leaving for ISBA 2018 in Edinburgh, I picked a Rebus book in my bookshelf,  book that happened to be The Naming of the Dead, which was published in 2006 and takes place in 2005, during the week of the G8 summit in Scotland and of the London Underground bombings. Quite a major week in recent British history! But also for Rebus and his colleague Siobhan Clarke, who investigate a sacrificial murder close, too close, to the location of the G8 meeting and as a result collide with superiors, secret services, protesters, politicians, and executives, including a brush with Bush ending up with his bike accident at Gleneagles, and ending up with both of them suspended from the force. But more than this close connection with true events in and around Edinburgh, the book is a masterpiece, maybe Rankin’s best, because of the depiction of the characters, who have even more depth and dimensions than in the other novels.  And for the analysis of the events of that week. Having been in Edinburgh at the time I started re-reading the book also made the description of the city much more vivid and realistic, as I could locate and sometimes remember some places. (The conclusion of some subplots may be less realistic than I would like them to be, but this is of very minor relevance.)

### Peter Coles - In the Dark

Glamorgan versus Somerset: Vitality Blast Twenty20

After a very nice little drinks reception in the School of Physics and Astronomy (at which I was given a very nice gift of wine) I joined the staff outing to Sophia Gardens to watch this evening’s Twenty20 cruise cricket between Glamorgan and Somerset.

The start was delayed by rain so we lingered in a pub on the way only to be caught on the hop when play actually started and missing the first few overs. Somerset batted well to reach 190 off their 20 overs, with Anderson hitting four big sixes in his 59.

Without Shaun Marsh, who will miss the rest of the season, the Glamorgan batting lineup seemed to have a very long tail and a lot rested on Khawaja and Ingram. Both scored runs quickly while they were in but neither could build a big score. Once those two were out, the Glamorgan innings faltered and they never looked like reaching Somerset’s total. The finished on 160 for 9, losing by 30 runs.

### ZapperZ - Physics and Physicists

Feynman's Lost Lecture
If you didn't buy the book or didn't read about it, here's a take on Feynman's Lost Lecture, presented by a guest on Minute Physics video.

Zz.

### Emily Lakdawalla - The Planetary Society Blog

The June solstice issue of The Planetary Report has arrived
The June solstice 2018 issue of The Planetary Report is about to mail and will arrive at Planetary Society members’ homes within days. Members who want to read it sooner can access the magazine online.

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

Summer days, academics and technological universities

The heatwave in the northern hemisphere may (or may not) be an ominous portend of things to come, but it’s certainly making for an enjoyable summer here in Ireland. I usually find it quite difficult to do any meaningful research when the sun is out, but things are a bit different when the good weather is regular.  Most days, I have breakfast in the village, a swim in the sea before work, a swim after work and a game of tennis to round off the evening. Tough life, eh.

Counsellor’s Strand in Dunmore East

So far, I’ve got one one conference proceeding written, one historical paper revamped and two articles refereed (I really enjoy the latter process, it’s so easy for academics to become isolated). Next week I hope to get back to that book I never seem to finish.

However, it would be misleading to portray a cosy image of a college full of academics beavering away over the summer. This simply isn’t the case around here – while a few researchers can be found in college this summer, the majority of lecturing staff decamped on June 20th and will not return until September 1st.

And why wouldn’t they? Isn’t that their right under the Institute of Technology contracts, especially given the heavy teaching loads during the semester? Sure – but I think it’s important to acknowledge that this is a very different set-up to the modern university sector, and doesn’t quite square with the move towards technological universities.

This week, the Irish newspapers are full of articles depicting the opening of Ireland’s first technological university, and apparently, the Prime Minister is anxious our own college should get a move on. Hmm. No mention of the prospect of a change in teaching duties, or increased facilities/time for research, as far as I can tell (I’d give a lot for an office that was fit for purpose).  So will the new designation just amount to a name change? And this is not to mention the scary business of the merging of different institutes of technology. Those who raise questions about this now tend to get cast as dismissed as resistors of progress. Yet the history of merging large organisations in Ireland hardly inspires confidence, not least because of a tendency for increased layers of bureaucracy to appear out of nowhere – HSE anyone?

### The n-Category Cafe

Compositionality: the Editorial Board

An editorial board has now been chosen for the journal Compositionality, and they’re waiting for people to submit papers.

We are happy to announce the founding editorial board of Compositionality, featuring established researchers working across logic, computer science, physics, linguistics, coalgebra, and pure category theory (see the full list below). Our steering board considered many strong applications to our initial open call for editors, and it was not easy narrowing down to the final list, but we think that the quality of this editorial board and the general response bodes well for our growing research community.

In the meantime, we hope you will consider submitting something to our first issue. Look out in the coming weeks for the journal’s official open-for-submissions announcement.

The editorial board of Compositionality:

• Corina Cristea, University of Southampton, UK

• Ross Duncan, University of Strathclyde, UK

• Andrée Ehresmann, University of Picardie Jules Verne, France

• Tobias Fritz, Max Planck Institute, Germany

• Neil Ghani, University of Strathclyde, UK

• Dan Ghica, University of Birmingham, UK

• Jeremy Gibbons, University of Oxford, UK

• Nick Gurski, Case Western Reserve University, USA

• Helle Hvid Hansen, Delft University of Technology, Netherlands

• Chris Heunen, University of Edinburgh, UK

• Aleks Kissinger, Radboud University, Netherlands

• Joachim Kock, Universitat Autònoma de Barcelona, Spain

• Martha Lewis, University of Amsterdam, Netherlands

• Samuel Mimram, École Polytechnique, France

• Simona Paoli, University of Leicester, UK

• Dusko Pavlovic, University of Hawaii, USA

• Christian Retoré, Université de Montpellier, France

• Mehrnoosh Sadrzadeh, Queen Mary University, UK

• Peter Selinger, Dalhousie University, Canada

• Pawel Sobocinski, University of Southampton, UK

• David Spivak, MIT, USA

• Jamie Vicary, University of Birmingham, UK

• Simon Willerton, University of Sheffield, UK

Best,
Joshua Tan, Brendan Fong, and Nina Otter
Executive editors, Compositionality

### Peter Coles - In the Dark

After Graduation

I didn’t get time to blog yesterday as I was involved with various festivities to with the graduation of students from the School of Physics & Astronomy at Cardiff University who, for some reason, shared a ceremony with students from the School of History, Archaeology and Religion. The ceremony was more-or-less my last official duty here at Cardiff. This morning I backed up my computer, returned my keys and removed my boxes of books and other stuff from the office of the Data Innovation Research Institute back to my house. This afternoon I gather there’ll be a small event to celebrate my departure, after which there’s a staff trip to see the cricket at Sophia Gardens (Glamorgan versus Somerset in the Vitality Blast).

Yesterday’s ceremony started at 12 noon and, as usual, was in St David’s Hall in Cardiff. When it was over we adjourned to the Main Building for a reception at which we were informed there would be unlimited Prosecco’. This turned out to be untrue, as the Prosecco ran out by about 5pm, at which point we moved to a local pub and thence for a late-night curry. It was all a bit excessive and I had a not inconsiderable hangover this morning. I suspect that was the case for many of the graduands too!

It was a very hot with all the graduation clobber, which is no doubt why such a large volume of liquid refreshment was consumed. The drinks were dispensed in a marquee which was sweltering inside. Anyway, here’s a pic of some of those students who received their degrees yesterday. I was actually there, but just out of shot to the right.

Graduation ceremonies are funny things. With all their costumes and weird traditions, they even seem a bit absurd. On the other hand, even in these modern times, we live with all kinds of  rituals and I don’t see why we shouldn’t celebrate academic achievement in this way. I love graduation ceremonies, actually. As the graduands go across the stage you realize that every one of them has a unique story to tell and a whole universe of possibilities in front of them. How their lives will unfold no-one can tell, but it’s a privilege to be there for one important milestone on their journey.

I always find graduation a bittersweet occasion. There’s joy and celebration, of course, but tempered by the realisation that many of the young people who you’ve seen around for three or for years, and whose faces you have grown accustomed to, will disappear into the big wide world never to be seen again.

Graduation of course isn’t just about dressing up. Nor is it only about recognising academic achievement. It’s also a rite of passage on the way to adulthood and independence, so the presence of the parents at the ceremony adds another emotional dimension to the goings-on. Although everyone is rightly proud of the achievement – either their own in the case of the graduands or that of others in the case of the guests – there’s also a bit of sadness to go with the goodbyes. It always seems that as a lecturer you are only just getting to know students by the time they graduate, but that’s enough to miss them when they go.

Anyway, all this is a roundabout way of saying congratulations once more to everyone who graduated yesterday, and I wish you all the very best for the future!

### ZapperZ - Physics and Physicists

Burton Richter Dies at 87
Another giant in our field, especially in elementary  particle physics, has passed away. Burton Richter, Nobel Laureate in physics, died on July 18, 2018.

Richter’s Nobel Prize-winning discovery of the J/psi subatomic particle, shared with MIT’s Samuel Ting, confirmed the existence of the charm quark. That discovery upended existing theories and forced a recalibration in theoretical physics that reverberated for years. It became known as the “November Revolution.” One Nobel committee member at the time described it as “the greatest discovery ever in the field of elementary particles.”

He would be shortchanged if all the public ever remembers him is for his Nobel Prize discovery, because he did a whole lot more in his lifetime.

Zz.

### Emily Lakdawalla - The Planetary Society Blog

Boldly advocating for more space science
Board member Robert Picardo burns some shoe leather on Capitol Hill with our advocacy team.

## July 19, 2018

### Christian P. Robert - xi'an's og

Le Monde puzzle [#1061]

A griddy Le Monde mathematical puzzle:

1. On a 4×5 regular grid, find how many nodes need to be turned on to see all 3×4 squares to have at least one active corner in case of one arbitrary node failing.
2.  Repeat for a 7×9 grid.

The question is open to simulated annealing, as in the following R code:

n=3;m=4;np=n+1;mp=m+1

cvr=function(grue){
grud=grue
obj=(max(grue)==0)
for (i in (1:length(grue))[grue==1]){
grud[i]=0
obj=max(obj,max((1-grud[-1,-1])*(1-grud[-np,-mp])*
(1-grud[-np,-1])*(1-grud[-1,-mp])))
grud[i]=1}
obj=99*obj+sum(grue)
return(obj)}

dumban=function(grid,T=1e3,temp=1,beta=.99){
obj=bez=cvr(grid)
sprk=grid
for (t in 1:T){
grue=grid
if (max(grue)==1){ grue[sample(rep((1:length(grid))[grid==1],2),1)]=0
}else{ grue[sample(1:(np*mp),np+mp)]=1}
jbo=cvr(grue)
if (bez>jbo){ bez=jbo;sprk=grue}
if (log(runif(1))<(obj-jbo)/temp){
grid=grue;obj=cvr(grid)}
temp=temp*beta
}
return(list(top=bez,sol=sprk))}


leading to

>  dumban(grid,T=1e6,temp=100,beta=.9999)
$top [1] 8$sol
[,1] [,2] [,3] [,4] [,5]
[1,]    0    1    0    1    0
[2,]    0    1    0    1    0
[3,]    0    1    0    1    0
[4,]    0    1    0    1    0


which sounds like a potential winner.

### Emily Lakdawalla - The Planetary Society Blog

How the Apollo missions transformed our understanding of the Moon’s origin
Where did the Moon come from? The origin of our cosmic neighbor is a fundamental question in planetary science.

### ZapperZ - Physics and Physicists

MinutePhysics Special Relativity Chapter 7
If you missed Chapter 6 of this series, check it out here.

In this chapter, the concept of spacetime intervals is presented. This is where we have "proper time" and "proper length".

Zz.

### Andrew Jaffe - Leaves on the Line

(Almost) The end of Planck

This week, we released (most of) the final set of papers from the Planck collaboration — the long-awaited Planck 2018 results (which were originally meant to be the “Planck 2016 results”, but everything takes longer than you hope…), available on the ESA website as well as the arXiv. More importantly for many astrophysicists and cosmologists, the final public release of Planck data is also available.

Anyway, we aren’t quite finished: those of you up on your roman numerals will notice that there are only 9 papers but the last one is “XII” — the rest of the papers will come out over the coming months. So it’s not the end, but at least it’s the beginning of the end.

And it’s been a long time coming. I attended my first Planck-related meeting in 2000 or so (and plenty of people had been working on the projects that would become Planck for a half-decade by that point). For the last year or more, the number of people working on Planck has dwindled as grant money has dried up (most of the scientists now analysing the data are doing so without direct funding for the work).

(I won’t rehash the scientific and technical background to the Planck Satellite and the cosmic microwave background (CMB), which I’ve been writing about for most of the lifetime of this blog.)

### Planck 2018: the science

So, in the language of the title of the first paper in the series, what is the legacy of Planck? The state of our science is strong. For the first time, we present full results from both the temperature of the CMB and its polarization. Unfortunately, we don’t actually use all the data available to us — on the largest angular scales, Planck’s results remain contaminated by astrophysical foregrounds and unknown “systematic” errors. This is especially true of our measurements of the polarization of the CMB, unfortunately, which is probably Planck’s most significant limitation.

The remaining data are an excellent match for what is becoming the standard model of cosmology: ΛCDM, or “Lambda-Cold Dark Matter”, which is dominated, first, by a component which makes the Universe accelerate in its expansion (Λ, Greek Lambda), usually thought to be Einstein’s cosmological constant; and secondarily by an invisible component that seems to interact only by gravity (CDM, or “cold dark matter”). We have tested for more exotic versions of both of these components, but the simplest model seems to fit the data without needing any such extensions. We also observe the atoms and light which comprise the more prosaic kinds of matter we observe in our day-to-day lives, which make up only a few percent of the Universe.

All together, the sum of the densities of these components are just enough to make the curvature of the Universe exactly flat through Einstein’s General Relativity and its famous relationship between the amount of stuff (mass) and the geometry of space-time. Furthermore, we can measure the way the matter in the Universe is distributed as a function of the length scale of the structures involved. All of these are consistent with the predictions of the famous or infamous theory of cosmic inflation), which expanded the Universe when it was much less than one second old by factors of more than 1020. This made the Universe appear flat (think of zooming into a curved surface) and expanded the tiny random fluctuations of quantum mechanics so quickly and so much that they eventually became the galaxies and clusters of galaxies we observe today. (Unfortunately, we still haven’t observed the long-awaited primordial B-mode polarization that would be a somewhat direct signature of inflation, although the combination of data from Planck and BICEP2/Keck give the strongest constraint to date.)

Most of these results are encoded in a function called the CMB power spectrum, something I’ve shown here on the blog a few times before, but I never tire of the beautiful agreement between theory and experiment, so I’ll do it again: (The figure is from the Planck “legacy” paper; more details are in others in the 2018 series, especially the Planck “cosmological parameters” paper.) The top panel gives the power spectrum for the Planck temperature data, the second panel the cross-correlation between temperature and the so-called E-mode polarization, the left bottom panel the polarization-only spectrum, and the right bottom the spectrum from the gravitational lensing of CMB photons due to matter along the line of sight. (There are also spectra for the B mode of polarization, but Planck cannot distinguish these from zero.) The points are “one sigma” error bars, and the blue curve gives the best fit model.

As an important aside, these spectra per se are not used to determine the cosmological parameters; rather, we use a Bayesian procedure to calculate the likelihood of the parameters directly from the data. On small scales (corresponding to 𝓁>30 since 𝓁 is related to the inverse of an angular distance), estimates of spectra from individual detectors are used as an approximation to the proper Bayesian formula; on large scales (𝓁<30) we use a more complicated likelihood function, calculated somewhat differently for data from Planck’s High- and Low-frequency instruments, which captures more of the details of the full Bayesian procedure (although, as noted above, we don’t use all possible combinations of polarization and temperature data to avoid contamination by foregrounds and unaccounted-for sources of noise).

Of course, not all cosmological data, from Planck and elsewhere, seem to agree completely with the theory. Perhaps most famously, local measurements of how fast the Universe is expanding today — the Hubble constant — give a value of H0 = (73.52 ± 1.62) km/s/Mpc (the units give how much faster something is moving away from us in km/s as they get further away, measured in megaparsecs (Mpc); whereas Planck (which infers the value within a constrained model) gives (67.27 ± 0.60) km/s/Mpc . This is a pretty significant discrepancy and, unfortunately, it seems difficult to find an interesting cosmological effect that could be responsible for these differences. Rather, we are forced to expect that it is due to one or more of the experiments having some unaccounted-for source of error.

The term of art for these discrepancies is “tension” and indeed there are a few other “tensions” between Planck and other datasets, as well as within the Planck data itself: weak gravitational lensing measurements of the distortion of light rays due to the clustering of matter in the relatively nearby Universe show evidence for slightly weaker clustering than that inferred from Planck data. There are tensions even within Planck, when we measure the same quantities by different means (including things related to similar gravitational lensing effects). But, just as “half of all three-sigma results are wrong”, we expect that we’ve mis- or under-estimated (or to quote the no-longer-in-the-running-for-the-worst president ever, “misunderestimated”) our errors much or all of the time and should really learn to expect this sort of thing. Some may turn out to be real, but many will be statistical flukes or systematic experimental errors.

(If you were looking a briefer but more technical fly-through the Planck results — from someone not on the Planck team — check out Renee Hlozek’s tweetstorm.)

### Planck 2018: lessons learned

So, Planck has more or less lived up to its advanced billing as providing definitive measurements of the cosmological parameters, while still leaving enough “tensions” and other open questions to keep us cosmologists working for decades to come (we are already planning the next generation of ground-based telescopes and satellites for measuring the CMB).

But did we do things in the best possible way? Almost certainly not. My colleague (and former grad student!) Joe Zuntz has pointed out that we don’t use any explicit “blinding” in our statistical analysis. The point is to avoid our own biases when doing an analysis: you don’t want to stop looking for sources of error when you agree with the model you thought would be true. This works really well when you can enumerate all of your sources of error and then simulate them. In practice, most collaborations (such as the Polarbear team with whom I also work) choose to un-blind some results exactly to be able to find such sources of error, and indeed this is the motivation behind the scores of “null tests” that we run on different combinations of Planck data. We discuss this a little in an appendix of the “legacy” paper — null tests are important, but we have often found that a fully blind procedure isn’t powerful enough to find all sources of error, and in many cases (including some motivated by external scientists looking at Planck data) it was exactly low-level discrepancies within the processed results that have led us to new systematic effects. A more fully-blind procedure would be preferable, of course, but I hope this is a case of the great being the enemy of the good (or good enough). I suspect that those next-generation CMB experiments will incorporate blinding from the beginning.

Further, although we have released a lot of software and data to the community, it would be very difficult to reproduce all of our results. Nowadays, experiments are moving toward a fully open-source model, where all the software is publicly available (in Planck, not all of our analysis software was available to other members of the collaboration, much less to the community at large). This does impose an extra burden on the scientists, but it is probably worth the effort, and again, needs to be built into the collaboration’s policies from the start.

That’s the science and methodology. But Planck is also important as having been one of the first of what is now pretty standard in astrophysics: a collaboration of many hundreds of scientists (and many hundreds more of engineers, administrators, and others without whom Planck would not have been possible). In the end, we persisted, and persevered, and did some great science. But I learned that scientists need to learn to be better at communicating, both from the top of the organisation down, and from the “bottom” (I hesitate to use that word, since that is where much of the real work is done) up, especially when those lines of hoped-for communication are usually between different labs or Universities, very often between different countries. Physicists, I have learned, can be pretty bad at managing — and at being managed. This isn’t a great combination, and I say this as a middle-manager in the Planck organisation, very much guilty on both fronts.

### Andrew Jaffe - Leaves on the Line

Loncon 3

Briefly (but not brief enough for a single tweet): I’ll be speaking at Loncon 3, the 72nd World Science Fiction Convention, this weekend (doesn’t that website have a 90s retro feel?).

At 1:30 on Saturday afternoon, I’ll be part of a panel trying to answer the question “What Is Science?” As Justice Potter Stewart once said in a somewhat more NSFW context, the best answer is probably “I know it when I see it” but we’ll see if we can do a little better than that tomorrow. My fellow panelists seem to be writers, curators, philosophers and theologians (one of whom purports to believe that the “the laws of thermodynamics prove the existence of God” — a claim about which I admit some skepticism…) so we’ll see what a proper physicist can add to the discussion.

At 8pm in the evening, for participants without anything better to do on a Saturday night, I’ll be alone on stage discussing “The Random Universe”, giving an overview of how we can somehow learn about the Universe despite incomplete information and inherently random physical processes.

There is plenty of other good stuff throughout the convention, which runs from 14 to 18 August. Imperial Astrophysics will be part of “The Great Cosmic Show”, with scientists talking about some of the exciting astrophysical research going on here in London. And Imperial’s own Dave Clements is running the whole (not fictional) science programme for the convention. If you’re around, come and say hi to any or all of us.

### The n-Category Cafe

The Duties of a Mathematician

What are the ethical responsibilities of a mathematician? I can think of many, some of which I even try to fulfill, but this document raises one that I have mixed feelings about:

Namely:

The ethical responsibility of mathematicians includes a certain duty, never precisely stated in any formal way, but of course felt by and known to serious researchers: to dedicate an appropriate amount of time to study each new groundbreaking theory or proof in one’s general area. Truly groundbreaking theories are rare, and this duty is not too cumbersome. This duty is especially applicable to researchers who are in the most active research period of their mathematical life and have already senior academic positions. In real life this informal duty can be taken to mean that a reasonable number of mathematicians in each major mathematical country studies such groundbreaking theories.

My first reaction to this claimed duty was quite personal: namely, that I couldn’t possibly meet it. My research is too thinly spread over too many fields to “study each new groundbreaking theory or proof” in my general area. While Fesenko says that “truly groundbreaking theories are rare, and this duty is not too cumbersome”, I feel the opposite. I’d really love to learn more about the Langlands program, and the amplitudohedron, and Connes’ work on the Riemann Hypothesis, and Lurie’s work on $\left(\infty ,1\right)\left(\infty,1\right)$-topoi, and homotopy type theory, and Monstrous Moonshine, and new developments in machine learning, and … many other things. But there’s not enough time!

More importantly, while it’s undeniably good to know what’s going on, that doesn’t make it a “duty”. I believe mathematicians should be free to study what they’re interested in.

But perhaps Fesenko has a specific kind of mathematician in mind, without mentioning it: not the larks who fly free, but the solid, established “gatekeepers” and “empire-builders”. These are the people who master a specific field, gain academic power, and strongly influence the field’s development, often by making pronouncements about what’s important and what’s not.

For such people to ignore promising developments in their self-proclaimed realm of expertise can indeed be damaging. Perhaps these people have a duty to spend a certain amount of time studying each new ground-breaking theory in their ambit. But I’m fundamentally suspicious of these people in the first place! So, I’m not eager to figure out their duties.

What do you think about “the duties of a mathematician”?

Of course I would be remiss not to mention the obvious, namely that Fesenko is complaining about the reception of Mochizuki’s work on inter-universal Teichmüller theory. If you read his whole article, that will be completely clear. But this is a controversial subject, and “hard cases make bad law”—so while it makes a fascinating read, I’d rather talk about the duties of a mathematician more generally. If you want to discuss what Fesenko has to say about inter-universal Teichmüller theory, Peter Woit’s blog might be a better place, since he’s jumped right into the middle of that conversation:

As for me, my joy is to learn new mathematics, figure things out, explain things, and talk to people about math. My duties include helping students who are having trouble, trying to make mathematics open-access, and coaxing mathematicians to turn their skills toward saving the planet. The difference is that joy makes me do things spontaneously, while duty taps me on the shoulder and says “don’t forget….”

## July 18, 2018

### Emily Lakdawalla - The Planetary Society Blog

Dawn Journal: Going Out on a High...Or Maybe a Low
Rapidly nearing the end of a unique decade-long interplanetary expedition, Dawn is taking phenomenal pictures of dwarf planet Ceres as it swoops closer to the ground than ever before.

### Peter Coles - In the Dark

Ongoing Hubble Constant Poll

Here are two interesting plots that I got via Renée Hložek on Twitter from the recent swathe of papers from Planck The first shows the tension’ between Planck’s parameter estimates direct’ measurements of the Hubble Constant (as exemplified by Riess et al. 2018); see my recent post for a discussion of the latter. Planck actually produces joint estimates for a set of half-a-dozen basic parameters from which estimates of others, including the Hubble constant, can be derived. The plot  below shows the two-dimensional region that is allowed by Planck if both the Hubble constant (H0) and the matter density parameter (ΩM) are allowed to vary within the limits allowed by various observations. The tightest contours come from Planck but other cosmological probes provide useful constraints that are looser but consistent; BAO’ refers to Baryon Acoustic Oscillations‘, and Pantheon’ is a sample of Type Ia supernovae.

You can see that the Planck measurements (blue) mean that a high value of the Hubble constant requires a low matter density but the allowed contour does not really overlap with the grey shaded horizontal regions. For those of you who like such things, the discrepancy is about 3.5σ..

Another plot you might find interesting is this one:

The solid line shows how the Hubble constant’ varies with redshift in the standard cosmological model; H0 is the present value of a redshift-dependent parameter H(z) that measures the rate at which the Universe is expanding. You will see that the Hubble parameter is larger at high redshift, but decreases as the expansion of the Universe slows down, until a redshift of around 0.5 and then it increases, indicating that the expansion of the Universe is accelerating.  Direct determinations of the expansion rate at high redshift are difficult, hence the large error bars, but the important feature is the gap between the direct determination at z=0 and what the standard model predicts. If the Riess et al. 2018 measurements are right, the expansion of the Universe seems to have been accelerating more rapidly than the standard model predicts.

So after that little update here’s a little poll I’ve been running for a while on whether people think this apparent discrepancy is serious or not. I’m interested to see whether these latest findings change the voting!

<noscript><a href="http://polldaddy.com/poll/9483425">Take Our Poll</a></noscript>

### ZapperZ - Physics and Physicists

Khan Academy's Photoelectric Effect Video Lesson
A lot of people use Khan Academy's video lessons. I know that they are quite popular, and I often time get asked about some of the material in the video, both by my students and also in online discussions. Generally, I have no problems with their videos, but I often wonder who exactly design the content of the videos, because I often find subtle issues and problems. It is not unusual for me to find that they were inaccurate in some things, and these are usually not the type of errors that say, an expert in such subjects would make.

I was asked about this photoelectric effect lesson by someone about a month ago. I've seen it before but never paid much attention to it till now. And now I think I should have looked at it closer, because there are a couple of misleading and inaccurate information about this.

Here is the video:

First, let's tackled the title here, because it is perpetuating a misconception.

Photoelectric effect | Electronic structure of atoms
First of all, the photoelectric effect doesn't have anything to do with "structure of atoms". It has, however, something to do with the structure of the solid metal! The work function, for example, is not part of an atom's energy level. Rather, it is due to the combination of all the atoms of the metal, forming this BANDS of energy. Such bands do not occur in individual atoms. This is why metals have conduction band and atoms do not.

We need to get people to understand that solid state physics is not identical to atomic/molecular physics. When many atoms get together to form a solid, their behavior as a conglomerate is different than their behavior as individual atoms. For many practical purpose, the atoms lose their individuality and instead, form a collective property. This is the most important message that you can learn from this.

And now, the content of the video. I guess the video is trying to tackle a very narrow topic on how to use Einstein's equation, but they are very sloppy on the language that they use. First of all, if you don't know anything else, from the video, you'd get the impression that a photon is an ordinary type of "particle", much like an electron. The illustration of a photon reinforced this erroneous picture. So let's be clear here. A "photon" is not a typical "particle" that we think of. It isn't defined by its "size" or shape. Rather, it is an entity that carries a specific amount of energy and momentum (and angular momentum). That's almost all that we can say without getting into further complications of QED.

But the most serious inaccuracy in the video is when it tackled the energy needed to liberate an electron from the metal. This energy was labelled as E_0. This was then equate to the work function of the metal.

E_0 is equal to the work function of the metal ONLY for the most energetic photoelectrons. It is not the work function for all the other photoelectrons. Photoelectrons are emitted with a range of energies. This is because they came from conduction electrons that are at the Fermi energy or below it. If they came from the Fermi energy, then they only have to overcome the work function. These will correspond to the most energetic photoelectrons. However, if they come from below the Fermi energy, then they have to overcome not only the work function, but also the binding energy. So the kinetic energy of these photoelectrons are not as high as the most energetic ones. So their "E_0" is NOT equal to the work function.

This is why when we have students do the photoelectric effect experiments in General Physics courses, we ask them to find the stopping potential, which is the potential that will stop the most energetic photoelectrons from reaching the anode. Only the info given by these most energetic photoelectrons will give you directly the work function.

Certainly, I don't think that this will affect the viewers ability to use the Einstein equation, which was probably the main purpose of the video. But there is an opportunity here to not mislead the viewers and make the video tighter and more accurate. It also might save many of us from having to explain to other people when they tried to go into this deeper (especially students of physics). For a video that is viewed by such a wide audience, this is not the type of inaccuracies that I expect for them to have missed.

Zz.

### Peter Coles - In the Dark

Ireland And The Roman Empire. Modern Politics Shaping The Ancient Past?

I’m here in Dublin Airport, not far from Drumanagh, the site discussed in the following post. I’m on my way back to Wales for, among other things, tomorrow’s graduation ceremony for students from the School of Physics & Astronomy at Cardiff University.

I thought I’d reblog the post here because it’s very interesting and it follows on from a comment thread relating to my post a few days ago about the current drought in Ireland which has revealed many previously unknown features of archaeological interest, and the (unrelated but also recent) discovery of a 5500 year-old passage tomb in County Lowth.

The site at Drumanagh is not related to either of those new discoveries, but it is fascinating because of the controversy about whether or not it is evidence of a Roman invasion of Ireland in the first century AD. I think the idea that no Romans ever set foot in Ireland during the occupation of Britain is hard to accept given the extensive trading links of the time, but there’s no evidence of a full-scale military invasion or lengthy period of occupation. The only unambiguously Roman finds at Drumanagh are coins and other artefacts which do not really indicate a military presence and there is no evidence there or anywhere else in Ireland of the buildings, roads or other infrastructure that one finds in Roman Britain.

My own opinion is that the Drumanagh site is more likely to have been some sort of trading post than a military fort, and it may even be entirely Celtic in origin. The position and overall character of the site seems more similar to Iron Age promontory forts than Roman military camps. I am, however, by no means an expert.

You can find a description of the Drumanagh site in its historical context here.

Way back in 1996, the Sunday Times newspaper in Britain ran an enthusiastic if awkwardly-phrased banner headline proclaiming that a “Fort discovery proves Romans invaded Ireland”. The “fort” in question was an archaeological site in north County Dublin known as Drumanagh, situated on a wave-eroded headland near the coastal village of Loughshinny. Nearly 900 metres long and 190 metres wide, the monument consists of a trio of parallel ditches protecting an oblong thumb of land jutting out into the ocean, the seaward sides of the irregular protrusion relying on the waters of the Irish Sea for defence. The location is fairly typical of a large number of Iron Age promontory settlements found in isolated spots throughout the country. However what made the area at Drumanagh of particular interest was the significant number of Roman artefacts found within its fields.

Unfortunately a comprehensive archaeological survey of the site has yet to be published due to questions over property rights and compensatory payments for finds, meaning most discoveries from the location have come through agricultural work or destructive raids by…

View original post 1,387 more words

### Clifford V. Johnson - Asymptotia

Muskovites Vs Anti-Muskovites…

Saw this split over Elon Musk coming over a year ago. This is panel from my graphic short story “Resolution” that appears in the 2018 SF anthology Twelve Tomorrows, edited by Wade Roush (There’s even an e-version now if you want fast access!) -cvj

The post Muskovites Vs Anti-Muskovites… appeared first on Asymptotia.

## July 17, 2018

### John Baez - Azimuth

Compositionality: the Editorial Board

The editors of this journal have an announcement:

We are happy to announce the founding editorial board of Compositionality, featuring established researchers working across logic, computer science, physics, linguistics, coalgebra, and pure category theory (see the full list below). Our steering board considered many strong applications to our initial open call for editors, and it was not easy narrowing down to the final list, but we think that the quality of this editorial board and the general response bodes well for our growing research community.

In the meantime, we hope you will consider submitting something to our first issue. Look out in the coming weeks for the journal’s official open-for-submissions announcement.

The editorial board of Compositionality:

• Corina Cristea, University of Southampton, UK
• Ross Duncan, University of Strathclyde, UK
• Andrée Ehresmann, University of Picardie Jules Verne, France
• Tobias Fritz, Max Planck Institute, Germany
• Neil Ghani, University of Strathclyde, UK
• Dan Ghica, University of Birmingham, UK
• Jeremy Gibbons, University of Oxford, UK
• Nick Gurski, Case Western Reserve University, USA
• Helle Hvid Hansen, Delft University of Technology, Netherlands
• Chris Heunen, University of Edinburgh, UK
• Aleks Kissinger, Radboud University, Netherlands
• Joachim Kock, Universitat Autònoma de Barcelona, Spain
• Martha Lewis, University of Amsterdam, Netherlands
• Samuel Mimram, École Polytechnique, France
• Simona Paoli, University of Leicester, UK
• Dusko Pavlovic, University of Hawaii, USA
• Christian Retoré, Université de Montpellier, France
• Mehrnoosh Sadrzadeh, Queen Mary University, UK
• Peter Selinger, Dalhousie University, Canada
• Pawel Sobocinski, University of Southampton, UK
• David Spivak, MIT, USA
• Jamie Vicary, University of Birmingham, UK
• Simon Willerton, University of Sheffield, UK

Best,
Josh, Brendan, and Nina
Executive editors, Compositionality

### Emily Lakdawalla - The Planetary Society Blog

How India built NavIC, the country's own GPS network
The country's satellite navigation system faced a long and difficult road, but it's finally operational.

### Peter Coles - In the Dark

Planck’s Last Papers

Well, they’ve been a little while coming but just today I heard that the final set of a dozen papers from the European Space Agency’s Planck mission are now available. You can find the latest ones, along with the all the others, here.

This final Legacy’ set of papers is sure to be a vital resource for many years to come and I can hear in my mind’s ear the sound of cosmologists all around the globe scurrying to download them!

I’m not sure when I’ll get time to read these papers, so if anyone finds any interesting nuggets therein please feel free to comment below!

## July 16, 2018

### Tommaso Dorigo - Scientificblogging

A Beautiful New Spectroscopy Measurement
What is spectroscopy ?
(A) the observation of ghosts by infrared visors or other optical devices
(B) the study of excited states of matter through observation of energy emissions

If you answered (A), you are probably using a lousy internet search engine; and btw, you are rather dumb. Ghosts do not exist.

Otherwise you are welcome to read on. We are, in fact, about to discuss a cutting-edge spectroscopy measurement, performed by the CMS experiment using lots of proton-proton collisions by the CERN Large Hadron Collider (LHC).

read more

## July 13, 2018

### John Baez - Azimuth

Applied Category Theory Course: Collaborative Design

In my online course we’re reading the fourth chapter of Fong and Spivak’s book Seven Sketches. Chapter 4 is about collaborative design: building big projects from smaller parts. This is based on work by Andrea Censi:

• Andrea Censi, A mathematical theory of co-design.

The main mathematical content of this chapter is the theory of enriched profunctors. We’ll mainly talk about enriched profunctors between categories enriched in monoidal preorders. The picture above shows what one of these looks like!

Here are my lectures so far:

### John Baez - Azimuth

Random Points on a Group

In Random Points on a Sphere (Part 1), we learned an interesting fact. You can take the unit sphere in $\mathbb{R}^n$, randomly choose two points on it, and compute their distance. This gives a random variable, whose moments you can calculate.

And now the interesting part: when n = 1, 2 or 4, and seemingly in no other cases, all the even moments are integers.

These are the dimensions in which the spheres are groups. We can prove that the even moments are integers because they are differences of dimensions of certain representations of these groups. Rogier Brussee and Allen Knutson pointed out that if we want to broaden our line of investigation, we can look at other groups. So that’s what I’ll do today.

If we take a representation of a compact Lie group $G,$ we get a map from group into a space of square matrices. Since there is a standard metric on any space of square matrices, this lets us define the distance between two points on the group. This is different than the distance defined using the shortest geodesic in the group: instead, we’re taking a straight-line path in the larger space of matrices.

If we randomly choose two points on the group, we get a random variable, namely the distance between them. We can compute the moments of this random variable, and today I’ll prove that the even moments are all integers.

So, we get a sequence of integers from any representation $\rho$ of any compact Lie group $G.$ So far we’ve only studied groups that are spheres:

• The defining representation of $\mathrm{O}(1) \cong S^0$ on the real numbers $\mathbb{R}$ gives the powers of 2.

• The defining representation of $\mathrm{U}(1) \cong S^1$ on the complex numbers $\mathbb{C}$ gives the central binomial coefficients $\binom{2n}{n}.$

• The defining representation of $\mathrm{Sp}(1) \cong S^3$ on the quaternions $\mathbb{H}$ gives the Catalan numbers.

It could be fun to work out these sequences for other examples. Our proof that the even moments are integers will give a way to calculate these sequences, not by doing integrals over the group, but by counting certain ‘random walks in the Weyl chamber’ of the group. Unfortunately, we need to count walks in a certain weighted way that makes things a bit tricky for me.

But let’s see why the even moments are integers!

If our group representation is real or quaternionic, we can either turn it into a complex representation or adapt my argument below. So, let’s do the complex case.

Let $G$ be a compact Lie group with a unitary representation $\rho$ on $\mathbb{C}^n.$ This means we have a smooth map

$\rho \colon G \to \mathrm{End}(\mathbb{C}^n)$

where $\mathrm{End}(\mathbb{C}^n)$ is the algebra of $n \times n$ complex matrices, such that

$\rho(1) = 1$

$\rho(gh) = \rho(g) \rho(h)$

and

$\rho(g) \rho(g)^\dagger = 1$

where $A^\dagger$ is the conjugate transpose of the matrix $A.$

To define a distance between points on $G$ we’ll give $\mathrm{End}(\mathbb{C}^n)$ its metric

$\displaystyle{ d(A,B) = \sqrt{ \sum_{i,j} \left|A_{ij} - B_{ij}\right|^2} }$

This clearly makes $\mathrm{End}(\mathbb{C}^n)$ into a $2n^2$-dimensional Euclidean space. But a better way to think about this metric is that it comes from the norm

$\displaystyle{ \|A\|^2 = \mathrm{tr}(AA^\dagger) = \sum_{i,j} |A_{ij}|^2 }$

where $\mathrm{tr}$ is the trace, or sum of the diagonal entries. We have

$d(A,B) = \|A - B\|$

I want to think about the distance between two randomly chosen points in the group, where ‘randomly chosen’ means with respect to normalized Haar measure: the unique translation-invariant probability Borel measure on the group. But because this measure and also the distance function are translation-invariant, we can equally well think about the distance between the identity 1 and one randomly chosen point $g$ in the group. So let’s work out this distance!

I really mean the distance between $\rho(g)$ and $\rho(1),$ so let’s compute that. Actually its square will be nicer, which is why we only consider even moments. We have

$\begin{array}{ccl} d(\rho(g),\rho(1))^2 &=& \|\rho(g) - \rho(1)\|^2 \\ \\ &=& \|\rho(g) - 1\|^2 \\ \\ &=& \mathrm{tr}\left((\rho(g) - 1)(\rho(g) - 1)^\dagger)\right) \\ \\ &=& \mathrm{tr}\left(\rho(g)\rho(g)^\dagger - \rho(g) - \rho(g)^\ast + 1\right) \\ \\ &=& \mathrm{tr}\left(2 - \rho(g) - \rho(g)^\dagger \right) \end{array}$

Now, any representation $\sigma$ of $G$ has a character

$\chi_\sigma \colon G \to \mathbb{C}$

defined by

$\chi_\sigma(g) = \mathrm{tr}(\sigma(g))$

and characters have many nice properties. So, we should rewrite the distance between $g$ and the identity using characters. We have our representation $\rho,$ whose character can be seen lurking in the formula we saw:

$d(\rho(g),\rho(1))^2 = \mathrm{tr}\left(2 - \rho(g) - \rho(g)^\dagger \right)$

But there’s another representation lurking here, the dual

$\rho^\ast \colon G \to \mathrm{End}(\mathbb{C}^n)$

given by

$\rho^\ast(g)_{ij} = \overline{\rho(g)_{ij}}$

This is a fairly lowbrow way of defining the dual representation, good only for unitary representations on $\mathbb{C}^n,$ but it works well for us here, because it lets us instantly see

$\mathrm{tr}(\rho(g)^\dagger) = \mathrm{tr}(\rho^\ast(g)) = \chi_{\rho^\ast}(g)$

This is useful because it lets us write our distance squared

$d(\rho(g),\rho(1))^2 = \mathrm{tr}\left(2 - \rho(g) - \rho(g)^\dagger \right)$

in terms of characters:

$d(\rho(g),\rho(1))^2 = 2n - \chi_\rho(g) - \chi_{\rho^\ast}(g)$

So, the distance squared is an integral linear combination of characters. (The constant function 1 is the character of the 1-dimensional trivial representation.)

And this does the job: it shows that all the even moments of our distance squared function are integers!

Why? Because of these two facts:

1) If you take an integral linear combination of characters, and raise it to a power, you get another integral linear combination of characters.

2) If you take an integral linear combination of characters, and integrate it over $G,$ you get an integer.

I feel like explaining these facts a bit further, because they’re part of a very beautiful branch of math, called character theory, which every mathematician should know. So here’s a quick intro to character theory for beginners. It’s not as elegant as I could make it; it’s not as simple as I could make it: I’ll try to strike a balance here.

There’s an abelian group $R(G)$ consisting of formal differences of isomorphism classes of representations of $G$, mod the relation

$[\rho] + [\sigma] = [\rho \oplus \sigma]$

Elements of $R(G)$ are called virtual representations of $G.$ Unlike actual representations we can subtract them. We can also add them, and the above formula relates addition in $R(G)$ to direct sums of representations.

We can also multiply them, by saying

$[\rho] [\sigma] = [\rho \otimes \sigma]$

and decreeing that multiplication distributes over addition and subtraction. This makes $R(G)$ into a ring, called the representation ring of $G.$

There’s a map

$\chi \colon R(G) \to C(G)$

where $C(G)$ is the ring of continuous complex-valued functions on $G.$ This map sends each finite-dimensional representation $\rho$ to its character $\chi_\rho.$ This map is one-to-one because we know a representation up to isomorphism if we know its character. This map is also a ring homomorphism, since

$\chi_{\rho \oplus \sigma} = \chi_\rho + \chi_\sigma$

and

$\chi_{\rho \otimes \sigma} = \chi_\rho \chi_\sigma$

These facts are easy to check directly.

We can integrate continuous complex-valued functions on $G,$ so we get a map

$\displaystyle{\int} \colon C(G) \to \mathbb{C}$

The first non-obvious fact in character theory is that we can compute inner products of characters as follows:

$\displaystyle{\int} \overline{\chi_\sigma} \chi_\rho = \dim(\mathrm{hom}(\sigma,\rho))$

where the expression at right is the dimension of the space of ‘intertwining operators’, or morphisms of representations, between the representation $\sigma$ and the representation $\rho.$

What matters most for us now is that this inner product is an integer. In particular, if $\chi_\rho$ is the character of any representation,

$\displaystyle{\int} \chi_\rho$

is an integer because we can take $\sigma$ to be the trivial representation in the previous formula, giving $\chi_\sigma = 1.$

Thus, the map

$R(G) \stackrel{\chi}{\longrightarrow} C(G) \stackrel{\int}{\longrightarrow} \mathbb{C}$

actually takes values in $\mathbb{Z}.$

Now, our distance squared function

$2n - \chi_\rho - \chi_{\rho^\ast} \in C(G)$

is actually the image under $\chi$ of an element of the representation ring, namely

$2n - [\rho] - [\rho^\ast]$

So the same is true for any of its powers—and when we integrate any of these powers we get an integer!

This stuff may seem abstract, but if you’re good at tensoring representations of some group, like $\mathrm{SU}(3),$ you should be able to use it to compute the even moments of the distance function on this group more efficiently than using the brute-force direct approach. Instead of complicated integrals we wind up doing combinatorics.

I would like to know what sequence of integers we get for $\mathrm{SU}(3).$ A much easier, less thrilling but still interesting example is $\mathrm{SO}(3).$ This is the 3-dimensional real projective space $\mathbb{R}\mathrm{P}^3,$ which we can think of as embedded in the 9-dimensional space of $3\times 3$ real matrices. It’s sort of cool that I could now work out the even moments of the distance function on this space by hand! But I haven’t done it yet.

### Clifford V. Johnson - Asymptotia

Radio Radio Summer Reading!

Friday will see me busy in the Radio world! Two things: (1) On the WNPR Connecticut morning show “Where We Live” they’ll be doing Summer reading recommendations. I’ll be on there live talking about my graphic non-fiction book The Dialogues: Conversations about the Nature of the Universe. Tune in either … Click to continue reading this post

The post Radio Radio Summer Reading! appeared first on Asymptotia.

## July 12, 2018

### Clifford V. Johnson - Asymptotia

Splashes

In case you’re wondering, after yesterday’s post… Yes I did find some time to do a bit of sketching. Here’s one that did not get finished but was fun for working the rust off… The caption from instagram says: Quick Sunday watercolour pencil dabbling … been a long time. This … Click to continue reading this post

The post Splashes appeared first on Asymptotia.

### Matt Strassler - Of Particular Significance

“Seeing” Double: Neutrinos and Photons Observed from the Same Cosmic Source

There has long been a question as to what types of events and processes are responsible for the highest-energy neutrinos coming from space and observed by scientists.  Another question, probably related, is what creates the majority of high-energy cosmic rays — the particles, mostly protons, that are constantly raining down upon the Earth.

As scientists’ ability to detect high-energy neutrinos (particles that are hugely abundant, electrically neutral, very light-weight, and very difficult to observe) and high-energy photons (particles of light, though not necessarily of visible light) have become more powerful and precise, there’s been considerable hope of getting an answer to these question.  One of the things we’ve been awaiting (and been disappointed a couple of times) is a violent explosion out in the universe that produces both high-energy photons and neutrinos at the same time, at a high enough rate that both types of particles can be observed at the same time coming from the same direction.

In recent years, there has been some indirect evidence that blazars — narrow jets of particles, pointed in our general direction like the barrel of a gun, and created as material swirls near and almost into giant black holes in the centers of very distant galaxies — may be responsible for the high-energy neutrinos.  Strong direct evidence in favor of this hypothesis has just been presented today.   Last year, one of these blazars flared brightly, and the flare created both high-energy neutrinos and high-energy photons that were observed within the same period, coming from the same place in the sky.

I have written about the IceCube neutrino observatory before; it’s a cubic kilometer of ice under the South Pole, instrumented with light detectors, and it’s ideal for observing neutrinos whose motion-energy far exceeds that of the protons in the Large Hadron Collider, where the Higgs particle was discovered.  These neutrinos mostly pass through Ice Cube undetected, but one in 100,000 hits something, and debris from the collision produces visible light that Ice Cube’s detectors can record.   IceCube has already made important discoveries, detecting a new class of high-energy neutrinos.

On Sept 22 of last year, one of these very high-energy neutrinos was observed at IceCube. More precisely, a muon created underground by the collision of this neutrino with an atomic nucleus was observed in IceCube.  To create the observed muon, the neutrino must have had a motion-energy tens of thousand times larger than than the motion-energy of each proton at the Large Hadron Collider (LHC).  And the direction of the neutrino’s motion is known too; it’s essentially the same as that of the observed muon.  So IceCube’s scientists knew where, on the sky, this neutrino had come from.

(This doesn’t work for typical cosmic rays; protons, for instance, travel in curved paths because they are deflected by cosmic magnetic fields, so even if you measure their travel direction at their arrival to Earth, you don’t then know where they came from. Neutrinos, beng electrically neutral, aren’t affected by magnetic fields and travel in a straight line, just as photons do.)

Very close to that direction is a well-known blazar (TXS-0506), four billion light years away (a good fraction of the distance across the visible universe).

The IceCube scientists immediately reported their neutrino observation to scientists with high-energy photon detectors.  (I’ve also written about some of the detectors used to study the very high-energy photons that we find in the sky: in particular, the Fermi/LAT satellite played a role in this latest discovery.) Fermi/LAT, which continuously monitors the sky, was already detecting high-energy photons coming from the same direction.   Within a few days the Fermi scientists had confirmed that TXS-0506 was indeed flaring at the time — already starting in April 2017 in fact, six times as bright as normal.  With this news from IceCube and Fermi/LAT, many other telescopes (including the MAGIC cosmic ray detector telescopes among others) then followed suit and studied the blazar, learning more about the properties of its flare.

Now, just a single neutrino on its own isn’t entirely convincing; is it possible that this was all just a coincidence?  So the IceCube folks went back to their older data to snoop around.  There they discovered, in their 2014-2015 data, a dramatic flare in neutrinos — more than a dozen neutrinos, seen over 150 days, had come from the same direction in the sky where TXS-0506 is sitting.  (More precisely, nearly 20 from this direction were seen, in a time period where normally there’d just be 6 or 7 by random chance.)  This confirms that this blazar is indeed a source of neutrinos.  And from the energies of the neutrinos in this flare, yet more can be learned about this blazar, and how it makes  high-energy photons and neutrinos at the same time.  Interestingly, so far at least, there’s no strong evidence for this 2014 flare in photons, except perhaps an increase in the number of the highest-energy photons… but not in the total brightness of the source.

The full picture, still emerging, tends to support the idea that the blazar arises from a supermassive black hole, acting as a natural particle accelerator, making a narrow spray of particles, including protons, at extremely high energy.  These protons, millions of times more energetic than those at the Large Hadron Collider, then collide with more ordinary particles that are just wandering around, such as visible-light photons from starlight or infrared photons from the ambient heat of the universe.  The collisions produce particles called pions, made from quarks and anti-quarks and gluons (just as protons are), which in turn decay either to photons or to (among other things) neutrinos.  And its those resulting photons and neutrinos which have now been jointly observed.

Since cosmic rays, the mysterious high energy particles from outer space that are constantly raining down on our planet, are mostly protons, this is evidence that many, perhaps most, of the highest energy cosmic rays are created in the natural particle accelerators associated with blazars. Many scientists have suspected that the most extreme cosmic rays are associated with the most active black holes at the centers of galaxies, and now we have evidence and more details in favor of this idea.  It now appears likely that that this question will be answerable over time, as more blazar flares are observed and studied.

The announcement of this important discovery was made at the National Science Foundation by Francis Halzen, the IceCube principal investigator, Olga Botner, former IceCube spokesperson, Regina Caputo, the Fermi-LAT analysis coordinator, and Razmik Mirzoyan, MAGIC spokesperson.

The fact that both photons and neutrinos have been observed from the same source is an example of what people are now calling “multi-messenger astronomy”; a previous example was the observation in gravitational waves, and in photons of many different energies, of two merging neutron stars.  Of course, something like this already happened in 1987, when a supernova was seen by eye, and also observed in neutrinos.  But in this case, the neutrinos and photons have energies millions and billions of times larger!

### John Baez - Azimuth

Random Points on a Sphere (Part 2)

This is the tale of a mathematical adventure. Last time our hardy band of explorers discovered that if you randomly choose two points on the unit sphere in 1-, 2- or 4-dimensional space and look at the probability distribution of their distances, then the even moments of this probability distribution are always integers. I gave a proof using some group representation theory.

On the other hand, with the help of Mathematica, Greg Egan showed that we can work out these moments for a sphere in any dimension by actually doing the bloody integrals.

He looked at the nth moment of the distance for two randomly chosen points in the unit sphere in $\mathbb{R}^d,$ and he got

$\displaystyle{ \text{moment}(d,n) = \frac{2^{d+n-2} \Gamma(\frac{d}{2}) \Gamma(\frac{1}{2} (d+n-1))}{\sqrt{\pi} \, \Gamma(d+ \frac{n}{2} - 1)} }$

This looks pretty scary, but you can simplify it using the relation between the gamma function and factorials. Remember, for integers we have

$\Gamma(n) = (n-1)!$

We also need to know $\Gamma$ at half-integers, which we can get knowing

$\Gamma(\frac{1}{2}) = \sqrt{\pi}$

and

$\Gamma(x + 1) = x \Gamma(x)$

Using these we can express moment(d,n) in terms of factorials, but the details depend on whether d and n are even or odd.

I’m going to focus on the case where both the dimension d and the moment number n are even, so let

$d = 2e, \; n = 2m$

In this case we get

$\text{moment}(2e,2m) = \displaystyle{\frac{\binom{2(e+m-1)} {m}}{\binom{e+m-1}{m}} }$

Here ‘we’ means that Greg Egan did all the hard work:

From this formula

$\text{moment}(2e,2m) = \displaystyle{\frac{\binom{2(e+m-1)} {m}}{\binom{e+m-1}{m}} }$

you can show directly that the even moments in 4 dimensions are Catalan numbers:

$\text{moment}(4,2m) = C_{m+1}$

while in 2 dimensions they are binomial coefficients:

$\mathrm{moment}(2,2m) = \displaystyle{ {2m \choose m} }$

More precisely, they are ‘central’ binomial cofficients, forming the middle column of Pascal’s triangle:

$1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, \dots$

So, it seems that with some real work one can get vastly more informative results than with my argument using group representation theory. The only thing you don’t get, so far, is an appealing explanation of why the even moments are integral in dimensions 1, 2 and 4.

The computational approach also opens up a huge new realm of questions! For example, are there any dimensions other than 1, 2 and 4 where the even moments are all integral?

I was especially curious about dimension 8, where the octonions live. Remember, 1, 2 and 4 are the dimensions of the associative normed division algebras, but there’s also a nonassociative normed division algebra in dimension 8: the octonions.

The d = 8 row seemed to have a fairly high fraction of integer entries:

I wondered if there were only finitely many entries in the 8th row that weren’t integers. Greg Egan did a calculation and replied:

The d=8 moments don’t seem to become all integers permanently at any point, but the non-integers become increasingly sparse.

He also got evidence suggesting that for any even dimension d, a large fraction of the even moments are integers. After some further conversation he found the nice way to think about this. Recall that

$\text{moment}(2e,2m) = \displaystyle{\frac{\binom{2(e+m-1)} {m}}{\binom{e+m-1}{m}} }$

If we let

$r = e-1$

then this moment is just

$\text{moment}(2r+2,2m) = \displaystyle{\frac{\binom{2(m+r)}{m}}{\binom{m+r}{m}} }$

so the question becomes: when is this an integer?

It’s good to think about this naively a bit. We can cancel out a bunch of stuff in that ratio of binomial coefficents and write it like this:

$\displaystyle{ \text{moment}(2r+2,2m) = \frac{(2r+m+1) \cdots (2r+2m)}{(r+1) \cdots (r+m)} }$

So when is this an integer? Let’s do the 8th moment in 4 dimensions:

$\text{moment}(4,8) = \displaystyle{ \frac{7 \cdot 8 \cdot 9 \cdot 10 }{2 \cdot 3 \cdot 4 \cdot 5} }$

This is an integer, namely the Catalan number 42: the Answer to the Ultimate Question of Life, the Universe, and Everything.  But apparently we had to be a bit ‘lucky’ to get an integer. For example, we needed the 10 on top to deal with the 5 on the bottom.

It seems plausible that our chances of getting an integer increase as the moment gets big compared to the dimension. For example, try the 4th moment in dimension 10:

$\text{moment}(10,4) = \displaystyle{ \frac{11 \cdot 12}{5 \cdot 6}}$

This not an integer, because we’re just not multiplying enough numbers to handle the prime 5 in the denominator. The 6th moment in dimension 10 is also not an integer. But if we try the 8th moment, we get lucky:

$\text{moment}(10,8) = \displaystyle{ \frac{13 \cdot 14 \cdot 15 \cdot 16}{5 \cdot 6 \cdot 7 \cdot 8}}$

This is an integer! We’ve got enough in the numerator to handle everything in the denominator.

Greg posted a question about this on MathOverflow:

• Greg Egan, When does doubling the size of a set multiply the number of subsets by an integer?, 9 July 2018.

He got a very nice answer from a mysterious figure named Lucia, who pointed out relevant results from this interesting paper:

• Carl Pomerance, Divisors of the middle binomial coefficient, American Mathematical Monthly 122 (2015), 636–644.

Using these, Lucia proved a result that implies the following:

Theorem. If we fix a sphere of some even dimension, and look at the even moments of the probability distribution of distances between randomly chosen points on that sphere, from the 2nd moment to the (2m)th, the fraction of these that are integers approaches 1 as m → ∞.

On the other hand, Lucia also believes Pomerance’s techniques can be used to prove a result that would imply this:

Conjecture. If we fix a sphere of some even dimension > 4, and consider the even moments of the probability distribution of distances between randomly chosen points on that sphere, infinitely many of these are not integers.

In summary: we’re seeing a more or less typical rabbit-hole in mathematics. We started by trying to understand how noncommutative quaternions are on average. We figured that out, but we got sidetracked by thinking about how far points on a sphere are on average. We started calculating, we got interested in moments of the probability distribution of distances, we noticed that the Catalan numbers show up, and we got pulled into some representation theory and number theory!

I wouldn’t say our results are earth-shaking, but we definitely had fun and learned a thing or two. One thing at least is clear. In pure math, at least, it pays to follow the ideas wherever they lead. Math isn’t really divided into different branches—it’s all connected!

### Afterword

Oh, and one more thing. Remember how this quest started with John D. Cook numerically computing the average of $|xy - yx|$ over unit quaternions? Well, he went on and numerically computed the average of $|(xy)z - x(yz)|$ over unit octonions!

• John D. Cook, How close is octonion multiplication to being associative?, 9 July 2018.

He showed the average is about 1.095, and he created this histogram:

Later, Greg Egan computed the exact value! It’s

$\displaystyle{ \frac{147456}{42875 \pi} \approx 1.0947335878 \dots }$

On Twitter, Christopher D. Long, whose handle is @octonion, pointed out the hidden beauty of this answer—it equals

$\displaystyle{ \frac{2^{14}3^2}{5^3 7^3 \pi} }$

Nice! Here’s how Greg did this calculation:

• Greg Egan, The average associator, 12 July 2018.

### Details

If you want more details on the proof of this:

Theorem. If we fix a sphere of some even dimension, and look at the even moments of the probability distribution of distances between randomly chosen points on that sphere, from the 2nd moment to the (2m)th, the fraction of these that are integers approaches 1 as m → ∞.

you should read Greg Egan’s question on Mathoverflow, Lucia’s reply, and Pomerance’s paper. Here is Greg’s question:

For natural numbers $m, r$, consider the ratio of the number of subsets of size $m$ taken from a set of size $2(m+r)$ to the number of subsets of the same size taken from a set of size $m+r$:

$\displaystyle{ R(m,r)=\frac{\binom{2(m+r)}{m}}{\binom{m+r}{m}} }$

For $r=0$ we have the central binomial coefficients, which of course are all integers:

$\displaystyle{ R(m,0)=\binom{2m}{m} }$

For $r=1$ we have the Catalan numbers, which again are integers:

$\displaystyle{ R(m,1)=\frac{\binom{2(m+1)}{m}}{m+1}=\frac{(2(m+1))!}{m!(m+2)!(m+1)}}$
$\displaystyle{ = \frac{(2(m+1))!}{(m+2)!(m+1)!}=C_{m+1}}$

However, for any fixed $r\ge 2$, while $R(m,r)$ seems to be mostly integral, it is not exclusively so. For example, with $m$ ranging from 0 to 20000, the number of times $R(m,r)$ is an integer for $r=$ 2,3,4,5 are 19583, 19485, 18566, and 18312 respectively.

I am seeking general criteria for $R(m,r)$ to be an integer.

Edited to add:

We can write:

$\displaystyle{ R(m,r) = \prod_{k=1}^m{\frac{m+2r+k}{r+k}} }$

So the denominator is the product of $m$ consecutive numbers $r+1, \ldots, m+r$, while the numerator is the product of $m$ consecutive numbers $m+2r+1,\ldots,2m+2r$. So there is a gap of $r$ between the last of the numbers in the denominator and the first of the numbers in the numerator.

Lucia replied:

Put $n=m+r$, and then we can write $R(m,r)$ more conveniently as

$\displaystyle{ R(m,r) = \frac{(2n)!}{m! (n+r)!} \frac{m! r!}{n!} = \frac{\binom{2n}{n} }{\binom{n+r}{r}}. }$

So the question essentially becomes one about which numbers $n+k$ for $k=1, \ldots, r$ divide the middle binomial coefficient $\binom{2n}{n}$. Obviously when $k=1$, $n+1$ always divides the middle binomial coefficient, but what about other values of $k$? This is treated in a lovely Monthly article of Pomerance:

• Carl Pomerance, Divisors of the middle binomial coefficient, American Mathematical Monthly 122 (2015), 636–644.

Pomerance shows that for any $k \ge 2$ there are infinitely many integers with $n+k$ not dividing $\binom{2n}{n}$, but the set of integers $n$ for which $n+k$ does divide $\binom{2n}{n}$ has density $1$. So for any fixed $r$, for a density $1$ set of values of $n$ one has that $(n+1), \ldots, (n+k)$ all divide $\binom{2n}{n}$, which means that their lcm must divide $\binom{2n}{n}$. But one can check without too much difficulty that the lcm of $n+1, \ldots, n+k$ is a multiple of $\binom{n+k}{k}$, and so for fixed $r$ one deduces that $R(m,r)$ is an integer for a set of values $m$ with density 1. (Actually, Pomerance mentions explicitly in (5) of his paper that $(n+1)(n+2)\cdots (n+k)$ divides $\binom{2n}{n}$ for a set of full density.)

I haven’t quite shown that $R(m,r)$ is not an integer infinitely often for $r\ge 2$, but I think this can be deduced from Pomerance’s paper (by modifying his Theorem 1).

I highly recommend Pomerance’s paper—you don’t need to care much about which integers divide

$\displaystyle{ \binom{2n}{n} }$

to find it interesting, because it’s full of clever ideas and nice observations.

### Clifford V. Johnson - Asymptotia

Retreated

Sorry I've been quiet on the blog for a few weeks. An unusually long gap, I think (although those of you following on instagram, twitter, Facebook and so forth have not noticed a gap). I've been hiding out at the Aspen Center for Physics for a while.

You've probably read things I've written about it here many times in past years, but if not, here's a movie that I produced/directed/designed/etc about it some time back. (You can use the search bar upper right to find earlier posts mentioning Aspen, or click here.)

Anyway, I arrived and pretty much immediately got stuck into an interesting project, as I had an idea that I just had to pursue. I filled up a whole notebook with computations and mumblings about ideas, and eventually a narrative (and a nice set of results) has emerged. So I've been putting those into some shape. I hope to tell you about it all soon. You'll be happy to know it involves black holes, entropy, thermodynamics, and quantum information [...] Click to continue reading this post

The post Retreated appeared first on Asymptotia.

## July 10, 2018

### John Baez - Azimuth

Random Points on a Sphere (Part 1)

John D. Cook, Greg Egan, Dan Piponi and I had a fun mathematical adventure on Twitter. It started when John Cook wrote a program to compute the probability distribution of distances $|xy - yx|$ where $x$ and $y$ were two randomly chosen unit quaternions:

• John D. Cook, How far is xy from yx on average for quaternions?, 5 July 2018.

Three things to note before we move on:

• Click the pictures to see the source and get more information—I made none of them!

• We’ll be ‘randomly choosing’ lots of points on spheres of various dimensions. Whenever we do this, I mean that they’re chosen independently, and uniformly with respect to the unique rotation-invariant Borel measure that’s a probability measure on the sphere. In other words: nothing sneaky, just the most obvious symmetrical thing!

• We’ll be talking about lots of distances between points on the unit sphere in $n$ dimensions. Whenever we do this, I mean the Euclidean distance in $\mathbb{R}^n$, not the length of the shortest path on the sphere connecting them.

Okay:

If you look at the histogram above, you’ll see the length $|xy - yx|$ is between 0 and 2. That’s good, since $xy$ and $yx$ are on the unit sphere in 4 dimensions. More interestingly, the mean looks bigger than 1. John Cook estimated it at 1.13.

Greg Egan went ahead and found that the mean is exactly

$\displaystyle{\frac{32}{9 \pi}} \approx 1.13176848421 \dots$

He did this by working out a formula for the probability distribution:

All this is great, but it made me wonder how surprised I should be. What’s the average distance between two points on the unit sphere in 4 dimensions, anyway?

Greg Egan worked this out too:

So, the mean distance $|x - y|$ for two randomly chosen unit quaternions is

$\displaystyle{\frac{64}{15 \pi}} \approx 1.35812218105\dots$

The mean of $|xy - yx|$ is smaller than this. In retrospect this makes sense, since I know what quaternionic commutators are like: for example the points $x = \pm 1$ at the ‘north and south poles’ of the unit sphere commute with everybody. However, we can now say the mean of $|xy - yx|$ is exactly

$\displaystyle{\frac{32}{9\pi} } \cdot \frac{15 \pi}{64} = \frac{5}{6}$

times the mean of $|x - y|,$ and there’s no way I could have guessed that.

While trying to get a better intuition for this, I realized that as you go to higher and higher dimensions, and you standing at the north pole of the unit sphere, the chance that a randomly chosen other point is quite near the equator gets higher and higher! That’s how high dimensions work. So, the mean value of $|x - y|$ should get closer and closer to $\sqrt{2}.$ And indeed, Greg showed that this is true:

The graphs here show the probability distributions of distances for randomly chosen pairs of points on spheres of various dimensions. As the dimension increases, the probability distribution gets more sharply peaked, and the mean gets closer to $\sqrt{2}.$

Greg wrote:

Here’s the general formula for the distribution, with plots for n=2,…,10. The mean distance does tend to √2, and the mean of the squared distance is always exactly 2, so the variance tends to zero.

But now comes the surprising part.

Dan Piponi looked at the probability distribution of distances $s = |x - y|$ in the 4-dimensional case:

$P(s) = \displaystyle{\frac{s^2\sqrt{4 - s^2}}{\pi} }$

and somehow noticed that its moments

$\int_0^2 P(s) s^{n} \, dx$

when n is even, are the Catalan numbers!

Now if you don’t know about moments of probability distributions you should go read about those, because they’re about a thousand times more important than anything you’ll learn here.

And if you don’t know about Catalan numbers, you should go read about those, because they’re about a thousand times more fun than anything you’ll learn here.

So, I’ll assume you know about those. How did Dan Piponi notice that the Catalan numbers

$C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, C_5 = 42, \dots$

were the even moments of this probability distribution? Maybe it’s because he recently managed to get ahold of Richard Stanley’s book on Amazon for just $11 instead of its normal price of$77.

(I don’t know how that happened. Some people write 7’s that look like 1’s, but….)

Anyway, you’ll notice that this strange phenomenon is all about points on the unit sphere in 4 dimensions. It doesn’t seem to involve quaternions anymore! So I asked if something similar happens in other dimensions, maybe giving us other interesting sequences of integers.

Greg Egan figured it out, and got some striking results:

Here d is the dimension of the Euclidean space containing our unit sphere, and Egan is tabulating the nth moment of the probability distribution of distances between two randomly chosen points on that sphere. The gnarly formula on top is a general expression for this moment in terms of the gamma function.

The obvious interesting feature of this table is that only for d = 2 and d = 4 rows are all the entries integers.

But Dan made another great observation: Greg left out the rather trivial d = 1 row, and that all the entries of this row would be integers too! Even better, d = 1, 2, and 4 are the dimensions of the associative normed division algebras: the real numbers, the complex numbers and the quaternions!

This made me eager to find a proof that all the even moments of the probability distribution of distances between points on the unit sphere in $\mathbb{R}^d$ are integers when $\mathbb{R}^d$ is an associative normed division algebra.

The first step is to notice the significance of even moments.

First, we don’t need to choose both points on the sphere randomly: we can fix one and let the other vary. So, we can think of the distance

$D(x) = |(x_1, \dots, x_d) - (1, \dots, 0)| = \sqrt{(x_1 - 1)^2 + x_2^2 + \cdots + x_d^2}$

as a function on the sphere, or more generally a function of $x \in \mathbb{R}^d.$ And when we do this we instantly notice that the square root is rather obnoxious, but all the even powers of the function $D$ are polynomials on $\mathbb{R}^d.$

Then, we notice that restricting polynomials from Euclidean space to the sphere is how we get spherical harmonics, so this problem is connected to spherical harmonics and ‘harmonic analysis’. The nth moment of the probability distribution of distances between points on the unit sphere in $\mathbb{R}^d$ is

$\int_{S^{d-1}} D^n$

where we are integrating with respect to the rotation-invariant probability measure on the sphere. We can rewrite this as an inner product in $L^2(S^{d-1}),$ namely

$\langle D^n , 1 \rangle$

where 1 is the constant function equal to 1 on the whole sphere.

We’re looking at the even moments, so let n = 2m. Now, why should

$\langle D^{2m} , 1 \rangle$

be an integer when d = 1, 2 and 4? Well, these are the cases where the sphere $S^{d-1}$ is a group! For d = 1,

$S^0 \cong \mathbb{Z}/2$

is the multiplicative group of unit real numbers, $\{\pm 1\}.$ For d = 2,

$S^1 \cong \mathrm{U}(1)$

is the multiplicative group of unit complex numbers. And for d = 4,

$S^3 \cong \mathrm{SU}(2)$

is the multiplicative group of unit quaternions.

These are compact Lie groups, and $L^2$ of a compact Lie group is very nice. Any finite-dimensional representation $\rho$ of a compact Lie group $G$ gives a function $\chi_\rho \in L^2(G)$ called its character, given by

$\chi_\rho(g) = \mathrm{tr}(\rho(g))$

And it’s well-known that for two representations $\rho$ and $\sigma,$ the inner product

$\langle \chi_\rho, \chi_\sigma \rangle$

is an integer! In fact it’s a natural number: just the dimension of the space of intertwining operators from $\rho$ to $\sigma.$ So, we should try to prove that

$\langle D^{2m} , 1 \rangle$

is an integer this way. The function 1 is the character of the trivial 1-dimensional representation, so we’re okay there. What about $D^{2m}?$

Well, there’s a way to take the mth tensor power $\rho^{\otimes m}$ of a representation $\rho$: you just tensor the representation with itself $m$ times. And then you can easily show

$\chi_{\rho^{\otimes m}} = (\chi_\rho)^m$

So, if we can show $D^2$ is the character of a representation, we’re done: $D^{2m}$ will be one as well, and the inner product

$\langle D^{2m}, 1 \rangle$

will be an integer! Great plan!

Unfortunately, $D^2$ is not the character of a representation.

Unless $\rho$ is the completely silly 0-dimensional representation we have

$\chi_\rho(1) = \mathrm{tr}(\rho(1)) = \dim(\rho) > 0$

where $1$ is the identity element of $G.$ But suppose we let $D(g)$ be the distance of $g$ from the identity element—the natural choice of ‘north pole’ when we make our sphere into a group. Then we have

$D(1)^2 = 0$

So $D^2$ can’t be a character. (It’s definitely not the character of the completely silly 0-dimensional representation: that’s zero.)

But there’s a well-known workaround. We can work with virtual representations, which are formal differences of representations, like this:

$\delta = \rho - \sigma$

The character of a virtual representation is defined in the obvious way

$\chi_\delta = \chi_\rho - \chi_\sigma$

Since the inner product of characters of two representations is a natural number, the inner product of characters of two virtual representations will be an integer. And we’ll be completely satisfied if we prove that

$\langle D^{2m}, 1 \rangle$

is an integer, since it’s obviously ≥ 0.

So, we just need to show that $D^{2m}$ is the character of a virtual representation. This will easily follow if we can show $D^2$ itself is the character of a virtual representation: you can tensor virtual representations, and then their characters multiply.

So, let’s do it! I’ll just do the quaternionic case. I’m doing it right now, thinking out loud here. I figure I should start with a really easy representation, take its character, compare that to our function $D^2,$ and then fix it by subtracting something.

Let $\rho$ be the spin-1/2 representation of $\mathrm{SU}(2),$ which just sends every matrix in $\mathrm{SU}(2)$ to itself. Every matrix in $\mathrm{SU}(2)$ is conjugate to one of the form

$g = \left(\begin{array}{cc} \exp(i\theta) & 0 \\ 0 & \exp(-i\theta) \end{array}\right)$

so we can just look at those, and we have

$\chi_\rho(g) = \mathrm{tr}(\rho(g)) = \mathrm{tr}(g) = 2 \cos \theta$

On the other hand, we can think of $g$ as a unit quaternion, and then

$g = \cos \theta + i \sin \theta$

where now $i$ stands for the quaternion of that name! So, its distance from 1 is

$D(g) = |\cos \theta + i \sin \theta - 1|$

and if we square this we get

$D(g)^2 = (1 - \cos \theta)^2 + \sin^2 \theta = 2 - 2 \cos \theta$

So, we’re pretty close:

$D(g)^2 = 2 - \chi_{\rho}$

In particular, this means $D^2$ is the character of the virtual representation

$(1 \oplus 1) - \rho$

where $1$ is the 1d trivial rep and $\rho$ is the spin-1/2 rep.

So we’re done!

At least we’re done showing the even moments of the distance between two randomly chosen points on the 3-sphere is an integer. The 1-sphere and 0-sphere cases are similar.

But course there’s another approach! We can just calculate the darn moments and see what we get. This leads to deeper puzzles, which we have not completely solved. But I’ll talk about these next time, in Part 2.

### CERN Bulletin

Summer is coming, enjoy our offers for the aquatic parks: Walibi & Aquaparc!

Summer is coming, enjoy our offers for the aquatic parks: Walibi & Aquaparc!

Walibi:

Tickets "Zone terrestre": 25 € instead of de 31 €.

Access to Aqualibi: 5 € instead of 8 € on presentation of your Staff Association member ticket.

Free for children under 100 cm, with limited access to the attractions.

Free car park.

*  *  *  *  *  *

Aquaparc:

Full day ticket:

• Children: 33 CHF instead of 39 CHF
• Adults: 33 CHF instead of 49 CHF

Free for children under 5.

### CERN Bulletin

Interfon

Cooperative open to international civil servants. We welcome you to discover the advantages and discounts negotiated with our suppliers either on our website www.interfon.fr or at our information office located at CERN, on the ground floor of bldg. 504, open Monday through Friday from 12.30 to 15.30.

### CERN Bulletin

Club de pétanque : résultats du Challenge Carteret 2018

Vingt-six joueurs était présent ce jeudi 5 juillet 2018 pour disputer le Challenge de notre regretté ami Claude Carteret qui vu le temps a été organisé au boulodrome de Saint Genis Pouilly.

Nos habitués de la table de marque père et fils Claude et David Jouve après trois parties parfois assez serrées proclamait vainqueur notre président du club Claude Cerruti avec trois parties gagnées devançant au goal-average Jean-Claude Frot de retour parmi nous et toujours aussi adroit.

Le troisième David Jouve qui cumulait les tâches de joueur et arbitre.

La première féminine Gabrielle Cerrutin, elle aussi joueuse battante et appliquée.

Cette soirée se terminait par un bon repas préparé par Sylvie Jouve et sa fille Jennifer que nous remercions infiniment.

Rendez-vous au prochain concours, Challenge Patrick Durand qui aura lieu le jeudi 26 juillet 2018.

## July 09, 2018

### CERN Bulletin

Reducing waste in the workplace

Paper, cardboard, PET, aluminium cans, glass, Nespresso capsules, wood and worksite waste: in 2016, CERN produced no less than 5700 tonnes of waste, about 50% of which was recycled. How can we improve on this?

Many measures are already in place at CERN to limit waste and encourage recycling. Several articles have been published to raise awareness among users, Cernois and visitors on ways to limit our waste: https://home.cern/fr/cern-people/updates/2018/05/much-less-plastic-thats-fantastic

Did you know?

NOVAE restaurants offer a 10 cent discount at the cash register for people who use their own cups/mugs.

Focusing on the ubiquitous and well-loved coffee break, note how much waste can be generated, from the sugar packets to coffee pods, plastic cutlery and especially disposable plastic or paper cups. In the same way our shopping outings are accompanied by reusable shopping bags, why not bring your own mug or cup for your morning or afternoon coffee? Also  think about the packaging of the products you consume (coffee, sugar, biscuits...) favouring larger quantities as opposed to single items is often cheaper and especially a source of less waste.

The Staff Association encourages these initiatives and would like to hear your ideas, and environmental concerns. Feel free to contact us by email: staff association@cern.ch or speak directly with your delegates.

### CERN Bulletin

Questions about your employment and working conditions at CERN? Contact your nearest staff association representative!

One of the Staff Association's Infom-Action Commission’s responsibilities is facilitating direct communication between members of the personnnel and the Association.

With the aim of finding an efficient means to identify staff association representatives, the commission worked closely with the SMB department and using the GIS portal, set-up a platform for you to look up your representative and their physical location on the CERN Site.

How to find and contact your representatives?

Your delegates are located all over CERN, on the Meyrin and Prevessin sites. Today, by going to the SA website (http://cern.ch/go/7hNM) you can easily locate your nearest delegate.

In one click, various information is provided such as e-mail address, telephone  number as well as group and department. Additional information is also available by clicking on the "more information" option.

Feel free to meet them!

### The n-Category Cafe

Beyond Classical Bayesian Networks

guest post by Pablo Andres-Martinez and Sophie Raynor

In the final installment of the Applied Category Theory seminar, we discussed the 2014 paper “Theory-independent limits on correlations from generalized Bayesian networks” by Henson, Lal and Pusey.

In this post, we’ll give a short introduction to Bayesian networks, explain why quantum mechanics means that one may want to generalise them, and present the main results of the paper. That’s a lot to cover, and there won’t be a huge amount of category theory, but we hope to give the reader some intuition about the issues involved, and another example of monoidal categories used in causal theory.

## Introduction

Bayesian networks are a graphical modelling tool used to show how random variables interact. A Bayesian network consists of a pair $\left(G,P\right)\left(G,P\right)$ of directed acyclic graph (DAG) $GG$ together with a joint probability distribution $PP$ on its nodes, satisfying the Markov condition. Intuitively the graph describes a flow of information.

The Markov condition says that the system doesn’t have memory. That is, the distribution on a given node $YY$ is only dependent on the distributions on the nodes $XX$ for which there is an edge $X\to YX \rightarrow Y$. Consider the following chain of binary events. In spring, the pollen in the air may cause someone to have an allergic reaction that may make them sneeze.

In this case the Markov condition says that given that you know that someone is having an allergic reaction, whether or not it is spring is not going to influence your belief about the likelihood of them sneezing. Which seems sensible.

Bayesian networks are useful

• as an inference tool, thanks to belief propagation algorithms,

• and because, given a Bayesian network $\left(G,P\right)\left(G,P\right)$, we can describe d-separation properties on $GG$ which enable us to discover conditional independences in $PP$.

It is this second point that we’ll be interested in here.

Before getting into the details of the paper, let’s try to motivate this discussion by explaining its title: “Theory-independent limits on correlations from generalized Bayesian networks" and giving a little more background to the problem it aims to solve.

Crudely put, the paper aims to generalise a method that assumes classical mechanics to one that holds in quantum and more general theories.

Classical mechanics rests on two intuitively reasonable and desirable assumptions, together called local causality,

• Causality:

Causality is usually treated as a physical primitive. Simply put it is the principle that there is a (partial) ordering of events in space time. In order to have information flow from event $AA$ to event $BB$, $AA$ must be in the past of $BB$.

Physicists often define causality in terms of a discarding principle: If we ignore the outcome of a physical process, it doesn’t matter what process has occurred. Or, put another way, the outcome of a physical process doesn’t change the initial conditions.

• Locality:

Locality is the assumption that, at any given instant, the values of any particle’s properties are independent of any other particle. Intuitively, it says that particles are individual entities that can be understood in isolation of any other particle.

Physicists usually picture particles as having a private list of numbers determining their properties. The principle of locality would be violated if any of the entries of such a list were a function whose domain is another particle’s property values.

In 1935 Einstein, Podolsky and Rosen showed that quantum mechanics (which was a recently born theory) predicted that a pair of particles could be prepared so that applying an action on one of them would instantaneously affect the other, no matter how distant in space they were, thus contradicting local causality. This seemed so unreasonable that the authors presented it as evidence that quantum mechanics was wrong.

But Einstein was wrong. In 1964, John S. Bell set the bases for an experimental test that would demonstrate that Einstein’s “spooky action at a distance” (Einstein’s own words), now known as entanglement, was indeed real. Bell’s experiment has been replicated countless of times and has plenty of variations. This video gives a detailed explanation of one of these experiments, for a non-physicist audience.

But then, if acting on a particle has an instantaneous effect on a distant point in space, one of the two principle above is violated: On one hand, if we acted on both particles at the same time, each action being a distinct event, both would be affecting each other’s result, so it would not be possible to decide on an ordering; causality would be broken. The other option would be to reject locality: a property’s value may be given by a function, so the resulting value may instantaneously change when the distant ‘domain’ particle is altered. In that case, the particles’ information was never separated in space, as they were never truly isolated, so causality is preserved.

Since causality is integral to our understanding of the world and forms the basis of scientific reasoning, the standard interpretation of quantum mechanics is to accept non-locality.

The definition of Bayesian networks implies a discarding principle and hence there is a formal sense in which they are causal (even if, as we shall see, the correlations they model do not always reflect the temporal order). Under this interpretation, the causal theory Bayesian networks describe is classical. Precisely, they can only model probability distributions that satisfy local causality. Hence, in particular, they are not sufficient to model all physical correlations.

The goal of the paper is to develop a framework that generalises Bayesian networks and d-separation results, so that we can still use graph properties to reason about conditional dependence under any given causal theory, be it classical, quantum, or even more general. In particular, this theory will be able to handle all physically observed correlations, and all theoretically postulated correlations.

Though category theory is not mentioned explicitly, the authors achieve their goal by using the categorical framework of operational probablistic theories (OPTs).

## Bayesian networks and d-separation

Consider the situation in which we have three Boolean random variables. Alice is either sneezing or she is not, she either has a a fever or she does not, and she may or may not have flu.

Now, flu can cause both sneezing and fever, that is

$P\left(\mathrm{sneezing}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right)\ne P\left(\mathrm{sneezing}\right)\phantom{\rule{thickmathspace}{0ex}}\text{and likewise}\phantom{\rule{thickmathspace}{0ex}}P\left(\mathrm{fever}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right)\ne P\left(\mathrm{fever}\right)P\left(sneezing \ | \ flu \right) \neq P\left( sneezing\right) \ \text\left\{ and likewise \right\} \ P\left(fever \ | \ flu \right) \neq P\left( fever\right)$

so we could represent this graphically as

Moreover, intuitively we wouldn’t expect there to be any other edges in the above graph. Sneezing and fever, though correlated - each is more likely if Alice has flu - are not direct causes of each other. That is,

$P\left(\mathrm{sneezing}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{fever}\right)\ne P\left(\mathrm{sneezing}\right)\phantom{\rule{thickmathspace}{0ex}}\text{but}\phantom{\rule{thickmathspace}{0ex}}P\left(\mathrm{sneezing}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{fever},\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right)=P\left(\mathrm{sneezing}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right).P\left(sneezing \ | \ fever \right) \neq P\left(sneezing\right) \ \text\left\{ but \right\} \ P\left(sneezing \ | \ fever, \ flu \right) = P\left(sneezing \ | \ flu\right).$

### Bayesian networks

Let $GG$ be a directed acyclic graph or DAG $GG$. (Here a directed graph is a presheaf on ($•⇉•\bullet \rightrightarrows \bullet$)).

The set $\mathrm{Pa}\left(Y\right)Pa\left(Y\right)$ of parents of a node $YY$ of $GG$ contains those nodes $XX$ of $GG$ such that there is a directed edge $X\to YX \to Y$.

So, in the example above $\mathrm{Pa}\left(\mathrm{flu}\right)=\varnothing Pa\left(flu\right) = \emptyset$ while $\mathrm{Pa}\left(\mathrm{fever}\right)=\mathrm{Pa}\left(\mathrm{sneezing}\right)=\left\{\mathrm{flu}\right\}Pa\left(fever\right) = Pa\left(sneezing\right) = \\left\{ flu \\right\}$.

To each node $XX$ of a directed graph $GG$, we may associate a random variable, also denoted $XX$. If $VV$ is the set of nodes of $GG$ and $\left({x}_{X}{\right)}_{X\in V}\left(x_X\right)_\left\{X \in V\right\}$ is a choice of value ${x}_{X}x_X$ for each node $XX$, such that $yy$ is the chosen value for $YY$, then $\mathrm{pa}\left(y\right)pa\left(y\right)$ will denote the $\mathrm{Pa}\left(Y\right)Pa\left(Y\right)$-tuple of values $\left({x}_{X}{\right)}_{X\in \mathrm{Pa}\left(Y\right)}\left(x_X\right)_\left\{X \in Pa\left(Y\right)\right\}$.

To define Bayesian networks, and establish the notation, let’s revise some probability basics.

Let $P\left(x,y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)P\left(x,y \ | \ z\right)$ mean $P\left(X=x\text{and}\phantom{\rule{thickmathspace}{0ex}}Y=y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}Z=z\right)P\left(X = x \text\left\{ and \right\} \ Y = y \ | \ Z = z\right)$, the probability that $XX$ has the value $xx$, and $YY$ has the value $yy$ given that $ZZ$ has the value $zz$. Recall that this is given by

$P\left(x,y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)=\frac{P\left(x,y,z\right)}{P\left(z\right)}.P\left(x,y \ |\ z\right) = \frac\left\{ P\left(x,y,z\right) \right\}\left\{P\left(z\right)\right\}.$

The chain rule says that, given a value $xx$ of $XX$ and sets of values $\Omega ,\Lambda \Omega, \Lambda$ of other random variables,

$P\left(x,\Omega \phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\Lambda \right)=P\left(x\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\Lambda \right)P\left(\Omega \phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}x,\Lambda \right).P\left(x, \Omega \ | \ \Lambda\right) = P\left( x \ | \ \Lambda\right) P\left( \Omega \ | \ x, \Lambda\right).$

Random variables $XX$ and $YY$ are said to be conditionally independent given $ZZ$, written $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp\!\!\!\!\!\!\!\perp Y \ | \ Z$, if for all values $xx$ of $XX$, $yy$ of $YY$ and $zz$ of $ZZ$

$P\left(x,y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)=P\left(x\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)P\left(y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right).P\left(x,y \ | \ z\right) = P\left(x \ | \ z\right) P\left(y \ | \ z\right).$

By the chain rule this is equivalent to

$P\left(x\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}y,z\right)=P\left(x\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right),\phantom{\rule{thickmathspace}{0ex}}\forall x,y,z.P\left(x \ | \ y,z \right) = P \left(x \ | \ z\right) , \ \forall x,y, z.$

More generally, we may replace $X,YX,Y$ and $ZZ$ with sets of random variables. So, in the special case that $ZZ$ is empty, then $XX$ and $YY$ are independent if and only if $P\left(x,y\right)=P\left(x\right)P\left(y\right)P\left(x, y\right) = P\left(x\right)P\left(y\right)$ for all $x,yx,y$.

#### Markov condition

A joint probability distribution $PP$ on the nodes of a DAG $GG$ is said to satisfy the Markov condition if for any set of random variable $\left\{{X}_{i}{\right\}}_{i=1}^{n}\\left\{X_i\\right\}_\left\{i = 1\right\}^n$ on the nodes of $GG$, with choice of values $\left\{{x}_{i}{\right\}}_{i=1}^{n}\\left\{x_i\\right\}_\left\{i = 1\right\}^n$

$P\left({x}_{i},\dots ,{x}_{n}\right)=\prod _{i=1}^{n}P\left({x}_{i}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{pa}\left({x}_{i}\right)\right).P\left(x_i, \dots, x_n\right) = \prod_\left\{i = 1\right\}^n P\left(x_i \ | \ \left\{pa\left(x_i\right)\right\}\right).$

So, for the flu, fever and sneezing example above, a distribution $PP$ satisfies the Markov condition if

$P\left(\mathrm{flu},\phantom{\rule{thickmathspace}{0ex}}\mathrm{fever},\phantom{\rule{thickmathspace}{0ex}}\mathrm{sneezing}\right)=P\left(\mathrm{fever}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right)P\left(\mathrm{sneezing}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right)P\left(\mathrm{flu}\right).P\left(flu, \ fever, \ sneezing\right) = P\left(fever \ | \ flu\right) P\left(sneezing \ | \ flu\right) P\left(flu\right).$

A Bayesian network is defined as a pair $\left(G,P\right)\left(G,P\right)$ of a DAG $GG$ and a joint probability distribution $PP$ on the nodes of $GG$ that satisfies the Markov condition with respect to $GG$. This means that each node in a Bayesian network is conditionally independent, given its parents, of any of the remaining nodes.

In particular, given a Bayesian network $\left(G,P\right)\left(G,P\right)$ such that there is a directed edge $X\to YX \to Y$, the Markov condition implies that

$\sum _{y}P\left(x,y\right)=\sum _{y}P\left(x\right)P\left(y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}x\right)=P\left(x\right)\sum _{y}P\left(y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}x\right)=P\left(x\right)\sum_\left\{y\right\} P\left(x,y\right) = \sum_y P\left(x\right) P\left(y \ | \ x\right) = P\left(x\right) \sum_y P\left(y \ | \ x\right) = P\left(x\right)$

which may be interpreted as a discard condition. (The ordering is reflected by the fact that we can’t derive $P\left(y\right)P\left(y\right)$ from ${\sum }_{x}P\left(x,y\right)={\sum }_{x}P\left(x\right)P\left(y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}x\right)\sum_\left\{x\right\} P\left(x,y\right) = \sum_x P\left(x\right) P\left(y \ | \ x\right)$.)

Let’s consider some simple examples.

Fork

In the example of flu, sneezing and fever above, the graph has a fork shape. For a probability distribution $PP$ to satisfy the Markov condition for this graph we must have

$P\left(x,y,z\right)=P\left(x\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)P\left(y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)P\left(z\right),\phantom{\rule{thickmathspace}{0ex}}\forall x,y,z.P\left(x, y, z\right) = P\left(x \ | \ z\right) P\left(y \ | \ z\right)P\left(z\right), \ \forall x,y,z.$

However, $P\left(x,y\right)\ne P\left(x\right)P\left(y\right)P\left(x,y\right) \neq P\left(x\right) P\left(y\right)$.

In other words, $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp\!\!\!\!\!\!\!\perp Y \ | \ Z$, though $XX$ and $YY$ are not independent. This makes sense, we wouldn’t expect sneezing and fever to be uncorrelated, but given that we know whether or not Alice has flu, telling us that she has fever isn’t going to tell us anything about her sneezing.

Collider

Reversing the arrows in the fork graph above gives a collider as in the following example.

Clearly whether or not Alice has allergies other than hayfever is independent of what season it is. So we’d expect a distribution on this graph to satisfy $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\varnothing X \perp\!\!\!\!\!\!\!\perp Y \ | \ \emptyset$. However, if we know that Alice is having an allergic reaction, and it happens to be spring, we will likely assume that she has some allergy, i.e. $XX$ and $YY$ are not conditionally independent given $ZZ$.

Indeed, the Markov condition and chain rule for this graph gives us $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\varnothing X \perp\!\!\!\!\!\!\!\perp Y \ | \ \emptyset$:

$P\left(x,y,z\right)=P\left(x\right)P\left(y\right)P\left(z\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}x,\phantom{\rule{thickmathspace}{0ex}}y\right)=P\left(z\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}x,\phantom{\rule{thickmathspace}{0ex}}y\right)P\left(x\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}y\right)P\left(y\right)\phantom{\rule{thickmathspace}{0ex}}\forall x,y,z.P\left(x, y, z\right) = P\left(x\right)P\left(y\right) P\left(z \ | \ x,\ y\right) = P\left(z \ | \ x,\ y\right) P\left( x\ | \ y\right) P\left(y\right) \ \forall x,y,z.$

from which we cannot derive $P\left(x\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)P\left(y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)=P\left(x,y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}z\right)P\left(x \ | \ z\right) P\left(y \ | \ z\right) = P\left(x,y \ | \ z\right)$. (However, it could still be true for some particular choice of probability distribution.)

Chain

Finally, let us return to the chain of correlations presented in the introduction.

Clearly the probabilities that it is spring and that Alice is sneezing are not independent, and indeed, we cannot derive $P\left(x,y\right)=P\left(x\right)P\left(y\right)P\left(x, y\right) = P\left(x\right) P\left(y\right)$. However observe that, by the chain rule, a Markov distribution on the chain graph must satisfy $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX\perp\!\!\!\!\!\!\!\perp Y \ | \ Z$. If we know Alice is having an allergic reaction that is not hayfever, whether or not she is sneezing is not going to affect our guess as to what season it is.

Crucially, in this case, knowing the season is also not going to affect whether we think Alice is sneezing. By definition, conditional independence of $XX$ and $YY$ given $ZZ$ is symmetric in $XX$ and $YY$. In other words, a joint distribution $PP$ on the variables $X,Y,ZX,Y,Z$ satisfies the Markov condition with respect to the chain graph

$X⟶Z⟶YX \longrightarrow Z \longrightarrow Y$

if and only if $PP$ satisfies the Markov condition on

$Y⟶Z⟶X.Y \longrightarrow Z \longrightarrow X .$

### d-separation

The above observations can be generalised to statements about conditional independences in any Bayesian network. That is, if $\left(G,P\right)\left(G,P\right)$ is a Bayesian network then the structure of $GG$ is enough to derive all the conditional independences in $PP$ that are implied by the graph $GG$ (in reality there may be more that have not been included in the network!).

Given a DAG $GG$ and a set of vertices $UU$ of $GG$, let $m\left(U\right)m\left(U\right)$ denote the union of $UU$ with all the vertices $vv$ of $GG$ such that there is a directed edge from $UU$ to $vv$. The set $W\left(U\right)W\left(U\right)$ will denote the non-inclusive future of $UU$, that is, the set of vertices $vv$ of $GG$ for which there is no directed (possibly trivial) path from $vv$ to $UU$.

For a graph $GG$, let $X,Y,ZX, Y, Z$ now denote disjoint subsets of the vertices of $GG$ (and their corresponding random variables). Set $W:=W\left(X\cup Y\cup Z\right)W := W\left(X \cup Y \cup Z\right)$.

Then $XX$ and $YY$ are said to be d-separated by $ZZ$, written $X\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp Y \ | \ Z$, if there is a partition $\left\{U,V,W,Z\right\}\\left\{U,V,W,Z\\right\}$ of the nodes of $GG$ such that

• $X\subseteq UX \subseteq U$ and $Y\subseteq VY \subseteq V$, and

• $m\left(U\right)\cap m\left(V\right)\subseteq W,m\left(U\right) \cap m\left(V\right) \subseteq W,$ in other words $UU$ and $VV$ have no direct influence on each other.

(This is lemma 19 in the paper.)

Now d-separation is really useful since it tells us everything there is to know about the conditional dependences on Bayesian networks with underlying graph $GG$. Indeed,

#### Theorem 5

• Soundness of d-separation (Verma and Pearl, 1988) If $PP$ is a Markov distribution with respect to a graph $GG$ then for all disjoint subsets $X,Y,ZX,Y,Z$ of nodes of $GG$ $X\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp Y \ | \ Z$ implies that $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp\!\!\!\!\!\!\!\perp Y \ | \ Z$.

• Completeness of d-separation (Meek, 1995) If $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp\!\!\!\!\!\!\!\perp Y \ | \ Z$ for all $PP$ Markov with respect to $GG$, then $X\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp Y \ | \ Z$.

We can combine the previous examples of fork, collider and chain graphs to get the following

A priori, Allergic reaction is conditionally independent of Fever. Indeed, we have the partition

which clearly satisfies d-separation. However, if Sneezing is known then $W=\varnothing W = \emptyset$, so Allergic reaction and Fever are not independent. Indeed, if we use the same sets $UU$ and $VV$ as before, then $m\left(U\right)\cap m\left(V\right)=\left\{\mathrm{Sneezing}\right\}m\left(U\right) \cap m\left(V\right) = \\left\{ Sneezing \\right\}$, so the condition for d-separation fails; and it does for any possible choice of $UU$ and $VV$. Interestingly, if Flu is also known, we again obtain conditional independence between Allergic reaction and Fever, as shown below.

Before describing the limitations of this setup and why we may want to generalise it, it is worth observing that Theorem 5 is genuinely useful computationally. Theorem 5 says that given a Bayesian network $\left(G,P\right)\left(G,P\right)$, the structure of $GG$ gives us a recipe to factor $PP$, thereby greatly increasing computation efficiency for Bayesian inference.

### Latent variables, hidden variables, and unobservables

In the context of Bayesian networks, there are two reasons that we may wish to add variables to a probabilistic model, even if we are not entirely sure what the variables signify or how they are distributed. The first reason is statistical and the second is physical.

Consider the example of flu, fever and sneezing discussed earlier. Although our analysis told us $\mathrm{Fever}\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp \mathrm{Sneezing}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{Flu}Fever \perp\!\!\!\!\!\!\!\perp Sneezing \ | \ Flu$, if we conduct an experiment we are likely to find:

$P\left(\mathrm{fever}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{sneezing},\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right)\ne P\left(\mathrm{fever}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\mathrm{flu}\right).P\left(fever \ | \ sneezing, \ flu\right) \neq P\left(fever \ | \ flu\right).$

The problem is caused by the graph not properly modelling reality, but a simplification of it. After all, there are a whole bunch of things that can cause sneezing and flu. We just don’t know what they all are or how to measure them. So, to make the network work, we may add a hypothetical latent variable that bunches together all the unknown joint causes, and equip it with a distribution that makes the whole network Bayesian, so that we are still able to perform inference methods like belief propagation.

On the other hand, we may want to add variables to a Bayesian network if we have evidence that doing so will provide a better model of reality.

For example, consider the network with just two connected nodes

Every distribution on this graph is Markov, and we would expect there to be a correlation between a road being wet and the grass next to it being wet as well, but most people would claim that there’s something missing from the picture. After all, rain could be a ‘common cause’ of the road and the grass being wet. So, it makes sense to add a third variable.

But maybe we can’t observe whether it has rained or not, only whether the grass and/or road are wet. Nonetheless, the correlation we observe suggests that they have a common cause. To deal with such cases, we could make the third variable hidden. We may not know what information is included in a hidden variable, nor its probability distribution.

All that matters is that the hidden variable helps to explain the observed correlations.

So, latent variables are a statistical tool that ensure the Markov condition holds. Hence they are inherently classical, and can, in theory, be known. But the universe is not classical, so, even if we lump whatever we want into as many classical hidden variables as we want and put them wherever we need, in some cases, there will still be empirically observed correlations that do not satisfy the Markov condition.

Most famously, Bell’s experiment shows that it is possible to have distinct variables $AA$ and $BB$ that exhibit correlations that cannot be explained by any classical hidden variable, since classical variables are restricted by the principle of locality.

In other words, though $A\perp B\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\Lambda A \perp B \ | \ \Lambda$,

$P\left(a\phantom{\rule{thickmathspace}{0ex}}|b,\phantom{\rule{thickmathspace}{0ex}}\lambda \right)\ne P\left(a\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\lambda \right).P\left(a \ | b,\ \lambda\right) \neq P\left(a \ | \ \lambda\right).$

Implicitly, this means that a classical $\Lambda \Lambda$ is not enough. If we want $P\left(a\phantom{\rule{thickmathspace}{0ex}}|b,\phantom{\rule{thickmathspace}{0ex}}\lambda \right)\ne P\left(a\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\lambda \right)P\left(a \ | b,\ \lambda\right) \neq P\left(a \ | \ \lambda\right)$ to hold, $\Lambda \Lambda$ must be a non-local (non-classical) variable. Quantum mechanics implies that we can’t possibly empirically find the value of a non-local variable (for similar reasons to the Heisenberg’s uncertainty principle), so non-classical variables are often called unobservables. In particular, it is irrelevant to question whether $A\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp B\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}\Lambda A \perp\!\!\!\!\!\!\!\perp B \ | \ \Lambda$, as we would need to know the value of $\Lambda \Lambda$ in order to condition over it.

Indeed, this is the key idea behind what follows. We declare certain variables to be unobservable and then insist that conditional (in)dependence only makes sense between observable variables conditioned over observable variables.

## Generalising classical causality

The correlations observed in the Bell experiment can be explained by quantum mechanics. But thought experiments such as the one described here suggest that theoretically, correlations may exist that violate even quantum causality.

So, given that graphical models and d-separation provide such a powerful tool for causal reasoning in the classical context, how can we generalise the Markov condition and Theorem 5 to quantum, and even more general causal theories? And, if we have a theory-independent Markov condition, are there d-separation results that don’t correspond to any given causal theory?

Clearly the first step in answering these questions is to fix a definition of a causal theory.

### Operational probabilistic theories

An operational theory is a symmetric monoidal category $\left(C,\otimes ,I\right)\left(\mathsf \left\{C\right\}, \otimes, I\right)$ whose objects are known as systems or resources. Morphisms are finite sets $f=\left\{{𝒞}_{i}{\right\}}_{i\in I}f = \\left\{\mathcal \left\{C\right\}_i\\right\}_\left\{i \in I\right\}$ called tests, whose elements are called outcomes. Tests with a single element are called deterministic, and for each system $A\in \mathrm{ob}\left(C\right)A \in ob \left(\mathsf \left\{C\right\}\right)$, the identity ${\mathrm{id}}_{A}\in \left(A,A\right)id_A \in \mathsf \left(A,A\right)$ is a deterministic test.

In this discussion, we’ll identify tests $\left\{{𝒞}_{i}{\right\}}_{i},\left\{{𝒟}_{j}{\right\}}_{j}\\left\{\mathcal \left\{C\right\}_i \\right\}_i , \\left\{\mathcal \left\{D\right\}_j\\right\}_j$ in $C\mathsf \left\{C\right\}$ if we may always replace one with the other without affecting the distributions in $C\left(I,I\right)\mathsf \left\{C\right\}\left(I, I\right)$.

Given $\left\{{𝒞}_{i}{\right\}}_{i}\in C\left(B,C\right)\\left\{\mathcal \left\{C\right\}_i \\right\}_i \in \mathsf \left\{C\right\}\left(B, C\right)$ and $\left\{{𝒟}_{j}\right\}\in C\left(A,B\right)\\left\{\mathcal \left\{D\right\}_j \\right\} \in \mathsf \left\{C\right\}\left(A, B\right)$, their composition $f\circ gf \circ g$ is given by

$\left\{{𝒞}_{i}\circ {𝒟}_{j}{\right\}}_{i,j}\in C\left(A,C\right).\\left\{ \mathcal \left\{C\right\}_i \circ \mathcal \left\{D\right\}_j \\right\}_\left\{i,j\right\} \in \mathsf \left\{C\right\}\left(A, C\right).$

First apply $𝒟\mathcal \left\{D\right\}$ with output $BB$ then apply $𝒞\mathcal \left\{C\right\}$ with outcome $CC$.

The monoidal composition $\left\{{𝒞}_{i}\otimes {𝒟}_{j}{\right\}}_{i,j}\in C\left(A\otimes C,B\otimes D\right)\\left\{ \mathcal \left\{C\right\}_i \otimes \mathcal \left\{D\right\}_j \\right\}_\left\{i, j\right\} \in \mathsf \left\{C\right\}\left(A \otimes C, B \otimes D\right)$ corresponds to applying $\left\{{𝒞}_{i}{\right\}}_{i}\in C\left(A,B\right)\\left\{\mathcal \left\{C\right\}_i\\right\}_i \in \mathsf \left\{C\right\}\left(A,B\right)$ and $\left\{{𝒟}_{j}{\right\}}_{j}\\left\{ \mathcal \left\{D\right\}_j \\right\}_j$ separately on $AA$ and $CC$.

An operational probabilistic theory or OPT is an operational theory such that every test $I\to II \to I$ is a probability distribution.

A morphism $\left\{{𝒞}_{i}{\right\}}_{i}\in C\left(A,I\right)\\left\{ \mathcal \left\{C\right\}_i \\right\}_i \in \mathsf \left\{C\right\}\left(A, I\right)$ is called an effect on $AA$. An OPT $C\mathsf \left\{C\right\}$ is called causal or a causal theory if, for each system $A\in \mathrm{ob}\left(C\right)A \in ob \left(\mathsf \left\{C\right\}\right)$, there is a unique deterministic effect ${\top }_{A}\in C\left(A,I\right)\top_A \in \mathsf \left\{C\right\}\left( A, I\right)$ which we call the discard of $AA$.

In particular, for a causal OPT $C\mathsf \left\{C\right\}$, uniqueness of the discard implies that, for all systems $A,B\in \mathrm{ob}\left(C\right)A, B \in ob \left(\mathsf \left\{C\right\}\right)$,

${\top }_{A}\otimes {\top }_{B}={\top }_{A\otimes B},\top_A \otimes \top_B = \top_\left\{A \otimes B\right\},$ and, given any determinstic test $𝒞\in C\left(A,B\right)\mathcal \left\{C\right\} \in \mathsf \left\{C\right\}\left(A, B\right)$,

${\top }_{B}\circ 𝒞={\top }_{A}.\top_B \circ \mathcal \left\{C\right\} = \top_A.$

The existence of a discard map allows a definition of causal morphisms in a causal theory. For example, as we saw in January when we discussed Kissinger and Uijlen’s paper, a test $\left\{{𝒞}_{i}{\right\}}_{i}\in C\left(A,B\right)\\left\{ \mathcal \left\{C\right\}_i \\right\}_i \in \mathsf \left\{C\right\} \left(A, B\right)$ is causal if

${\top }_{B}\circ \left\{{𝒞}_{i}{\right\}}_{i}={\top }_{A}\in C\left(A,I\right).\top_B \circ \\left\{ \mathcal \left\{C\right\}_i \\right\}_i = \top_A \in \mathsf \left\{C\right\}\left( A, I\right).$

In other words, for a causal test, discarding the outcome is the same as not performing the test. Intuitively it is not obvious why such morphisms should be called causal. But this definition enables the formulation of a non-signalling condition that describes the conditions under which the possibility of cause-effect correlation is excluded, in particular, it implies the impossibility of time travel.

#### Examples

The category $\mathrm{Mat}\left({ℝ}_{+}\right)Mat\left(\mathbb \left\{R\right\}_+\right)$ of natural numbers and with $\mathrm{Mat}\left({ℝ}_{+}\right)\left(m,n\right)Mat\left(\mathbb \left\{R\right\}_+\right)\left(m,n\right)$ the set of $n×mn \times m$ matrices, has the structure of a causal OPT. The causal morphisms in $\mathrm{Mat}\left({ℝ}_{+}\right)Mat\left(\mathbb \left\{R\right\}_+\right)$ are the stochastic maps (the matrices whose columns sum to 1). This category describes classical probability theory.

The category $\mathrm{CPM}\mathsf\left\{CPM\right\}$ of sets of linear operators on Hilbert spaces and completely positive maps between them is an OPT and describes quantum relations. The causal morphisms are the trace preserving completely positive maps.

Finally, Boxworld is the theory that allows to describe any correlation between two variables as the cause of some resource of the theory in the past.

### Generalised Bayesian networks

So, we’re finally ready to give the main construction and results of the paper. As mentioned before, to get a generalised d-separation result, the idea is that we will distinguish observable and unobservable variables, and simply insist that conditional independence is only defined relative to observable variables.

To this end, a generalised DAG or GDAG is a DAG $GG$ together with a partition on the nodes of $GG$ into two subsets called observed and unobserved. We’ll represent observed nodes by triangles, and unobserved nodes by circles. An edge out of an (un)observed node will be called (un)observed and represented by a (solid) dashed arrow.

In order to get a generalisation of Theorem 5, we still need to come up with a sensible generalisation of the Markov property which will essentially say that at an observed node that has only observed parents, the distribution must be Markov. However, if an observed node has an unobserved parent, the latter’s whole history is needed to describe the distribution.

To state this precisely, we will associate a causal theory $\left(C,\otimes ,I\right)\left(\mathsf \left\{C\right\}, \otimes, I\right)$ to a GDAG $GG$ via an assignment of systems to edges of $GG$ and tests to nodes of $GG$, such that the observed edges of $GG$ will ‘carry’ only the outcomes of classical tests (so will say something about conditional probability) whereas unobserved edges will carry only the output system.

Precisely, such an assignment $PP$ satisfies the generalised Markov condition (GMC) and is called a generalised Markov distribution if

• Each unobserved edge corresponds to a distinct system in the theory.

• If we can’t observe what is happening at a node, we can’t condition over it: To each unobserved node and each value of its observed parents, we assign a deterministic test from the system defined by the product of its incoming (unobserved) edges to the system defined by the product of its outgoing (unobserved) edges.

• Each observed node $XX$ is an observation test, i.e. a morphism in $C\left(A,I\right)\mathsf \left\{C\right\}\left(A, I\right)$ for the system $A\in \mathrm{ob}\left(C\right)A \in ob\left( \mathsf \left\{C\right\}\right)$ corresponding to the product of the systems assigned to the unobserved input edges of $XX$. Since $C\mathsf \left\{C\right\}$ is a causal theory, this says that $XX$ is assigned a classical random variable, also denoted $XX$, and that if $YY$ is an observed node, and has observed parent $XX$, the distribution at $YY$ is conditionally dependent on the distribution at $XX$ (see here for details).

• It therefore follows that each observed edge is assigned the trivial system $II$.

• The joint probability distribution on the observed nodes of $GG$ is given by the morphism $C\left(I,I\right)\mathsf \left\{C\right\}\left(I, I\right)$ that results from these assignments.

A generalised Bayesian network consists of a GDAG $GG$ together with a generalised Markov distribution $PP$ on $GG$.

#### Example

Consider the following GDAG

Let’s build its OPT morphism as indictated by the generalised Markov condition.

The observed node $XX$ has no incoming edges so it corresponds to a $C\left(I,I\right)\mathsf \left\{C\right\}\left(I, I\right)$ morphism, and thus we assign a probability distribution to it.

The unobserved node A depends on $XX$, and has no unobserved inputs, so we assign a deterministic test $A\left(x\right):I\to AA\left(x\right): I \to A$ for each value $xx$ of $XX$.

The observed node $YY$ has one incoming unobserved edge and no incoming observed edges so we assign to it a test $Y:A\to IY: A \to I$ such that, for each value $xx$ of $XX$, $Y\circ A\left(x\right)Y \circ A\left(x\right)$ is a probability distribution.

Building up the rest of the picture gives an OPT diagram of the form

which is a $C\left(I,I\right)\mathsf \left\{C\right\}\left(I, I\right)$ morphism that defines the joint probability distribution $P\left(x,y,z,w\right)P\left(x,y,z,w\right)$. We now have all the ingredients to state Theorem 22, the generalised d-separation theorem. This is the analogue of Theorem 5 for generalised Markov distributions.

#### Theorem 22

Given a GDAG $GG$ and subsets $X,Y,ZX,Y, Z$ of observed nodes

• if a probability distribution $PP$ is generalised Markov relative to $GG$ then $X\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}Z⇒X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp Y \ | \ Z \Rightarrow X\perp\!\!\!\!\!\!\!\perp Y \ | \ Z$.

• If $X\perp \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX\perp\!\!\!\!\!\!\!\perp Y \ | \ Z$ holds for all generalised Markov probability distributions on $GG$, then $X\perp Y\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}ZX \perp Y \ | \ Z$.

Note in particular that there is no change in the definition of d-separation: d-separation of a GDAG $GG$ is simply d-separation with respect to its underlying DAG. There is also no change in the definition of conditional independence. Now, however, we restrict to statements of conditional independence with respect to observed nodes only. This enables the generalised soundness and completeness statements of the theorem.

The proof of soundness uses uniqueness of discarding, and completeness follows since generalised Markov is a stronger condition on a distribution than classically Markov.

### Classical distributions on GDAGs

Theorem 22 is all well and good. But does it really generalise the classical case? That is, can we recover Theorem 5 for all classical Bayesian networks from Theorem 22?

As a first step, Proposition 17 states that if all the nodes of a generalised Bayesian network are observed, then it is a classical bayesian network. In fact, this follows pretty immediately from the definitions.

Moreover, it is easily checked that, given a classical Bayesian network, even if it has hidden or latent variables, it can still be expressed directly as a generalised Bayesian network with no unobserved nodes.

In fact, Theorem 22 generalises Theorem 5 in a stricter sense. That is, the generalised Bayesian network setup together with classical causality adds nothing extra to the theory of classical Bayesian networks. If a generalised Markov distribution is classical (then hidden and latent variables may be represented by unobserved nodes), it can be viewed as a classical Bayesian network. More precisely, Lemma 18 says that, given any generalised Bayesian network $\left(G,P\right)\left(G,P\right)$ with underlying DAG $G\prime G\text{'}$ and distribution $P\in 𝒞P \in \mathcal \left\{C\right\}$, we can construct a classical Bayesian network $\left(G\prime ,P\prime \right)\left(G\text{'}, P\text{'}\right)$ such that $P\prime P\text{'}$ agrees with $PP$ on the observed nodes.

It is worth voicing a note of caution. The authors themselves mention in the conclusion that the construction based on GDAGs with two types of nodes is not entirely satisfactory. The problem is that, although the setups and results presented here do give a generalisation of Theorem 5, they do not, as such, provide a way of generalising Bayesian networks as they are used for probabilistic inference to non-classical settings. For example, belief propagation works through observed nodes, but there is no apparent way of generalising it for unobserved nodes.

## Theory independence

More generally, given a GDAG $GG$, we can look at the set of distributions on $GG$ that are generalised Markov with respect to a given causal theory. Of particular importance are the following.

• The set $𝒞\mathcal \left\{C\right\}$ of generalised Markov distributions in $\mathrm{Mat}\left({ℝ}_{+}\right)Mat\left(\mathbb \left\{R\right\}_+\right)$ on $GG$.

• The set $𝒬\mathcal \left\{Q\right\}$ of generalised Markov distributions in $\mathrm{CPM}\mathsf\left\{CPM\right\}$ on $GG$.

• The set $𝒢\mathcal \left\{G\right\}$ of all generalised Markov distributions on $GG$. (This is the set of generalised Markov distributions in Boxworld.)

Moreover, we can distinguish another class of distributions on $GG$, by not restricting to d-seperation of observed nodes, but considering distributions that satisfy the observable conditional independences given by any d-separation properties on the graph. Theorem 22 implies, in particular that $G\subseteq IG \subseteq I$.

And, so, since $\mathrm{Mat}\left({ℝ}_{+}\right)Mat\left(\mathbb \left\{R\right\}_+\right)$ embeds into $\mathrm{CPM}\mathsf\left\{CPM\right\}$, we have $𝒞\subseteq 𝒬\subseteq 𝒢\subseteq ℐ\mathcal \left\{C\right\} \subseteq \mathcal \left\{Q\right\} \subseteq \mathcal \left\{G\right\} \subseteq \mathcal \left\{I\right\}$.

This means that one can ask for which graphs (some or all of) these inequalities are strict, and the last part of the paper explores these questions. In the original paper, a sufficient condition is given for graphs to satisfy $𝒞\ne ℐ\mathcal \left\{C\right\} \neq \mathcal \left\{I\right\}$. I.e. for these graphs it is guaranteed that the causal structure admits correlations that are non-local. Moreover the authors show that their condition is necessary for small enough graphs.

Another interesting result is that there exist graphs for which $𝒢\ne ℐ\mathcal \left\{G\right\} \neq \mathcal \left\{I\right\}$. This means that using a theory of resources, whatever theory it may be, to explain correlations imposes constraints that are stronger than those imposed by the relations themselves.

## What next?

This setup represents one direction for using category theory to generalise Bayesian networks. In our group work at the ACT workshop, we considered another generalisation of Bayesian networks, this time staying within the classical realm. Namely, building on the work of Bonchi, Gadducci, Kissinger, Sobocinski, and Zanasi, we gave a functorial Markov condition on directed graphs admitting cycles. Hopefully we’ll present this work here soon.

### Lubos Motl - string vacua and pheno

Spin correlations at ATLAS: tops deviate by 3.2 or 3.7 sigma
After some time, we saw an LHC preprint with an intriguing deviation from the Standard Model predictions. It appeared in the preprint
Measurements of top-quark pair spin correlations in the $$e\mu$$ channel at $$\sqrt s = 13\TeV$$ using $$pp$$ collisions in the ATLAS detector
You should also see a 27-page-long presentation by Reinhild Peters.

To make the story short, the measured correlation between the top quark spins – in events with one $$e^\pm$$ and one oppositely charged $$\mu^\mp$$ at the end and a top quark pair in the middle – exceeds the theoretical prediction by 3.7 standard deviations if you pretend that the theoretical prediction is exact, or 3.2 sigma if you choose some sensible nonzero error margin for the theoretical prediction.

The chance that a deviation of this size appears by chance is comparable to 1 in 1,000.

It may be a fluke – after all, ATLAS and CMS have measured a thousand of similar numbers so one of them may deviate by a large deviation that seems like "one in one thousand cases". As always, there's some possibility that the top quarks' spin correlation is enhanced by some physics beyond the Standard Model. It could be many things, I have no idea what should be the default explanation. If the top quarks sometimes came from some new spinless or spin-one intermediate particles, you could move the spin correlation up or down, respectively.

The LHC (Les Horribles Cernettes) girls have sung a song about the spins of quarks. You are invited to listen to the song, measure the correlations yourself, and determine whether the deviation from the Standard Model is exciting enough.

## July 08, 2018

### Marco Frasca - The Gauge Connection

ICHEP 2018

The great high-energy physics conference ICHEP 2018 is over and, as usual, I spend some words about it. The big collaborations of CERN presented their last results. I think the most relevant of this is about the evidence ($3\sigma$) that the Standard Model is at odds with the measurement of spin correlation between top-antitop pair of quarks. More is given in the ATLAS communicate. As expected, increasing precision proves to be rewarding.

About the Higgs particle, after the important announcement about the existence of the ttH process, both ATLAS and CMS are pursuing further their improvement of precision. About the signal strength they give the following results. For ATLAS (see here)

$\mu=1.13\pm 0.05({\rm stat.})\pm 0.05({\rm exp.})^{+0.05}_{-0.04}({\rm sig. th.})\pm 0.03({\rm bkg. th})$

and CMS (see here)

$\mu=1.17\pm 0.06({\rm stat.})^{+0.06}_{-0.05}({\rm sig. th.})\pm 0.06({\rm other syst.}).$

The news is that the error is diminished and both agrees. They show a small tension, 13% and 17% respectively, but the overall result is consistent with the Standard Model.

When the different contributions are unpacked in the respective contributions due to different processes, CMS claims some tensions in the WW decay that should be taken under scrutiny in the future (see here). They presented the results from $35.9{\rm fb}^{-1}$ data and so, there is no significant improvement, for the moment, with respect to Moriond conference this year. The situation is rather better for the ZZ decay where no tension appears and the agreement with the Standard Model is there in all its glory (see here). Things are quite different, but not too much, for ATLAS as in this case they observe some tensions but these are all below $2\sigma$ (see here). For the WW decay, ATLAS does not see anything above $1\sigma$ (see here).

So, although there is something to take under attention with the increase of data, that will reach $100 {\rm fb}^{-1}$ this year, but the Standard Model is in good health with respect to the Higgs sector even if there is a lot to be answered yet and precision measurements are the main tool. The correlation in the tt pair is absolutely promising and we should hope this will be confirmed a discovery.

## July 04, 2018

### The n-Category Cafe

Symposium on Compositional Structures

There’s a new conference series, whose acronym is pronounced “psycho”. It’s part of the new trend toward the study of “compositionality” in many branches of thought, often but not always using category theory:

• First Symposium on Compositional Structures (SYCO1), School of Computer Science, University of Birmingham, 20-21 September, 2018. Organized by Ross Duncan, Chris Heunen, Aleks Kissinger, Samuel Mimram, Simona Paoli, Mehrnoosh Sadrzadeh, Pawel Sobocinski and Jamie Vicary.

The Symposium on Compositional Structures is a new interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students.

More details below! Our very own David Corfield is one of the invited speakers.

The Symposium on Compositional Structures is a new interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students.

Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere.

While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related to any of the following areas, in particular from the perspective of category theory:

• logical methods in computer science, including classical and quantum programming, type theory, concurrency, natural language processing and machine learning;
• graphical calculi, including string diagrams, Petri nets and reaction networks;
• languages and frameworks, including process algebras, proof nets, type theory and game semantics;
• abstract algebra and pure category theory, including monoidal category theory, higher category theory, operads, polygraphs, and relationships to homotopy theory;
• quantum algebra, including quantum computation and representation theory;
• tools and techniques, including rewriting, formal proofs and proof assistants, and game theory;
• industrial applications, including case studies and real-world problem descriptions.

This new series aims to bring together the communities behind many previous successful events which have taken place over the last decade, including “Categories, Logic and Physics”, “Categories, Logic and Physics (Scotland)”, “Higher-Dimensional Rewriting and Applications”, “String Diagrams in Computation, Logic and Physics”, “Applied Category Theory”, “Simons Workshop on Compositionality”, and the “Peripatetic Seminar in Sheaves and Logic”.

The steering committee hopes that SYCO will become a regular fixture in the academic calendar, running regularly throughout the year, and becoming over time a recognized venue for presentation and discussion of results in an informal and friendly atmosphere. To help create this community, in the event that more good-quality submissions are received than can be accommodated in the timetable, we may choose to defer some submissions to a future meeting, rather than reject them. This would be done based on submission order, giving an incentive for early submission, and avoiding any need to make difficult choices between strong submissions. Deferred submissions would be accepted for presentation at any future SYCO meeting without the need for peer review. This will allow us to ensure that speakers have enough time to present their ideas, without creating an unnecessarily competitive atmosphere. Meetings would be held sufficiently frequently to avoid a backlog of deferred papers.

# Invited Speakers

• David Corfield, Department of Philosophy, University of Kent: “The ubiquity of modal type theory”.

• Jules Hedges, Department of Computer Science, University of Oxford: “Compositional game theory”

# Important Dates

All times are anywhere-on-earth.

• Submission deadline: Sunday 5 August 2018
• Author notification: Monday 13 August 2018
• Travel support application deadline: Monday 20 August 2018
• Symposium dates: Thursday 20 September and Friday 21 September 2018

# Submissions

Submission is by EasyChair, via the following link:

Submissions should present research results in sufficient detail to allow them to be properly considered by members of the programme committee, who will assess papers with regards to significance, clarity, correctness, and scope. We encourage the submission of work in progress, as well as mature results. There are no proceedings, so work can be submitted even if it has been previously published, or has been submitted for consideration elsewhere. There is no specific formatting requirement, and no page limit, although for long submissions authors should understand that reviewers may not be able to read the entire document in detail.

# Funding

Some funding is available to cover travel and subsistence costs, with a priority for PhD students and junior researchers. To apply for this funding, please contact the local organizer Jamie Vicary at j.o.vicary@bham.ac.uk by the deadline given above, with a short statement of your travel costs and funding required.

# Programme Committee

The symposium managed by the following people, who also serve as the programme committee.

• Ross Duncan, University of Strathclyde
• Chris Heunen, University of Edinburgh
• Aleks Kissinger, Radboud University Nijmegen
• Samuel Mimram, École Polytechnique
• Simona Paoli, University of Leicester
• Mehrnoosh Sadrzadeh, Queen Mary, University of London
• Pawel Sobocinski, University of Southampton
• Jamie Vicary, University of Birmingham and University of Oxford (local organizer)

### Tommaso Dorigo - Scientificblogging

Chasing The Higgs Self Coupling: New CMS Results
Happy Birthday Higgs boson! The discovery of the last fundamental particle of the Standard Model was announced exactly 6 years ago at CERN (well, plus one day, since I decided to postpone to July 5 the publication of this post...).

In the Standard Model, the theory of fundamental interactions among elementary particles which enshrines our current understanding of the subnuclear world,  particles that constitute matter are fermionic: they have a haif-integer value of a quantity we call spin; and particles that mediate interactions between those fermions, keeping them together and governing their behaviour, are bosonic: they have an integer value of spin.

read more

## July 03, 2018

### Lubos Motl - string vacua and pheno

David Gross: make America great again
The first string theory's formula, the Veneziano amplitude, was introduced to physics in 1968, i.e. half a century ago.

In that year of 1968, Czechoslovakia tried its "socialism with a human face", and the experiment was terminated by the Warsaw Pact tanks in August (next month, we will "celebrate" that). Meanwhile, the youth in the West tried a seemingly similar revolution. Only in recent years, I was forced to realize that what started in the West was really going in the opposite direction than the 1968 Prague Spring.

At any rate, the annual conference, Strings 2018, didn't forget about the Veneziano far-reaching playing with the Euler Beta function. Veneziano was present. The 2-hour-long panel discussion at the end of the conference is arguably the most layman-friendly video produced by the conference. One frustrating fact is that the video only has 1500 views as of this moment. No journalists were interested in the conference.

Some 500 string theorists (perhaps 1/2 of the world's currently employed professional string theorists) gathered in the tropical Okinawa, Japan. I feel absolutely confident that among gatherings of 500-1,000 people, the annual string conferences have by far the highest average IQ of the participants. No Bilderberg, Davos, freemason meetings, and even the parties of the old Nobel prize winners could compete. If the U.S. invaded and bombed Okinawa again, like in April 1945, and all the people who are there would be killed, the set of the world's people with the IQ above 160 would be detectably decimated.

You may watch the panel discussion which recalls some topics and/or controversies. You may also try to read Tetragraviton's earlier sketch of the conference.

Although David Gross pretends that he doesn't love Donald Trump too much, it's still true that the great minds think alike. Before 1:54:40, he lists the next four places to host the annual conferences as Brussels, Capetown, Vacuum, and Vienna, where Vacuum represents a pause in 2021. It would be a shame to have a hole.

Who can fill the hole? Gross shows the organizers and in the late 20th century, the U.S. were much more important, relatively speaking. After 2000, the conferences spread out of the U.S. OK, he gradually converges to the – not quite a priori obvious – punch line: Make America great again. A larger number of conferences should be held in the U.S. again. Gross also tells Cambridge, MA and Palo Alto, CA to feel no pressure. ;-)

America is still the world's most attractive place for investment. It's the most likely country to create places with a huge concentration of high brainpower – like in the Silicon Valley. In Europe and other places, we often place limits and we moderate things. We like to vote for the The Party of Moderate Progress Within the Bounds of the Law founded by Jaroslav Hašek, the author of the Good soldier Švejk. But in America, there are no limits. Or you can take it to the limit. Or you can get to the Moon – in fact, literally.

This is how many of us have understood America's WOW factor. Maybe in the computer technologies, it's still true. But I think that since 2000 or so, America started to lose this WOW factor. In fact, I think that the U.S. became the main source of the political correctness and related toxic diseases that are gradually poisoning and devouring the Western civilization.

Some of the "outsourcing" of the string conferences was purely due to the political correctness – those second-class, third-class, and other places that aren't quite as good as America shouldn't think that they're worse than America. Well, almost all of them still are. But such games can never remain just games. I think that some of the outsourcing is real. America has lost much of its motivation to lead the world.

Donald Trump obviously isn't the man who should be expected to revitalize the string theory research. But helpfully enough, David Gross accepted the role of the Donald in string theory. Well, he's had this role for some 33 years, I think. You know, folks in America should realize that they should still be the bosses of the world because things won't work too well without them.

Already since the early 21st century, I grew very disillusioned with America. I noticed that some of the garbage that almost defines the European Union exists in the U.S. as well – and in some cases, America harbors more hardcore versions of the low-quality humans and their pathetic excuses to keep the world as a network of muddy sycophants connected to a stagnant bureaucratic structure.

For example, when I translated Brian Greene's first bestseller to Czech, it was a great success in Czechia and the feedback was almost entirely enthusiastic. In some sense, I brought a piece of America to Czechia. There was one exception who wrote some tirade against string theory, against Brian Greene, against myself, and several other related entities. I hadn't been quite familiar with that shocking šithead before – but the exposure has taught me a very speedy lesson. This "Prof" Jiří Chýla – at that time, the boss of Particle Physics at the Czech Academy of Sciences who found his job appropriate for spending days and trying to harm a Rutgers grad student and his book with a 30-page-long rant (which he didn't, thank God) – was the best example of the communist era crap – the frogs sitting on all the springs – that keeps the Czech institutions uncompetitive in theoretical physics.

He is a symbol of the culture of old men who haven't produced any ideas that anyone in the world would give a damn about – at least no such idea since a 30-citation paper they wrote as postdocs 40 years ago, but that one wasn't important, either. But they want to mask this uselessness and pretend that they're doing pretty much the same thing as the best people in the world so they are hiring younger people who just lick their aßes, much like Mr Chýla licks the aßes of the jerks in the European Union all the time (who pay him the money because it's so wonderful to steal lots of money from the European taxpayers and pour it on useless parasites such as Mr Chýla).

Shortly after 2000, I was learning about some younger people – people of my generation – who found Mr Chýla's behavior and character OK and I just lost my emotional attachment to them, too. I just can't understand how someone may be so incredibly morally fudged up. Everyone who defends the likes of Mr Chýla is just scum.

For years, I had thought that the influence of fudged up individuals of Mr Chýla's type on the institutions is an artifact of my nation's not being so good as other nations. Our DNA is perhaps not so good and the communism has screwed our social structure and morality, too. But I no longer think that there's something very special about Czechia here. It was an unreasonable and unfair "masochist racism". I think that the likes of Peter W*it, Sabine Hossenfelder, and many others are pretty much analogous pieces of crap as Mr Chýla – and they and their pathetic excuses for their own inadequacy got comparably influential in the U.S. and Western Europe. All this filth just like to spread ludicrous propaganda that they're on par with the actual physicists – much like communism was producing propaganda that we were better off than the capitalist world – even though they must know that all these things are ludicrous lies. They're forming alternative structures that try to conquer the environment.

In the panel discussion, lots of questions were asked. My understanding is that all the questions were posed by actual registered participants of the conference. Nevertheless, the "plurality" (Gross' word) was about the falsifiability, gloom, and all this garbage. (Other repeated questions dealt with the existence of de Sitter solutions and other "more normal" or "professional" topics that have existed at previous conferences, too.) Most of the authors of such questions were probably some young participants. But where is the mankind going? It's not trivial to fly to Okinawa (even the U.S. troops in April 1945 would agree) and it's not trivial to do the other things needed to host a participant. Does it make any sense to fly to these islands if you have such serious doubts about the very justification of the field?

I don't really believe that the Millennial generation will advance any things that are actually hard enough, like string theory. Shiraz Minwalla said that the field was healthy, diverse, and people allow the evidence to take them wherever it goes. They shouldn't listen to anybody else, that's how things should be. It sounds nice. I think that there's still some individual stubbornness in Shiraz's attitude and I think that a big part of my generation is close to it.

But the individual stubbornness is exactly what the Millennials are completely lacking – so Shiraz's bullish description talks about something that will go extinct with the older currently alive generations. I've met some great exceptions but their percentage – even among the folks who should be intellectual elites of the Millennials in one way or another – is just insanely tiny. Almost all the Millennials want to be obedient, behave as members of a herd of stupid sheep, and say how smart and original they are by being stupid sheep in the herd. They want to join a club of one million holders of the Bitcoin who just buy the Bitcoin for their parents' money (and they think that being in a bunch of millions of people who do an exercise turns you into an "elite" of a new kind, wow) and say that this is how they change the world. Or they want to parrot pathetic lies about multiculturalism, environmentalism, and several other prominent delusions for the stupid masses.

This attitude isn't compatible with serious research in cutting-edge theoretical physics. It's incompatible with lots of other things that are needed for some true progress of the mankind, science, and technology.

Another topic: Don Garbutt recorded a nice track called "String Theory" with a 2012 animation showing the scales from the Planck scale to the observable Universe. There are lots of funny things of many sizes – e.g. Italy and Pluto are neighbors somewhere in the middle.

## June 28, 2018

### Lubos Motl - string vacua and pheno

James Wells' anti-naturalness quackery
Sabine Hossenfelder celebrates a preprint titled
Naturalness, Extra-Empirical Theory Assessments, and the Implications of Skepticism
and rightfully so because its author, James Wells, could literally shake her hand right away and join her personal movement of crackpots. Wells' paper isn't just wrong – it's incredibly stupid. Thankfully, he only sent it to the "History and Philosophy of Physics" subarchive although it was cross-listed to the professional subarchives. (Maybe the arXiv moderators should be thanked for correctly classifying this paper as social sciences, pseudosciences, and humanities.)

OK, Wells (like Hossenfelder) wants to eliminate naturalness – and any "extra-empirical quality" – from science. Do you really think it's possible? Not really. Let us discuss the abstract carefully.
Naturalness is an extra-empirical quality that aims to assess plausibility of a theory.
It's a proposed definition or classification and it's fair enough.

Now,
Finetuning measures are one way to quantify the task.
Very well. Everyone knows that. The following sentence says:
However, knowing statistical distributions on parameters appears necessary for rigor.
Yup. If you want to precisely (it's a better word than "rigorously") calculate a fine-tuning measure or another quantity telling you how much a theory is fine-tuned, you need statistical distributions on the space of possible theories and on their parameter spaces.
Such meta-theories are not known yet.
Strictly speaking, it may be true because there's no precise or rigorous prescription to calculate the probability of some values of parameters or the probability of one theory consistent with observations or another.

However, what Wells completely misses is that some i.e. not precise and not rigorous prescription to compare two theories has to be used, anyway, otherwise the scientific method as a whole would be impossible. Without this type of – imprecise or not rigorous – thinking, we couldn't say whether evolution or creationism is a better theory of the origin of species. We wouldn't be able to say anything.

Again, I must quote Feynman's monologue about the flying saucers. All the statements that science produces are of the form that one statement is more likely and another one is less likely etc. All such probabilities always depend on the priors, not only on the evidence. It's unavoidable. If you ban sentences that "flying saucers are unlikely" (because you find the dependence on the prior probabilities "unscientific"), and Feynman's antagonist wanted to ban them, then you are banning science as a whole.

So it's not true that such meta-theories are not known yet. They are known, they are imprecise and not rigorous, but they are absolutely essential for science and successful, too.
A critical discussion of these issues is presented, including their possible resolutions in fixed points.
He includes a technical discussion of fixed points (scaling-invariant field theories) but claims that all "extra-empirical reasoning" is unacceptable in their context, too.
Skepticism of naturalness's utility remains credible, as is skepticism to any extra-empirical theory assessment (SEETA) that claims to identify "more correct" theories that are equally empirically adequate.
This skepticism is as credible as creationism and all other wrong approaches to science – in fact, this skepticism is a key part of them. Otherwise, it's great that he invented a new acronym. Brain-dead journalists will surely boast about their ability to copy and hype this new meaningless acronym.

A crucial proposal for "a new kind of science" appears here:
Specifically to naturalness, SEETA implies that one must accept all concordant theory points as a priori equally plausible, with the practical implication that a theory can never have its plausibility status diminished by even a "massive reduction" of its viable parameter space as long as a single theory point still survives.
Wow. You know, saying that all theories are "equally possible" means that they have the same probability, namely $$p$$. But a problem is that they're mutually exclusive and there are infinitely many of them. It follows that$\sum_{i=1}^\infty p \leq 1.$ Their total probability is at most equal to one. I chose the $$\leq$$ sign to emphasize that we're only summing over the known theories and there may be additional ones that have a chance to be correct. But the left hand side above is equal to $$p\cdot \infty$$ and the only allowed value of $$p$$ that obeys the inequality above is $$p=0$$. If all theories in an infinite list were equally plausible, then all of them would be strictly ruled out, too!

In reality, the theories are also parameterized by continuous parameters so the sum above should be replaced or supplemented with an integral. With an integral, the statement that they are "equally plausible" becomes ill-defined because, as Wells admitted, he doesn't have any measure. He wants to use the absence of a canonical measure as a "weapon against others" but overlooks that it's a weapon against his own claims, too.

If he forbids you to use any measure, then his statement that two points (or regions) at a continuous parameter space are "equally plausible" becomes nonsensical.
A second implication of SEETA suggests that only falsifiable theories allow their plausibility status to change, but only after discovery or after null experiments with total theory coverage.
Excellent. If this rule is interpreted literally, you really can't eliminate creationism or any wrong theory. In those seven days He had to create all the species, He could have used tools with a sufficient number of parameters so that He created the correct DNA of all the species we need. If you can't exclude "all creationist models" and every single one of them, you can't really say that creationism is very unlikely, Wells (just like Feynman's antagonist) tells us.

Many of us say that evolution is a far better theory of the origin of species than creationism. Why? Because the fine-tuning that creationism needs to agree with the observed details is massive. And when the required fine-tuning is massive, it just doesn't really matter what's the "precise" or "rigorous" way to quantify it. Any sensible way to quantify it will still conclude that it is massive. Now, the word "sensible" in the previous sentence also fails to be defined precisely. But at some moment, you have to stop with these complaints, otherwise you just can't get anywhere in science.

That's why Wells' claim that you should completely abandon naturalness and "extra-empirical criteria" just because they're not perfectly precise is so unbelievably idiotic. You could try to apply his fundamentalist attitude in any other context. Child porn cannot be precisely defined, either. Does it mean that we can't ban it? Well, Justice Potter Stewart defined porn by saying that "I know it when I see it".

That's really the point in the discussion of naturalness, too. There may be some marginal cases in which the absence of a precise definition or quantification will make it impossible to reliably decide whether something is porn or whether something is natural. But in a huge fraction of the cases that are relevant for law enforcement officials and for physicists, the quantities labeling the "amount of porn" or the "naturalness" end up being so far from the "disputable lines" that the imprecision won't matter at all. In so many cases, we will say: "This is porn." We will say it even without a rigorous definition of "porn". And in the same way, we will say that a creationist model explaining some DNA sequences is "unnatural" even though we don't really have a canonical, unique, universal, ultimate, precise definition of "naturalness of a hypothesis about the origin of species", either!

So when some theories are really heavily unnatural, we simply see it. And we need this judgment, despite its lack of rigor and precision, to do science. We have always needed it. We couldn't decide even about the basic questions if we banned this "extra-empirical" reasoning. Everyone who questions the need for this imprecise or "extra-empirical" reasoning is absolutely deluded.

Sometimes the implausibility of a theory – like creationism – is understood informally, intuitively, and qualitatively. Sometimes, especially in fundamental physics, we need a bit more quantitative treatment. This treatment is not rigorous or precise but it's more quantitative than the arguments we need to criticize creationism. So we assume that the distributions are some natural uniform distributions mostly spanning the values of dimensionless parameters that are of order one. The detailed choice doesn't really matter when something is really unnatural! In most cases, we have pretty good arguments to say that the choice of a uniform distribution for $$g$$ or $$g^2$$ or $$1/g^2$$ is more natural than the other two etc.

The last sentence of the abstract is very cute, too:
And a third implication of SEETA is that theory preference then becomes not about what theory is more correct but what theory is practically more advantageous, such as fewer parameters, easier to calculate, or has new experimental signatures to pursue.
The only problem is that a genuine scientist, pretty much by definition, looks for the more correct or more likely theories. He wants to answer the questions such as whether creationism or evolution is a better theory of the origin of species, whether a proton is composed of smaller particles, whether there is a Higgs boson, and millions of other things.

So the correctness or probability of different possibilities simply has to be compared, otherwise you're not doing science at all. You're not producing any scientific results, any laws, nothing. By promoting SEETA, Wells pretends to be "more scientific" but in reality, he wants to throw the key baby of the scientific method out with the bath water.

A real scientist is working to find the truth about Nature – which means the (more) correct and (more) likely theories that explain our observations. If he's looking for a theory that is "practically more advantageous, such as fewer parameters, easier to calculate, or has new experimental signatures to pursue", then he is simply not a scientist in the proper sense. He's a utilitarian of a sort.

Theory A may be simpler to calculate with than theory B. But that doesn't mean that it's more correct or more likely to be true.

At the beginning of the abstract, Wells declared his goal to liberate science from all the distributions and "extra-empirical" judgments. But in the last sentence, he contradicts himself and basically admits it's impossible. So he also has some "extra-empirical" criteria, after all. The only difference is that his criteria aren't designed to look for more likely theories. He's looking for more convenient (and similar adjectives that are not equivalent to the truth) ideas.

There is an overlap between his criteria and the criteria of physicists who are actually looking for the truth about Nature. For example, both seem to prefer "theories with a small number of parameters". But Wells only picks this criterion because of some convenience. In proper physics, we may actually justify why we start with a theory with a fewer parameters. Why is it so? Because theories with a larger number of parameters are either
1. much less likely than the theory with a few parameters because most of the "new parameter space" spoils the predictions – because additional parameters have to be adjusted and it's unlikely that it's done right, or
2. the theory with fewer parameters may be considered as a "subset" of more complex theories, so if you study the simpler theory in this sense, you're not wasting your time – most of your work may be recycled once you deal with the possible more complex theories (the whole paradigm of effective field theories is a broad subcategory of this phenomenon)
These arguments aren't "rigorously proven" to be correct but if we didn't use any "extra-empirical" guides at all, we just couldn't possibly make a single decision in science, ever, because an arbitrarily wrong theory may always be modified, engineered, and tuned to be formally compatible with the data.

His list of the "preferred extra-empirical" criteria includes
simplicity, testability, falsifiability, naturalness, calculability, and diversity.
None of them actually tries to be equivalent to the validity of a theory, the probability that it's correct, which is why those aren't really scientific criteria. But in this list, the last entry, "diversity", must have shocked many readers just like it has shocked me. What kind of diversity? Does he want to prefer papers written by black or female or transsexual authors? ;-)
Or, a scientist may wish to widen her vision of observable consequences of concordant theories in order to cast a wider experimental net, which would lead her to pursue diverse theories over simple theories.
Well, the choice of pronouns indicates that he really wants to prefer theories by female authors, even if he never makes this statement explicitly. Well, I am sure you still hope that he doesn't actually talk about the identity politics. Another sentence says the following about diversity:
No theory of theory preference will be given here, except to say that “diversity” has a strong claim to a quality for preference.
It's rather hard to figure out what he means by the "diversity of a single theory". We usually understand "diversity" as a property of whole sets or groups (e.g. groups of people), not the individual elements or members. But a few sentences later, we read:
A few examples out of many in the literature that have the quality of diversity at least going for it are clockwork theories [19, 20] and theories of superlight dark matter (see, e.g., [21, 22]). These theories lead to new experiments, or new experimental analyses, that may not have been performed otherwise.
He just picks some – not really terribly motivated – theories, clockwork theories and superlight dark matter, and wants to prefer them because they have a "quality of diversity". The last sentence explains that by "diversity", he means that the theory "leads" to new experiments or new analyses.

It's just nutty. Theories never "lead" to experiments. Experimenters may decide to build an experiment but it's their practical decision that doesn't follow in any logical way from a theory. An experimenter needs some creativity, practical skills including some intuition for the economy of some efforts, knowledge of the established theory as well as proposed hypotheses to go beyond them, and good luck to successfully decide which things are interesting to be tested or measured and how he can find something interesting or new.

There's no "straightforward" way to derive these experimenters' decisions from any theory by itself. There's surely no "rigorous" way to do so – but you see the double standards. Other people's criteria have to be "rigorous", otherwise they need to be thrown away. But his criteria may be totally non-rigorous. What the fudge?

So if an experimenter is inspired by some theory, and the experiment may only be justified by a clockwork theory or a theory of superlight dark matter, good for him. But the experimenter isn't guaranteed to find the damn new effect. And if the new effect is only predicted by some very special theory, or one theory among hundreds, then – sorry to say – it probably makes it less likely, not more likely, that the experiment will lead to some interesting results. Such a dependence of the new effect on some very special theory is clearly an argument (not an indisputable one, but still an argument) against the experiment if the experimenter is rational.

Wells clearly wants to invalidate the self-evidently rational reasoning above. How does he invalidate it? If a theory C predicts something that no other theory predicts, this theory will be declared "more important" because it passes a test from "diversity". Holy crap. Even if he talks about some technical features of theories, their predictions, the logic of his reasoning is almost isomorphic to affirmative action, reverse racism, and reverse sexism, indeed. For all purposes, clockwork theories are transsexual Muslims and the superlight dark matter is a female vegan who loves steaks. And that's why he wants to make them more widespread. But from a rational viewpoint, what he calls "diversity" should be viewed as a negative trait, at least a negative recommendation for an experimenter.

His "extra preferences" are absolutely irrational from the viewpoint of the search for the truth and due to their similarity to the toxic left-wing identity politics, every decent physicist must immediately vomit when he hears about Wells' proposals for "new criteria". If you fail to vomit, you are probably not a good physicist.

Sorry but as long as science remains science, it is looking for the truth i.e. for theories that are more likely to be true or compatible with a body of observations. And this is always evaluated by meritocratic criteria using justifiable probability distributions. Because the final theory of everything isn't known yet, these probability distributions and criteria aren't totally precise and rigorously defined. But they're parts of the required theorist's toolkit, they're being tested by the experiments as well, and their current form as believed by the best theorists are good enough – and good scientists also spend some time by trying to improve and refine them. At any rate, they're vastly better than the pseudoscientific and borderline political new criteria proposed by Wells that have nothing whatever to do with the chances of the theories to be true.

And that's the memo.

### The n-Category Cafe

Elmendorf's Theorem

I want to tell you about Elmendorf’s theorem on equivariant homotopy theory. This theorem played a key role in a recent preprint I wrote with Hisham Sati and Urs Schreiber:

We figured out how to apply this theorem in mathematical physics. But Elmendorf’s theorem by itself is a gem of homotopy theory and deserves to be better known. Here’s what it says, roughly: given any $GG$-space $XX$, the equivariant homotopy type of $XX$ is determined by the ordinary homotopy types of the fixed point subspaces ${X}^{H}X^H$, where $HH$ runs over all subgroups of $GG$. I don’t know how to intuitively motivate this fact; I would like to know, and if any of you have ideas, please comment. Below the fold, I will spell out the precise theorem, and show you how it gives us a way to define a $GG$-equivariant version of any homotopy theory.

We know that in ordinary homotopy theory, there are two kinds of spaces we can study. We can study CW-complexes up to homotopy equivalence, or we can study topological spaces up to weak homotopy equivalence. Weak homotopy equivalence is morally the right kind of equivalence, but Whitehead’s theorem tells us that for the nicer kind of space, the CW-complex, weak homotopy equivalence is the same as strong homotopy equivalence. Moreover, the CW-approximation theorem says that any space is weak homotopy equivalent to a CW-complex. So, they’re really two ways of studying the same thing. One is more flexible, the other more concrete.

NB. In this post, I’ll use the adjective “strong” to contrast homotopy equivalence with weak homotopy equivalence. People usually call strong homotopy equivalence just homotopy equivalence.

Now let $GG$ be a compact Lie group. For $GG$-spaces, we can also define both strong and weak homotopy equivalence. The strong homotopy equivalence is the obvious thing: you have two equivariant maps $f:X\to Yf \colon X \to Y$ and $g:Y\to Xg \colon Y \to X$, that are inverse to each other up to equivariant homotopies $\eta :fg⇒{1}_{Y}\eta \colon f g \Rightarrow 1_Y$ and $\eta \prime :gf⇒{1}_{X}\eta\text{'} \colon g f \Rightarrow 1_X$. This lets us consider $GG$-spaces up to homotopy equivalence. But as for spaces, the morally correct notion of equivalence is weak homotopy equivalence, and this is much stranger: a $GG$-equivariant map $f:X\to Yf \colon X \to Y$ is a equivariant weak homotopy equivalence if it restricts to an ordinary weak homotopy equivalence between the fixed points spaces, $f:{X}^{H}\to {Y}^{H}f \colon X^H \to Y^H$, for all closed subgroups $H\subseteq GH \subseteq G$.

Why on earth should these two notions of equivalence be so different? The equivariant Whitehead theorem justifies this, though again I don’t have a good intuitive explanation for why it should be true. To state this theorem, first I have to tell you what a $GG$-CW-complex is. We can construct them much as we do ordinary CW-complexes, except they are built from cells of the form:

${D}^{n}×G/H D^n \times G/H $

where ${D}^{n}D^n$ is the $nn$-disk with the trivial $GG$ action, and $G/HG/H$ is a coset space of $GG$ with the left $GG$ action. These cells are then glued together by $GG$-equivariant attaching maps, just like an ordinary CW-complex. The result is a $GG$-CW-complex. The equivariant Whitehead theorem, due to Bredon, then says that for any pair of $GG$-CW-complexes, they are weak homotopy equivalent if and only if they are strong homotopy equivalent.

This suggests the key insight behind Elmendorf’s theorem: that we can study $GG$-spaces simply by looking at ${X}^{H}X^H$ for all closed subgroups $H\subseteq GH \subseteq G$. But this operation, of taking a subgroup $HH$ to a space ${X}^{H}X^H$, actually defines a functor:

$X:{\mathrm{Orb}}_{G}^{\mathrm{op}}\to \mathrm{Spaces}. X \colon Orb_G^\left\{op\right\} \to Spaces . $

Here, the domain of this contravariant functor is the orbit category ${\mathrm{Orb}}_{G}Orb_G$. This is the category with:

• objects the coset spaces $G/HG/H$, for each closed subgroup $H\subseteq GH \subseteq G$.
• morphisms the $GG$-equivariant maps.

This is called the orbit category thanks to the elementary fact that any orbit in any $GG$-space is of the form $G/HG/H$, for a closed subgroup $HH$ the stabilizer of some point in the orbit.

Since the functor associated to $XX$ is contravariant, it is a presheaf on the orbit category ${\mathrm{Orb}}_{G}Orb_G$, valued in the category of spaces, $\mathrm{Spaces}Spaces$. The assignment taking a $GG$-space $XX$ to the presheaf with value ${X}^{H}X^H$ on the orbit space $G/HG/H$ defines an embedding:

$y:G\mathrm{Spaces}\to \mathrm{PSh}\left({\mathrm{Orb}}_{G},\mathrm{Spaces}\right) y \colon G Spaces \to PSh\left(Orb_G, Spaces\right) $

from the category $G\mathrm{Spaces}G Spaces$ of $GG$-spaces into the category of all presheaves on ${\mathrm{Orb}}_{G}Orb_G$. This is a souped up version of the Yoneda embedding: ${\mathrm{Orb}}_{G}Orb_G$ is a subcategory of $G\mathrm{Spaces}G Spaces$, and the embedding above is just Yoneda when restricted to this subcategory.

It turns out this embedding doesn’t change the homotopy theory at all, as long as we choose the correct weak equivalences on the right hand side: we choose them to be the levelwise weak equivalences. That is, two presheaves $XX$ and $YY$ are weak equivalent if there is a natural transformation $f:X⇒Yf \colon X \Rightarrow Y$ whose components ${f}^{H}:{X}^{H}\to {Y}^{H}f^H \colon X^H \to Y^H$ are ordinary weak equivalences of spaces. With this choice of weak equivalences, the homotopy theory of presheaves on ${\mathrm{Orb}}_{G}Orb_G$ is the same as that of $G\mathrm{Spaces}G Spaces$. That’s Elmendorf’s theorem:

Theorem (Elmendorf). There is an equivalence of homotopy theories $G\mathrm{Spaces}\simeq \mathrm{PSh}\left({\mathrm{Orb}}_{G},\mathrm{Spaces}\right). G Spaces \simeq PSh\left(Orb_G, Spaces\right) . $ In the direction $G\mathrm{Spaces}\to \mathrm{PSh}\left({\mathrm{Orb}}_{G},\mathrm{Spaces}\right)G Spaces \to PSh\left(Orb_G, Spaces\right)$, this equivalence is simply the embedding $yy$.

You can read more about Elmendorf’s theorem in the original paper:

A much more modern treatment is in Andrew Blumberg’s lectures on equivariant homotopy theory. The theorem is so foundational to the topic that it first appears in Section 1.2 of these notes, and Section 1.3 is devoted to it:

Let us step back and appreciate what this theorem has bought us. Besides being a really nice reformulation from a categorical point of view, it gives us a paradigm for constructing equivariant homotopy theories more generally. That is, if we have a homotopy theory in the guise of a category $𝒞\mathcal\left\{C\right\}$ with weak equivalences, then you might go ahead and define the equivariant homotopy theory of $𝒞\mathcal\left\{C\right\}$ to be: $G𝒞=\mathrm{PSh}\left({\mathrm{Orb}}_{G},𝒞\right) G \mathcal\left\{C\right\} = PSh\left(Orb_G, \mathcal\left\{C\right\}\right) $ where the weak equivalences are the levelwise weak equivalences, as in Elmendorf.

For instance, if $𝒞\mathcal\left\{C\right\}$ is a model of rational homotopy theory ${\mathrm{Spaces}}_{ℚ}Spaces_\left\{\mathbb\left\{Q\right\}\right\}$, then $GG$-equivariant rational homotopy ought to be: $\mathrm{PSh}\left({\mathrm{Orb}}_{G},{\mathrm{Spaces}}_{ℚ}\right). PSh\left(Orb_G, Spaces_\left\{\mathbb\left\{Q\right\}\right\}\right) . $ This is precisely what one finds in the literature, at least in the case when $GG$ is a finite group:

This paper actually came before Elmendorf’s - perhaps it served as inspiration!

Or, if you want to get more adventurous, you can define “rational super homotopy theory”, a supersymmetric version of rational homotopy theory, modeled by some category with weak equivalences called ${\mathrm{SuperSpace}}_{ℚ}SuperSpace_\left\{\mathbb\left\{Q\right\}\right\}$. Then the $GG$-equivariant rational super homotopy theory ought to be: $G{\mathrm{SuperSpace}}_{ℚ}=\mathrm{PSh}\left({\mathrm{Orb}}_{G},{\mathrm{SuperSpace}}_{ℚ}\right). G SuperSpace_\left\{\mathbb\left\{Q\right\}\right\} = PSh\left(Orb_G, SuperSpace_\left\{\mathbb\left\{Q\right\}\right\}\right) . $ This is the homotopy theory where the work in our preprint takes place! We use Elmendorf’s theorem to get our hands on what physicists call “black branes”. These turn out to be the fixed point subspaces ${X}^{H}X^H$, for $XX$ a particular rational superspace equipped with an action.

To close, let me ask if you or anyone you know has a nice conceptual explanation for Elmendorf’s theorem, or at the very least for the equivariant Whitehead theorem:

Question. What is an intuitive reason that equivariant homotopy types are captured by the homotopy types of their fixed point subspaces?

## June 27, 2018

### Lubos Motl - string vacua and pheno

Vafa, quintessence vs Gross, Silverstein
It has been one year since Strings 2017 ended in the Israeli capital (yes, I mean Jerusalem, that's where Czechia has the honorary consulate) and Strings 2018 in Okinawa (list of titles) is here.

The Japanese organizers have tried an original reform of the YouTube activity. They post the whole days as unified videos.

Most of you want to delay your dinner by 3 hours and 46 minutes – so you should watch the video above for the food to taste better.

Cumrun Vafa whom I know much better than the other 20 speakers in the video starts to speak at 31:44 and his topic is a paper released on Monday,
De Sitter Space and the Swampland (Obied, Ooguri, Spodyneiko, Vafa)
Recall that Vafa's Swampland is a giant parameter space of effective field theories that cannot be realized within a consistent theory of quantum gravity i.e. within string/M-theory. Only a tiny island inside this Swampland, namely the stringy Landscape, is compatible with quantum gravity. String/M-theory makes lots of very strong predictions – namely that we don't live in the Swampland. We have to live in the special hospitable Landscape.

Along with his friend Donald Trump, Cumrun Vafa decided to drain the Swampland. ;-)

OK, these comments became ludicrous too early so let us be more serious for a while. In the paper I mentioned, Vafa and pals have proposed a new inequality obeyed by the potential energy $$V$$ in every consistent theory of quantum gravity$V \leq \frac{|\nabla V|}{c}$ which excludes too high positive cosmological constants and de Sitter spaces in particular. The gradient is calculated on the field space, with the metric given by the kinetic terms of the scalar fields. Use 4D Planck units (Einstein frame) if others aren't good enough.

It's a fun proposal – a potential cousin of the Weak Gravity Conjecture, but a younger and less justified one (at least so far). Note that $$V_0\leq 0$$ constraining the minimum would be hard to justify and the appearance of the minimum $$V_0$$ only would be unnatural. So in a sense, Vafa and co-authors have found and supported a stronger version of that inequality that places an upper bound on the potential energy density at each point of the configuration space. Neat.

For a while after I saw the paper, I was worried about the infinite tower of massive stringy fields which make the gradient arbitrarily high – because the mass is high. But then I understood my mistake. If the vacua are stabilized, the gradient from the inequality by Vafa et al. is strictly zero and they make the bold statement that the cosmological constant in stable vacua has to be zero (Minkowski) or negative (AdS).

I think it's a cool inequality and people should investigate the arguments for and against and the consequences. Such a simple inequality could indeed summarize the absence of persuasive constructions of metastable de Sitter spaces within string/M-theory.

Now, I am confident that Cumrun Vafa is one of the staunchest believers in string/M-theory in the world – I might have a hard time to compete with him even when it comes to the strength of the belief. So doesn't he think that this "no-go theorem" for the positive cosmological constant excludes string theory because the positive cosmological constant has been observed?

A good question, indeed. Well, it turns out that Cumrun Vafa has become a quintessence believer. So the cosmological constant isn't really constant, some scalar field is rolling, and its relative constancy only emulates the cosmological constant. As David Gross reminds everybody around 0:58:00 in the video above, quintenessence has a widely believed problem: it seems to predict time-dependent fundamental constants such as the fine-structure constant (the conditions are changing with time).

David Gross came to Okinawa from California.

These predicted variations in time don't have a very good reason to be small. Because experiments exist that prove that the fine-structure constant was the same within an impressive accuracy a few bilion years ago, quintessence seems to be basically excluded.

Cumrun believes that the thing can work in some way but I haven't quite absorbed his new belief – it's still rather non-standard for me.

(Along with Steinhardt and two other authors, Vafa posted an even more recent paper against inflation that tries to suggest that even inflation is in tension with the Swampland rules of quantum gravity. Cumrun has discussed this paper in his talk, too. He would probably replace inflation with the string gas cosmology or something like that – well, it's possible to research it but I think that the string gas cosmology is even less likely to be true than quintessence.)

Around 1:01:00, Eva Silverstein joins David Gross in the criticisms. Both of them have offered some lore that somewhat contradicts Vafa's picture, to put it mildly. Let me say the following: I have gone through similar thoughts, have been exposed to similar arguments, and I also tend to think that the lore is more likely than not.

But something is still wrong with the overall picture of dark energy and other things in string theory so I think it's right to be open-minded. By constantly repeating the lore – and David Gross does it rather often – one may force the quantum Zeno effect on the string researchers. They won't have the opportunity to make a jump that is almost certainly needed.

Cumrun Vafa knows quite something and when he becomes a quitenessence believer, I think it's useful to be interested in the mental processes that have led him to this transformation. Gross repeats some lore but there is really no rock-solid argument against quintessence. In fact, I think that some people could sensibly argue that swampland-like inequalities are actually more solid consequences of a theoretical framework – and you may view them as consequences of either quantum gravity or string/M-theory (which are ultimately equivalent but the phrases "sound" different) – than Gore's lore [I meant Gross' lore but I kept the typo because it looked funny] that disfavors quintessence models.

I have had similar feelings when David Gross was rather heavily attacking (repeated TRF guest blogger) Gordon Kane. You know, I have shared Gross' observation that in his M-theory phenomenology, Kane had to make numerous additional assumptions, some of which were inspired by rather detailed empirical observations (about possible ways to extend the Standard Model to a theory that also agrees with some cosmological criteria).

So while I have never believed that Kane has derived the necessity of his kind of models from the first principles – and indeed, it seems that his prediction of a gluino below $$2\TeV$$ has been invalidated – there was something that I disliked about Gross' criticism. It just sounded to me that Gross wanted to discourage the people from thinking about specific scenarios, specific classes of models with some extra assumptions that just happen to look natural.

Well, I would even put it in this way: Gross apparently wanted everybody to treat the whole "landscape" as a set of equal elements and avoid the focusing on any particular elements or subgroups of the vacua because that would be a discrimination. In December, Gross dreamed about the early death of Donald Trump and a month later, he rather brutally attacked mainstream conservative Indians.

But those things are fine, no one cares what David Gross thinks about politics – everyone more or less correctly assumes he's just another cloned leftist in the Academia. However, I feel that a similar constant imposition of his lore and group think is something he does in physics, too. And it isn't right. People like Kane must have the freedom to study and propose realistic M-theory compactifications; and people like Vafa must have the freedom to investigate quintessence within string/M-theory. People must combine and recombine assumptions, pick privileged classes of vacua that look more promising to them given these assumptions and the observations, and so on. If those choices are discrimination, then it is a basic moral duty of a physicist to discriminate at basically all times!

I don't understand his reasoning but I find it (less likely than 50% but) conceivable that Cumrun has some reasons to think that the time variation of the constants might be compatible with their quintessence picture. On the other hand, Vafa admits that they don't solve the cosmological constant problem – why the "apparent" current vacuum energy density is so small.

But the inequality they propose – especially if you assume that it wants to be near-saturated – eliminates the "double fine-tuning" that you would need in generic quintessence models, those that were studied before their Swampland findings. In regular quintessence, $$V\approx 10^{-122}$$ and $$|V|\approx 10^{-122}$$ are two independent fine-tunings. With the near-saturated Swampland inequality, these two fine-tunings reduce to basically one, they are not independent. So you could say that with the Swampland findings, if established, the quintessence is as natural as the cosmological constant (one fine-tuning by 122 orders of magnitude). Vafa has made additional comparisons of naturalness within the standard or their axiomatic system and theirs seems to win.

Eva Silverstein has joined the polemics with her pet topic that I have been aware of since 2005: the affirmative action for supercritical strings. Supercritical string vacua must be treated on par with the critical string theory's compactifications, there can't be any discrimination. Please: Not again! (You know me as someone who passionately argues about an extremely diverse spectrum of topics. But I think that I have actually never argued as tensely about string theory with a string theorist as I did with Silverstein in 2005.)

The discussion at Strings 2018 made it clear that she has tried to convince Cumrun about that supercritical affirmative action and Cumrun has rejected in a very similar way – and maybe for similar reasons – as I did. Sorry, Eva, but supercritical string theory simply isn't an equally likely or convincing picture of the real world as presented by string theory as the critical string theory is.

The prediction of the critical dimension, e.g. $$D=10$$ for the weakly coupled superstring, is one of the first heroic predictions of string theory that obviously go beyond the predictive power of quantum field theories. Some supercritical string world sheet CFTs may be defined but it is much less clear whether these theories may be completed to non-perturbative consistent theories.

In the case of critical superstring theory, S-dualities etc. make it extremely likely that the theory is fully consistent at any finite coupling (because it seems OK at zero and infinity). But in the supercritical case, there are no known S-dualities like that and lots of other arguments in favor of non-perturbative consistency that work in critical string theory simply can't be applied to supercritical string theory.

Moreover, the terms proportional to $$(D-10)$$ appear in the beta-function for the dilaton, a scalar field that plays a preferred role at weak coupling but that should become a generic scalar field at a stronger coupling. The beta-function for the dilaton dictates the Euler-Lagrange equation of motion that one would derive from varying the dilaton in the effective action.

I actually find it likely that there exists a swampland-style inequality similar to the one that Cumrun just discussed that says that the other terms $$t$$ in the equation $$(D-10)c + t = 0$$ cannot be large enough to actually beat the "wrong dimension term" for $$D\neq 10$$. As far as I know, all these questions are rather difficult and convincing justifications for one answer or another just don't exist. In fact, I can imagine that this statement excluding the supercritical string theory's stabilized vacua directly follows exactly from the inequality that Vafa et al. propose: they say that the gradient terms, and $$t$$ contains those, are always too small to beat the constant terms $$V$$ – and $$(D-10)c$$ might simply be a constant term that needs to be beaten but is too large.

We don't have a proof in either direction. But that's exactly why Cumrun's open-minded approach "I am not saying you must be wrong, Eva, but what I say might also be right" is exactly the right one.

Eva's claim that all the supercritical vacua, perhaps those in $$D=2018$$, are as established and as consistent as the $$D=10$$ or $$D=11$$ vacua is just plain silly. This claim utterly disagrees with the composition of the string theory literature where most of the "good properties" depend on the critical dimension (and/or on supersymmetry, and supersymmetry is also possible in the critical spacetime dimension only). Silverstein's claim about the "equality" of critical and supercritical vacua is just some unjustified ideology at this point. Maybe the research will change towards the "equality between critical and supercritical vacua" in the future but it's a pure speculation; the critical string theory is much more established today and some future advances may also eliminate the supercritical string theory altogether.

Moreover, such a full legitimization of the supercritical vacua would probably lead to a much more hopeless proliferation of the "landscape of possibilities" than the regular landscape of the critical string/M-theory. The very dimension $$D$$ would be unlimited and the complexity and diversity of possible compactifications would dramatically increase with $$D$$, too. Maybe mathematics or Nature may make the search for the right vacuum much more difficult than we thought (and it's been hard enough for some two decades). But this is just a possibility, not an established fact. It's totally sensible to do research dependent on the working hypothesis that supercritical string theory is a curiosity in perturbative string theory that may be given one page of a textbook – but otherwise it's worthless, unusable, inconsistent rubbish in the string model building!

Cumrun clearly agrees with this working hypothesis of mine while Eva – without real evidence – is trying to make this assumption politically incorrect. She would love to impose a duty on everyone to spend the same time with supercritical string theory as with critical string theory. That's just a counterproductive pressure that should be ignored by Cumrun and others because the outcomes of such an extra rule would almost certain be tragic.

## June 25, 2018

### Sean Carroll - Preposterous Universe

On Civility

Alex Wong/Getty Images

White House Press Secretary Sarah Sanders went to have dinner at a local restaurant the other day. The owner, who is adamantly opposed to the policies of the Trump administration, politely asked her to leave, and she did. Now (who says human behavior is hard to predict?) an intense discussion has broken out concerning the role of civility in public discourse and our daily life. The Washington Post editorial board, in particular, called for public officials to be allowed to eat in peace, and people have responded in volume.

I don’t have a tweet-length response to this, as I think the issue is more complex than people want to make it out to be. I am pretty far out to one extreme when it comes to the importance of engaging constructively with people with whom we disagree. We live in a liberal democracy, and we should value the importance of getting along even in the face of fundamentally different values, much less specific political stances. Not everyone is worth talking to, but I prefer to err on the side of trying to listen to and speak with as wide a spectrum of people as I can. Hell, maybe I am even wrong and could learn something.

On the other hand, there is a limit. At some point, people become so odious and morally reprehensible that they are just monsters, not respected opponents. It’s important to keep in our list of available actions the ability to simply oppose those who are irredeemably dangerous/evil/wrong. You don’t have to let Hitler eat in your restaurant.

This raises two issues that are not so easy to adjudicate. First, where do we draw the line? What are the criteria by which we can judge someone to have crossed over from “disagreed with” to “shunned”? I honestly don’t know. I tend to err on the side of not shunning people (in public spaces) until it becomes absolutely necessary, but I’m willing to have my mind changed about this. I also think the worry that this particular administration exhibits authoritarian tendencies that could lead to a catastrophe is not a completely silly one, and is at least worth considering seriously.

More importantly, if the argument is “moral monsters should just be shunned, not reasoned with or dealt with constructively,” we have to be prepared to be shunned ourselves by those who think that we’re moral monsters (and those people are out there).  There are those who think, for what they take to be good moral reasons, that abortion and homosexuality are unforgivable sins. If we think it’s okay for restaurant owners who oppose Trump to refuse service to members of his administration, we have to allow staunch opponents of e.g. abortion rights to refuse service to politicians or judges who protect those rights.

The issue becomes especially tricky when the category of “people who are considered to be morally reprehensible” coincides with an entire class of humans who have long been discriminated against, e.g. gays or transgender people. In my view it is bigoted and wrong to discriminate against those groups, but there exist people who find it a moral imperative to do so. A sensible distinction can probably be made between groups that we as a society have decided are worthy of protection and equal treatment regardless of an individual’s moral code, so it’s at least consistent to allow restaurant owners to refuse to serve specific people they think are moral monsters because of some policy they advocate, while still requiring that they serve members of groups whose behaviors they find objectionable.

The only alternative, as I see it, is to give up on the values of liberal toleration, and to simply declare that our personal moral views are unquestionably the right ones, and everyone should be judged by them. That sounds wrong, although we do in fact enshrine certain moral judgments in our legal codes (murder is bad) while leaving others up to individual conscience (whether you want to eat meat is up to you). But it’s probably best to keep that moral core that we codify into law as minimal and widely-agreed-upon as possible, if we want to live in a diverse society.

This would all be simpler if we didn’t have an administration in power that actively works to demonize immigrants and non-straight-white-Americans more generally. Tolerating the intolerant is one of the hardest tasks in a democracy.

## June 24, 2018

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

7th Robert Boyle Summer School

This weekend saw the 7th Robert Boyle Summer School, an annual 3-day science festival in Lismore, Co. Waterford in Ireland. It’s one of my favourite conferences – a select number of talks on the history and philosophy of science, aimed at curious academics and the public alike, with lots of time for questions and discussion after each presentation.

The Irish-born scientist and aristocrat Robert Boyle

Lismore Castle in Co. Waterford , the birthplace of Robert Boyle

Born in Lismore into a wealthy landowning family, Robert Boyle became one of the most important figures in the Scientific Revolution. A contemporary of Isaac Newton and Robert Hooke, he is recognized the world over for his scientific discoveries, his role in the rise of the Royal Society and his influence in promoting the new ‘experimental philosophy’ in science.

This year, the theme of the conference was ‘What do we know – and how do we know it?’. There were many interesting talks such as Boyle’s Theory of Knowledge by Dr William Eaton, Associate Professor of Early Modern Philosophy at Georgia Southern University: The How, Who & What of Scientific Discovery by Paul Strathern, author of a great many books on scientists and philosophers such as the well-known Philosophers in 90 Minutes series: Scientific Enquiry and Brain StateUnderstanding the Nature of Knowledge by Professor William T. O’Connor, Head of Teaching and Research in Physiology at the University of Limerick Graduate Entry Medical School: The Promise and Peril of Big Data by Timandra Harkness, well-know media presenter, comedian and writer. For physicists, there was a welcome opportunity to hear the well-known American philosopher of physics Robert P. Crease present the talk Science Denial: will any knowledge do? The full programme for the conference can be found here.

All in all, a hugely enjoyable summer school, culminating in a garden party in the grounds of Lismore castle, Boyle’s ancestral home. My own contribution was to provide the music for the garden party – a flute, violin and cello trio, playing the music of Boyle’s contemporaries, from Johann Sebastian Bach to Turlough O’ Carolan. In my view, the latter was a baroque composer of great importance whose music should be much better known outside Ireland.

Images from the garden party in the grounds of Lismore Castle

## June 23, 2018

### Clifford V. Johnson - Asymptotia

Google Talk!

I think that I forgot to post this link when it came out some time ago. I gave a talk at Google when I passed though London last Spring. There was a great Q & A session too - the Google employees were really interested and asked great questions. I talked in some detail about the book (The Dialogues), why I made it, how I made it, and what I was trying to do with the whole project. For a field that is supposed to be quite innovative (and usually is), I think that, although there are many really great non-fiction science books by Theoretical Physicists, we offer a rather narrow range of books to the general public, and I'm trying to broaden the spectrum with The Dialogues. In the months since the book has come out, people have been responding really positively to the book, so that's very encouraging (and thank you!). It's notable that it is a wide range of people, from habitual science book readers to people who say they've never picked up a science book before... That's a really great sign!

Here's the talk on YouTube:

Direct link here. Embed below: [...] Click to continue reading this post

The post Google Talk! appeared first on Asymptotia.

## June 22, 2018

### Jester - Resonaances

Both g-2 anomalies
Two months ago an experiment in Berkeley announced a new ultra-precise measurement of the fine structure constant α using interferometry techniques. This wasn't much noticed because the paper is not on arXiv, and moreover this kind of research is filed under metrology, which is easily confused with meteorology. So it's worth commenting on why precision measurements of α could be interesting for particle physics. What the Berkeley group really did was to measure the mass of the cesium-133 atom, achieving the relative accuracy of 4*10^-10, that is 0.4 parts par billion (ppb). With that result in hand, α can be determined after a cavalier rewriting of the high-school formula for the Rydberg constant:
Everybody knows the first 3 digits of the Rydberg constant, Ry≈13.6 eV, but actually it is experimentally known with the fantastic accuracy of 0.006 ppb, and the electron-to-atom mass ratio has also been determined precisely. Thus the measurement of the cesium mass can be translated into a 0.2 ppb measurement of the fine structure constant: 1/α=137.035999046(27).

You may think that this kind of result could appeal only to a Pythonesque chartered accountant. But you would be wrong. First of all, the new result excludes  α = 1/137 at 1 million sigma, dealing a mortal blow to the field of epistemological numerology. Perhaps more importantly, the result is relevant for testing the Standard Model. One place where precise knowledge of α is essential is in calculation of the magnetic moment of the electron. Recall that the g-factor is defined as the proportionality constant between the magnetic moment and the angular momentum. For the electron we have
Experimentally, ge is one of the most precisely determined quantities in physics,  with the most recent measurement quoting a= 0.00115965218073(28), that is 0.0001 ppb accuracy on ge, or 0.2 ppb accuracy on ae. In the Standard Model, ge is calculable as a function of α and other parameters. In the classical approximation ge=2, while the one-loop correction proportional to the first power of α was already known in prehistoric times thanks to Schwinger. The dots above summarize decades of subsequent calculations, which now include O(α^5) terms, that is 5-loop QED contributions! Thanks to these heroic efforts (depicted in the film  For a Few Diagrams More - a sequel to Kurosawa's Seven Samurai), the main theoretical uncertainty for the Standard Model prediction of ge is due to the experimental error on the value of α. The Berkeley measurement allows one to reduce the relative theoretical error on adown to 0.2 ppb:  ae = 0.00115965218161(23), which matches in magnitude the experimental error and improves by a factor of 3 the previous prediction based on the α measurement with rubidium atoms.

At the spiritual level, the comparison between the theory and experiment provides an impressive validation of quantum field theory techniques up to the 13th significant digit - an unimaginable  theoretical accuracy in other branches of science. More practically, it also provides a powerful test of the Standard Model. New particles coupled to the electron may contribute to the same loop diagrams from which ge is calculated, and could shift the observed value of ae away from the Standard Model predictions. In many models, corrections to the electron and muon magnetic moments are correlated. The latter famously deviates from the Standard Model prediction by 3.5 to 4 sigma, depending on who counts the uncertainties. Actually, if you bother to eye carefully the experimental and theoretical values of ae beyond the 10th significant digit you can see that they are also discrepant, this time at the 2.5 sigma level. So now we have two g-2 anomalies! In a picture, the situation can be summarized as follows:

If you're a member of the Holy Church of Five Sigma you can almost preach an unambiguous discovery of physics beyond the Standard Model. However, for most of us this is not the case yet. First, there is still some debate about the theoretical uncertainties entering the muon g-2 prediction. Second, while it is quite easy to fit each of the two anomalies separately, there seems to be no appealing model to fit both of them at the same time.  Take for example the very popular toy model with a new massive spin-1 Z' boson (aka the dark photon) kinetically mixed with the ordinary photon. In this case Z' has, much like the ordinary photon, vector-like and universal couplings to electron and muons. But this leads to a positive contribution to g-2, and it does not fit well the ae measurement which favors a new negative contribution. In fact, the ae measurement provides the most stringent constraint in part of the parameter space of the dark photon model. Conversely, a Z' boson with purely axial couplings to matter does not fit the data as it gives a negative contribution to g-2, thus making the muon g-2 anomaly worse. What might work is a hybrid model with a light Z' boson having lepton-flavor violating interactions: a vector coupling to muons and a somewhat smaller axial coupling to electrons. But constructing a consistent and realistic model along these lines is a challenge because of other experimental constraints (e.g. from the lack of observation of μ→eγ decays). Some food for thought can be found in this paper, but I'm not sure if a sensible model exists at the moment. If you know one you are welcome to drop a comment here or a paper on arXiv.

More excitement on this front is in store. The muon g-2 experiment in Fermilab should soon deliver first results which may confirm or disprove the muon anomaly. Further progress with the electron g-2 and fine-structure constant measurements is also expected in the near future. The biggest worry is that, if the accuracy improves by another two orders of magnitude, we will need to calculate six loop QED corrections...

## June 16, 2018

### Tommaso Dorigo - Scientificblogging

On The Residual Brightness Of Eclipsed Jovian Moons
While preparing for another evening of observation of Jupiter's atmosphere with my faithful 16" dobsonian scope, I found out that the satellite Io will disappear behind the Jovian shadow tonight. This is a quite common phenomenon and not a very spectacular one, but still quite interesting to look forward to during a visual observation - the moon takes some time to fully disappear, so it is fun to follow the event.
This however got me thinking. A fully eclipsed jovian moon should still be able to reflect back some light picked up from the still lit other satellites - so it should not, after all, appear completely dark. Can a calculation be made of the effect ? Of course - and it's not that difficult.

read more

## June 12, 2018

### Axel Maas - Looking Inside the Standard Model

How to test an idea
As you may have guessed from reading through the blog, our work is centered around a change of paradigm: That there is a very intriguing structure of the Higgs and the W/Z bosons. And that what we observe in the experiments are actually more complicated than what we usually assume. That they are not just essentially point-like objects.

This is a very bold claim, as it touches upon very basic things in the standard model of particle physics. And the interpretation of experiments. However, it is at the same time a necessary consequence if one takes the underlying more formal theoretical foundation seriously. The reason that there is not a huge clash is that the standard model is very special. Because of this both pictures give almost the same prediction for experiments. This can also be understood quantitatively. That is where I have written a review about. It can be imagined in this way:

Thus, the actual particle, which we observe, and call the Higgs is actually a complicated object made from two Higgs particles. However, one of those is so much eclipsed by the other that it looks like just a single one. And a very tiny correction to it.

So far, this does not seem to be something where it is necessary to worry about.

However, there are many and good reasons to believe that the standard model is not the end of particle physics. There are many, many blogs out there, which explain the reasons for this much better than I do. However, our research provides hints that what works so nicely in the standard model, may work much less so in some extensions of the standard model. That there the composite nature makes huge differences for experiments. This was what came out of our numerical simulations. Of course, these are not perfect. And, after all, unfortunately we did not yet discover anything beyond the standard model in experiments. So we cannot test our ideas against actual experiments, which would be the best thing to do. And without experimental support such an enormous shift in paradigm seems to be a bit far fetched. Even if our numerical simulations, which are far from perfect, support the idea. Formal ideas supported by numerical simulations is just not as convincing as experimental confirmation.

So, is this hopeless? Do we have to wait for new physics to make its appearance?

Well, not yet. In the figure above, there was 'something'. So, the ideas make also a statement that even within the standard model there should be a difference. The only question is, what is really the value of a 'little bit'? So far, experiments did not show any deviations from the usual picture. So 'little bit' needs indeed to be really rather small. But we have a calculation prescription for this 'little bit' for the standard model. So, at the very least what we can do is to make a calculation for this 'little bit' in the standard model. We should then see if the value of 'little bit' may already be so large that the basic idea is ruled out, because we are in conflict with experiment. If this is the case, this would raise a lot of question on the basic theory, but well, experiment rules. And thus, we would need to go back to the drawing board, and get a better understanding of the theory.

Or, we get something which is in agreement with current experiment, because it is smaller then the current experimental precision. But then we can make a statement how much better experimental precision needs to become to see the difference. Hopefully the answer will not be so much that it will not be possible within the next couple of decades. But this we will see at the end of the calculation. And then we can decide, whether we will get an experimental test.

Doing the calculations is actually not so simple. On the one hand, they are technically challenging, even though our method for it is rather well under control. But it will also not yield perfect results, but hopefully good enough. Also, it depends strongly on the type of experiment how simple the calculations are. We did a first few steps, though for a type of experiment not (yet) available, but hopefully in about twenty years. There we saw that not only the type of experiment, but also the type of measurement matters. For some measurements the effect will be much smaller than for others. But we are not yet able to predict this before doing the calculation. There, we need still much better understanding of the underlying mathematics. That we will hopefully gain by doing more of these calculations. This is a project I am currently pursuing with a number of master students for various measurements and at various levels. Hopefully, in the end we get a clear set of predictions. And then we can ask our colleagues at experiments to please check these predictions. So, stay tuned.

By the way: This is the standard cycle for testing new ideas and theories. Have an idea. Check that it fits with all existing experiments. And yes, this may be very, very many. If your idea passes this test: Great! There is actually a chance that it can be right. If not, you have to understand why it does not fit. If it can be fixed, fix it, and start again. Or have a new idea. And, at any rate, if it cannot be fixed, have a new idea. When you got an idea which works with everything we know, use it to make a prediction where you get a difference to our current theories. By this you provide an experimental test, which can decide whether your idea is the better one. If yes: Great! You just rewritten our understanding of nature. If not: Well, go back to fix it or have a new idea. Of course, it is best if we have already an experiment which does not fit with our current theories. But there we are at this stage a little short off. May change again. If your theory has no predictions which can be testable in any foreseeable future experimentally. Well, that is a good question how to deal with this, and there is not yet a consensus how to proceed.

## June 10, 2018

### Tommaso Dorigo - Scientificblogging

Modeling Issues Or New Physics ? Surprises From Top Quark Kinematics Study
Simulation, noun:
1. Imitation or enactment
2. The act or process of pretending; feigning.
3. An assumption or imitation of a particular appearance or form; counterfeit; sham.

Well, high-energy physics is all about simulations.

We have a theoretical model that predicts the outcome of the very energetic particle collisions we create in the core of our giant detectors, but we only have approximate descriptions of the inputs to the theoretical model, so we need simulations.

read more

## June 09, 2018

### Jester - Resonaances

Dark Matter goes sub-GeV
It must have been great to be a particle physicist in the 1990s. Everything was simple and clear then. They knew that, at the most fundamental level, nature was described by one of the five superstring theories which, at low energies, reduced to the Minimal Supersymmetric Standard Model. Dark matter also had a firm place in this narrative, being identified with the lightest neutralino of the MSSM. This simple-minded picture strongly influenced the experimental program of dark matter detection, which was almost entirely focused on the so-called WIMPs in the 1 GeV - 1 TeV mass range. Most of the detectors, including the current leaders XENON and LUX, are blind to sub-GeV dark matter, as slow and light incoming particles are unable to transfer a detectable amount of energy to the target nuclei.

Sometimes progress consists in realizing that you know nothing Jon Snow. The lack of new physics at the LHC invalidates most of the historical motivations for WIMPs. Theoretically, the mass of the dark matter particle could be anywhere between 10^-30 GeV and 10^19 GeV. There are myriads of models positioned anywhere in that range, and it's hard to argue with a straight face that any particular one is favored. We now know that we don't know what dark matter is, and that we should better search in many places. If anything, the small-scale problem of the 𝞚CDM cosmological model can be interpreted as a hint against the boring WIMPS and in favor of light dark matter. For example, if it turns out that dark matter has significant (nuclear size) self-interactions, that can only be realized with sub-GeV particles.

It takes some time for experiment to catch up with theory, but the process is already well in motion. There is some fascinating progress on the front of ultra-light axion dark matter, which deserves a separate post. Here I want to highlight the ongoing  developments in direct detection of dark matter particles with masses between MeV and GeV. Until recently, the only available constraint in that regime was obtained by recasting data from the XENON10 experiment - the grandfather of the currently operating XENON1T.  In XENON detectors there are two ingredients of the signal generated when a target nucleus is struck:  ionization electrons and scintillation photons. WIMP searches require both to discriminate signal from background. But MeV dark matter interacting with electrons could eject electrons from xenon atoms without producing scintillation. In the standard analysis, such events would be discarded as background. However,  this paper showed that, recycling the available XENON10 data on ionization-only events, one can exclude dark matter in the 100 MeV ballpark with the cross section for scattering on electrons larger than ~0.01 picobarn (10^-38 cm^2). This already has non-trivial consequences for concrete models; for example, a part of the parameter space of milli-charged dark matter is currently best constrained by XENON10.

It is remarkable that so much useful information can be extracted by basically misusing data collected for another purpose (earlier this year the DarkSide-50 recast their own data in the same manner, excluding another chunk of the parameter space).  Nevertheless, dedicated experiments will soon  be taking over. Recently, two collaborations published first results from their prototype detectors:  one is SENSEI, which uses 0.1 gram of silicon CCDs, and the other is SuperCDMS, which uses 1 gram of silicon semiconductor.  Both are sensitive to eV energy depositions, thanks to which they can extend the search region to lower dark matter mass regions, and set novel limits in the virgin territory between 0.5 and 5 MeV.  A compilation of the existing direct detection limits is shown in the plot. As you can see, above 5 MeV the tiny prototypes cannot yet beat the XENON10 recast. But that will certainly change as soon as full-blown detectors are constructed, after which the XENON10 sensitivity should be improved by several orders of magnitude.

Should we be restless waiting for these results? Well, for any single experiment the chance of finding nothing are immensely larger than that of finding something. Nevertheless, the technical progress and the widening scope of searches offer some hope that the dark matter puzzle may be solved soon.

## June 08, 2018

### Jester - Resonaances

Massive Gravity, or You Only Live Twice
Proving Einstein wrong is the ultimate ambition of every crackpot and physicist alike. In particular, Einstein's theory of gravitation -  the general relativity -  has been a victim of constant harassment. That is to say, it is trivial to modify gravity at large energies (short distances), for example by embedding it in string theory, but it is notoriously difficult to change its long distance behavior. At the same time, motivations to keep trying go beyond intellectual gymnastics. For example, the accelerated expansion of the universe may be a manifestation of modified gravity (rather than of a small cosmological constant).

In Einstein's general relativity, gravitational interactions are mediated by a massless spin-2 particle - the so-called graviton. This is what gives it its hallmark properties: the long range and the universality. One obvious way to screw with Einstein is to add mass to the graviton, as entertained already in 1939 by Fierz and Pauli. The Particle Data Group quotes the constraint m ≤ 6*10^−32 eV, so we are talking about the De Broglie wavelength comparable to the size of the observable universe. Yet even that teeny mass may cause massive troubles. In 1970 the Fierz-Pauli theory was killed by the van Dam-Veltman-Zakharov (vDVZ) discontinuity. The problem stems from the fact that a massive spin-2 particle has 5 polarization states (0,±1,±2) unlike a massless one which has only two (±2). It turns out that the polarization-0 state couples to matter with the similar strength as the usual polarization ±2 modes, even in the limit where the mass goes to zero, and thus mediates an additional force which differs from the usual gravity. One finds that, in massive gravity, light bending would be 25% smaller, in conflict with the very precise observations of stars' deflection around the Sun. vDV concluded that "the graviton has rigorously zero mass". Dead for the first time...

The second coming was heralded soon after by Vainshtein, who noticed that the troublesome polarization-0 mode can be shut off in the proximity of stars and planets. This can happen in the presence of graviton self-interactions of a certain type. Technically, what happens is that the polarization-0 mode develops a background value around massive sources which, through the derivative self-interactions, renormalizes its kinetic term and effectively diminishes its interaction strength with matter. See here for a nice review and more technical details. Thanks to the Vainshtein mechanism, the usual predictions of general relativity are recovered around large massive source, which is exactly where we can best measure gravitational effects. The possible self-interactions leading a healthy theory without ghosts have been classified, and go under the name of the dRGT massive gravity.

There is however one inevitable consequence of the Vainshtein mechanism. The graviton self-interaction strength grows with energy, and at some point becomes inconsistent with the unitarity limits that every quantum theory should obey. This means that massive gravity is necessarily an effective theory with a limited validity range and has to be replaced by a more fundamental theory at some cutoff scale 𝞚. This is of course nothing new for gravity: the usual Einstein gravity is also an effective theory valid at most up to the Planck scale MPl～10^19 GeV.  But for massive gravity the cutoff depends on the graviton mass and is much smaller for realistic theories. At best,
So the massive gravity theory in its usual form cannot be used at distance scales shorter than ～300 km. For particle physicists that would be a disaster, but for cosmologists this is fine, as one can still predict the behavior of galaxies, stars, and planets. While the theory certainly cannot be used to describe the results of table top experiments,  it is relevant for the  movement of celestial bodies in the Solar System. Indeed, lunar laser ranging experiments or precision studies of Jupiter's orbit are interesting probes of the graviton mass.

Now comes the latest twist in the story. Some time ago this paper showed that not everything is allowed  in effective theories.  Assuming the full theory is unitary, causal and local implies non-trivial constraints on the possible interactions in the low-energy effective theory. These techniques are suitable to constrain, via dispersion relations, derivative interactions of the kind required by the Vainshtein mechanism. Applying them to the dRGT gravity one finds that it is inconsistent to assume the theory is valid all the way up to 𝞚max. Instead, it must be replaced by a more fundamental theory already at a much lower cutoff scale,  parameterized as 𝞚 = g*^1/3 𝞚max (the parameter g* is interpreted as the coupling strength of the more fundamental theory). The allowed parameter space in the g*-m plane is showed in this plot:

Massive gravity must live in the lower left corner, outside the gray area  excluded theoretically  and where the graviton mass satisfies the experimental upper limit m～10^−32 eV. This implies g* ≼ 10^-10, and thus the validity range of the theory is some 3 order of magnitude lower than 𝞚max. In other words, massive gravity is not a consistent effective theory at distance scales below ～1 million km, and thus cannot be used to describe the motion of falling apples, GPS satellites or even the Moon. In this sense, it's not much of a competition to, say, Newton. Dead for the second time.

Is this the end of the story? For the third coming we would need a more general theory with additional light particles beyond the massive graviton, which is consistent theoretically in a larger energy range, realizes the Vainshtein mechanism, and is in agreement with the current experimental observations. This is hard but not impossible to imagine. Whatever the outcome, what I like in this story is the role of theory in driving the progress, which is rarely seen these days. In the process, we have understood a lot of interesting physics whose relevance goes well beyond one specific theory. So the trip was certainly worth it, even if we find ourselves back at the departure point.

## June 07, 2018

### Jester - Resonaances

Can MiniBooNE be right?
The experimental situation in neutrino physics is confusing. One one hand, a host of neutrino experiments has established a consistent picture where the neutrino mass eigenstates are mixtures of the 3 Standard Model neutrino flavors νe, νμ, ντ. The measured mass differences between the eigenstates are Δm12^2 ≈ 7.5*10^-5 eV^2 and Δm13^2 ≈ 2.5*10^-3 eV^2, suggesting that all Standard Model neutrinos have masses below 0.1 eV. That is well in line with cosmological observations which find that the radiation budget of the early universe is consistent with the existence of exactly 3 neutrinos with the sum of the masses less than 0.2 eV. On the other hand, several rogue experiments refuse to conform to the standard 3-flavor picture. The most severe anomaly is the appearance of electron neutrinos in a muon neutrino beam observed by the LSND and MiniBooNE experiments.

This story begins in the previous century with the LSND experiment in Los Alamos, which claimed to observe νμνe antineutrino oscillations with 3.8σ significance.  This result was considered controversial from the very beginning due to limitations of the experimental set-up. Moreover, it was inconsistent with the standard 3-flavor picture which, given the masses and mixing angles measured by other experiments, predicted that νμνe oscillation should be unobservable in short-baseline (L ≼ km) experiments. The MiniBooNE experiment in Fermilab was conceived to conclusively prove or disprove the LSND anomaly. To this end, a beam of mostly muon neutrinos or antineutrinos with energies E~1 GeV is sent to a detector at the distance L~500 meters away. In general, neutrinos can change their flavor with the probability oscillating as P ~ sin^2(Δm^2 L/4E). If the LSND excess is really due to neutrino oscillations, one expects to observe electron neutrino appearance in the MiniBooNE detector given that L/E is similar in the two experiments. Originally, MiniBooNE was hoping to see a smoking gun in the form of an electron neutrino excess oscillating as a function of L/E, that is peaking at intermediate energies and then decreasing towards lower energies (possibly with several wiggles). That didn't happen. Instead, MiniBooNE finds an excess increasing towards low energies with a similar shape as the backgrounds. Thus the confusion lingers on: the LSND anomaly has neither been killed nor robustly confirmed.

In spite of these doubts, the LSND and MiniBooNE anomalies continue to arouse interest. This is understandable: as the results do not fit the 3-flavor framework, if confirmed they would prove the existence of new physics beyond the Standard Model. The simplest fix would be to introduce a sterile neutrino νs with the mass in the eV ballpark, in which case MiniBooNE would be observing the νμνsνe oscillation chain. With the recent MiniBooNE update the evidence for the electron neutrino appearance increased to 4.8σ, which has stirred some commotion on Twitter and in the blogosphere. However, I find the excitement a bit misplaced. The anomaly is not really new: similar results showing a 3.8σ excess of νe-like events were already published in 2012.  The increase of the significance is hardly relevant: at this point we know anyway that the excess is not a statistical fluke, while a systematic effect due to underestimated backgrounds would also lead to a growing anomaly. If anything, there are now less reasons than in 2012 to believe in the sterile neutrino origin the MiniBooNE anomaly, as I will argue in the following.

What has changed since 2012? First, there are new constraints on νe appearance from the OPERA experiment (yes, this OPERA) who did not see any excess νe in the CERN-to-Gran-Sasso νμ beam. This excludes a large chunk of the relevant parameter space corresponding to large mixing angles between the active and sterile neutrinos. From this point of view, the MiniBooNE update actually adds more stress on the sterile neutrino interpretation by slightly shifting the preferred region towards larger mixing angles...  Nevertheless, a not-too-horrible fit to all appearance experiments can still be achieved in the region with Δm^2~0.5 eV^2 and the mixing angle sin^2(2θ) of order 0.01.

Next, the cosmological constraints have become more stringent. The CMB observations by the Planck satellite do not leave room for an additional neutrino species in the early universe. But for the parameters preferred by LSND and MiniBooNE, the sterile neutrino would be abundantly produced in the hot primordial plasma, thus violating the Planck constraints. To avoid it, theorists need to deploy a battery of  tricks (for example, large sterile-neutrino self-interactions), which makes realistic models rather baroque.

But the killer punch is delivered by disappearance analyses. Benjamin Franklin famously said that only two things in this world were certain: death and probability conservation. Thus whenever an electron neutrino appears in a νμ beam, a muon neutrino must disappear. However, the latter process is severely constrained by long-baseline neutrino experiments, and recently the limits have been further strengthened thanks to the MINOS and IceCube collaborations. A recent combination of the existing disappearance results is available in this paper.  In the 3+1 flavor scheme, the probability of a muon neutrino transforming into an electron  one in a short-baseline experiment is
where U is the 4x4 neutrino mixing matrix.  The Uμ4 matrix elements controls also the νμ survival probability
The νμ disappearance data from MINOS and IceCube imply |Uμ4|≼0.1, while |Ue4|≼0.25 from solar neutrino observations. All in all, the disappearance results imply that the effective mixing angle sin^2(2θ) controlling the νμνsνe oscillation must be much smaller than 0.01 required to fit the MiniBooNE anomaly. The disagreement between the appearance and disappearance data had already existed before, but was actually made worse by the MiniBooNE update.
So the hypothesis of a 4th sterile neutrino does not stand scrutiny as an explanation of the MiniBooNE anomaly. It does not mean that there is no other possible explanation (more sterile neutrinos? non-standard interactions? neutrino decays?). However, any realistic model will have to delve deep into the crazy side in order to satisfy the constraints from other neutrino experiments, flavor physics, and cosmology. Fortunately, the current confusing situation should not last forever. The MiniBooNE photon background from π0 decays may be clarified by the ongoing MicroBooNE experiment. On the timescale of a few years the controversy should be closed by the SBN program in Fermilab, which will add one near and one far detector to the MicroBooNE beamline. Until then... years of painful experience have taught us to assign a high prior to the Standard Model hypothesis. Currently, by far the most plausible explanation of the existing data is an experimental error on the part of the MiniBooNE collaboration.

## June 01, 2018

### Jester - Resonaances

WIMPs after XENON1T
After today's update from the XENON1T experiment, the situation on the front of direct detection of WIMP dark matter is as follows

WIMP can be loosely defined as a dark matter particle with mass in the 1 GeV - 10 TeV range and significant interactions with ordinary matter. Historically, WIMP searches have stimulated enormous interest because this type of dark matter can be easily realized in models with low scale supersymmetry. Now that we are older and wiser, many physicists would rather put their money on other realizations, such as axions, MeV dark matter, or primordial black holes. Nevertheless, WIMPs remain a viable possibility that should be further explored.

To detect WIMPs heavier than a few GeV, currently the most successful strategy is to use huge detectors filled with xenon atoms, hoping one of them is hit by a passing dark matter particle. Xenon1T beats the competition from the LUX and Panda-X experiments because it has a bigger gun tank. Technologically speaking, we have come a long way in the last 30 years. XENON1T is now sensitive to 40 GeV WIMPs interacting with nucleons with the cross section of 40 yoctobarn (1 yb = 10^-12 pb = 10^-48 cm^2). This is 6 orders of magnitude better than what the first direct detection experiment in the Homestake mine could achieve back in the 80s. Compared to the last year, the  limit is better by a factor of two at the most sensitive mass point. At high mass the improvement is somewhat smaller than expected due to a small excess of events observed by XENON1T, which is probably just a 1 sigma upward fluctuation of the background.

What we are learning about WIMPs is how they can (or cannot) interact with us. Of course, at this point in the game we don't see qualitative progress, but rather incremental quantitative improvements. One possible scenario is that WIMPs experience one of the Standard Model forces,  such as the weak or the Higgs force. The former option is strongly constrained by now. If WIMPs had interacted in the same way as our neutrino does, that is by exchanging a Z boson,  it would have been found in the Homestake experiment. Xenon1T is probing models where the dark matter coupling to the Z boson is suppressed by a factor cχ ~ 10^-3 - 10^-4 compared to that of an active neutrino. On the other hand, dark matter could be participating in weak interactions only by exchanging W bosons, which can happen for example when it is a part of an SU(2) triplet. In the plot you can see that XENON1T is approaching but not yet excluding this interesting possibility. As for models using the Higgs force, XENON1T is probing the (subjectively) most natural parameter space where WIMPs couple with order one strength to the Higgs field.

And the arms race continues. The search in XENON1T will go on until the end of this year, although at this point a discovery is extremely unlikely. Further progress is expected on a timescale of a few years thanks to the next generation xenon detectors XENONnT and LUX-ZEPLIN, which should achieve yoctobarn sensitivity. DARWIN may be the ultimate experiment along these lines, in the sense that there is no prefix smaller than yocto it will reach the irreducible background from atmospheric neutrinos, after which new detection techniques will be needed.  For dark matter mass closer to 1 GeV, several orders of magnitude of pristine parameter space will be covered by the SuperCDMS experiment. Until then we are kept in suspense. Is dark matter made of WIMPs? And if yes, does it stick above the neutrino sea?

### Tommaso Dorigo - Scientificblogging

MiniBoone Confirms Neutrino Anomaly
Neutrinos, the most mysterious and fascinating of all elementary particles, continue to puzzle physicists. 20 years after the experimental verification of a long-debated effect whereby the three neutrino species can "oscillate", changing their nature by turning one into the other as they propagate in vacuum and in matter, the jury is still out to decide what really is the matter with them. And a new result by the MiniBoone collaboration is stirring waters once more.

read more

## May 26, 2018

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

A festschrift at UCC

One of my favourite academic traditions is the festschrift, a conference convened to honour the contribution of a senior academic. In a sense, it’s academia’s version of an Oscar for lifetime achievement, as scholars from all around the world gather to pay tribute their former mentor, colleague or collaborator.

Festschrifts tend to be very stimulating meetings, as the diverging careers of former students and colleagues typically make for a diverse set of talks. At the same time, there is usually a unifying theme based around the specialism of the professor being honoured.

And so it was at NIALLFEST this week, as many of the great and the good from the world of Einstein’s relativity gathered at University College Cork to pay tribute to Professor Niall O’Murchadha, a theoretical physicist in UCC’s Department of Physics noted internationally for seminal contributions to general relativity.  Some measure of Niall’s influence can be seen from the number of well-known theorists at the conference, including major figures such as Bob WaldBill UnruhEdward Malec and Kip Thorne (the latter was recently awarded the Nobel Prize in Physics for his contribution to the detection of gravitational waves). The conference website can be found here and the programme is here.

University College Cork: probably the nicest college campus in Ireland

As expected, we were treated to a series of high-level talks on diverse topics, from black hole collapse to analysis of high-energy jets from active galactic nuclei, from the initial value problem in relativity to the search for dark matter (slides for my own talk can be found here). To pick one highlight, Kip Thorne’s reminiscences of the forty-year search for gravitational waves made for a fascinating presentation, from his description of early designs of the LIGO interferometer to the challenge of getting funding for early prototypes – not to mention his prescient prediction that the most likely chance of success was the detection of a signal from the merger of two black holes.

All in all, a very stimulating conference. Most entertaining of all were the speakers’ recollections of Niall’s working methods and his interaction with students and colleagues over the years. Like a great piano teacher of old, one great professor leaves a legacy of critical thinkers dispersed around their world, and their students in turn inspire the next generation!

## May 21, 2018

### Andrew Jaffe - Leaves on the Line

Leon Lucy, R.I.P.

I have the unfortunate duty of using this blog to announce the death a couple of weeks ago of Professor Leon B Lucy, who had been a Visiting Professor working here at Imperial College from 1998.

Leon got his PhD in the early 1960s at the University of Manchester, and after postdoctoral positions in Europe and the US, worked at Columbia University and the European Southern Observatory over the years, before coming to Imperial. He made significant contributions to the study of the evolution of stars, understanding in particular how they lose mass over the course of their evolution, and how very close binary stars interact and evolve inside their common envelope of hot gas.

Perhaps most importantly, early in his career Leon realised how useful computers could be in astrophysics. He made two major methodological contributions to astrophysical simulations. First, he realised that by simulating randomised trajectories of single particles, he could take into account more physical processes that occur inside stars. This is now called “Monte Carlo Radiative Transfer” (scientists often use the term “Monte Carlo” — after the European gambling capital — for techniques using random numbers). He also invented the technique now called smoothed-particle hydrodynamics which models gases and fluids as aggregates of pseudo-particles, now applied to models of stars, galaxies, and the large scale structure of the Universe, as well as many uses outside of astrophysics.

Leon’s other major numerical contributions comprise advanced techniques for interpreting the complicated astronomical data we get from our telescopes. In this realm, he was most famous for developing the methods, now known as Lucy-Richardson deconvolution, that were used for correcting the distorted images from the Hubble Space Telescope, before NASA was able to send a team of astronauts to install correcting lenses in the early 1990s.

For all of this work Leon was awarded the Gold Medal of the Royal Astronomical Society in 2000. Since then, Leon kept working on data analysis and stellar astrophysics — even during his illness, he asked me to help organise the submission and editing of what turned out to be his final papers, on extracting information on binary-star orbits and (a subject dear to my heart) the statistics of testing scientific models.

Until the end of last year, Leon was a regular presence here at Imperial, always ready to contribute an occasionally curmudgeonly but always insightful comment on the science (and sociology) of nearly any topic in astrophysics. We hope that we will be able to appropriately memorialise his life and work here at Imperial and elsewhere. He is survived by his wife and daughter. He will be missed.

## May 14, 2018

### Sean Carroll - Preposterous Universe

Intro to Cosmology Videos

In completely separate video news, here are videos of lectures I gave at CERN several years ago: “Cosmology for Particle Physicists” (May 2005). These are slightly technical — at the very least they presume you know calculus and basic physics — but are still basically accurate despite their age.

Update: I originally linked these from YouTube, but apparently they were swiped from this page at CERN, and have been taken down from YouTube. So now I’m linking directly to the CERN copies. Thanks to commenters Bill Schempp and Matt Wright.

## May 10, 2018

### Sean Carroll - Preposterous Universe

User-Friendly Naturalism Videos

Some of you might be familiar with the Moving Naturalism Forward workshop I organized way back in 2012. For two and a half days, an interdisciplinary group of naturalists (in the sense of “not believing in the supernatural”) sat around to hash out the following basic question: “So we don’t believe in God, what next?” How do we describe reality, how can we be moral, what are free will and consciousness, those kinds of things. Participants included Jerry Coyne, Richard Dawkins, Terrence Deacon, Simon DeDeo, Daniel Dennett, Owen Flanagan, Rebecca Newberger Goldstein, Janna Levin, Massimo Pigliucci, David Poeppel, Nicholas Pritzker, Alex Rosenberg, Don Ross, and Steven Weinberg.

Happily we recorded all of the sessions to video, and put them on YouTube. Unhappily, those were just unedited proceedings of each session — so ten videos, at least an hour and a half each, full of gems but without any very clear way to find them if you weren’t patient enough to sift through the entire thing.

No more! Thanks to the heroic efforts of Gia Mora, the proceedings have been edited down to a number of much more accessible and content-centered highlights. There are over 80 videos (!), with a median length of maybe 5 minutes, though they range up to about 20 minutes and down to less than one. Each video centers on a particular idea, theme, or point of discussion, so you can dive right into whatever particular issues you may be interested in. Here, for example, is a conversation on “Mattering and Secular Communities,” featuring Rebecca Goldstein, Dan Dennett, and Owen Flanagan.

The videos can be seen on the workshop web page, or on my YouTube channel. They’re divided into categories:

A lot of good stuff in there. Enjoy!

## March 29, 2018

### Robert Helling - atdotde

Machine Learning for Physics?!?
Today was the last day of a nice workshop here at the Arnold Sommerfeld Center organised by Thomas Grimm and Sven Krippendorf on the use of Big Data and Machine Learning in string theory. While the former (at this workshop mainly in the form of developments following Kreuzer/Skarke and taking it further for F-theory constructions, orbifolds and the like) appears to be quite advanced as of today, the latter is still in its very early days. At best.

I got the impression that for many physicists that have not yet spent too much time with this, deep learning and in particular deep neural networks are expected to be some kind of silver bullet that can answer all kinds of questions that humans have not been able to answer despite some effort. I think this hope is at best premature and looking at the (admittedly impressive) examples where it works (playing Go, classifying images, speech recognition, event filtering at LHC) these seem to be more like those problems where humans have at least a rough idea how to solve them (if it is not something that humans do everyday like understanding text) and also roughly how one would code it but that are too messy or vague to be treated by a traditional program.

So, during some of the less entertaining talks I sat down and thought about problems where I would expect neural networks to perform badly. And then, if this approach fails even in simpler cases that are fully under control one should maybe curb the expectations for the more complex cases that one would love to have the answer for. In the case of the workshop that would be guessing some topological (discrete) data (that depends very discontinuously on the model parameters). Here a simple problem would be a 2-torus wrapped by two 1-branes. And the computer is supposed to compute the number of matter generations arising from open strings at the intersections, i.e. given two branes (in terms of their slope w.r.t. the cycles of the torus) how often do they intersect? Of course these numbers depend sensitively on the slope (as a real number) as for rational slopes $p/q$ and $m/n$ the intersection number is the absolute value of $pn-qm$. My guess would be that this is almost impossible to get right for a neural network, let alone the much more complicated variants of this simple problem.

Related but with the possibility for nicer pictures is the following: Can a neural network learn the shape of the Mandelbrot set? Let me remind those of you who cannot remember the 80ies anymore, for a complex number c you recursively apply the function
$f_c(z)= z^2 +c$
starting from 0 and ask if this stays bounded (a quick check shows that once you are outside $|z| < 2$ you cannot avoid running to infinity). You color the point c in the complex plane according to the number of times you have to apply f_c to 0 to leave this circle. I decided to do this for complex numbers x+iy in the rectangle -0.74
I have written a small mathematica program to compute this image. Built into mathematica is also a neural network: You can feed training data to the function Predict[], for me these were 1,000,000 points in this rectangle and the number of steps it takes to leave the 2-ball. Then mathematica thinks for about 24 hours and spits out a predictor function. Then you can plot this as well:

There is some similarity but clearly it has no idea about the fractal nature of the Mandelbrot set. If you really believe in magic powers of neural networks, you might even hope that once it learned the function for this rectangle one could extrapolate to outside this rectangle. Well, at least in this case, this hope is not justified: The neural network thinks the correct continuation looks like this:
Ehm. No.

All this of course with the caveat that I am no expert on neural networks and I did not attempt anything to tune the result. I only took the neural network function built into mathematica. Maybe, with a bit of coding and TensorFlow one can do much better. But on the other hand, this is a simple two dimensional problem. At least for traditional approaches this should be much simpler than the other much higher dimensional problems the physicists are really interested in.

### Axel Maas - Looking Inside the Standard Model

Asking questions leads to a change of mind
In this entry, I would like to digress a bit from my usual discussion of our physics research subject. Rather, I would like to talk a bit about how I do this kind of research. There is a twofold motivation for me to do this.

One is that I am currently teaching, together with somebody from the philosophy department, a course on science philosophy of physics. It cam to me as a surprise that one thing the students of philosophy are interested in is, how I think. What are the objects, or subjects, and how I connect them when doing research. Or even when I just think about a physics theory. The other is the review I have have recently written. Both topics may seem unrelated at first. But there is deep connection. It is less about what I have written in the review, but rather what led me up to this point. This requires some historical digression in my own research.

In the very beginning, I started out with doing research on the strong interactions. One of the features of the strong interactions is that the supposed elementary particles, quarks and gluons, are never seen separately, but only in combinations as hadrons. This is a phenomenon which is called confinement. It always somehow presented as a mystery. And as such, it is interesting. Thus, one question in my early research was how to understand this phenomenon.

Doing that I came across an interesting result from the 1970ies. It appears that a, at first sight completely unrelated, effect is very intimately related to confinement. At least in some theories. This is the Brout-Englert-Higgs effect. However, we seem to observe the particles responsible for and affected by the Higgs effect. And indeed, at that time, I was still thinking that the particles affected by the Brout-Englert-Higgs effect, especially  the Higgs and the W and Z bosons, are just ordinary, observable particles. When one reads my first paper of this time on the Higgs, this is quite obvious. But then there was the results of the 1970ies. It stated that, on a very formal level, there should be no difference between confinement and the Brout-Englert-Higgs effect, in a very definite way.

Now the implications of that serious sparked my interest. But I thought this would help me to understand confinement, as it was still very ingrained into me that confinement is a particular feature of the strong interactions. The mathematical connection I just took as a curiosity. And so I started to do extensive numerical simulations of the situation.

But while trying to do so, things which did not add up started to accumulate. This is probably most evident in a conference proceeding where I tried to put sense into something which, with hindsight, could never be interpreted in the way I did there. I still tried to press the result into the scheme of thinking that the Higgs and the W/Z are physical particles, which we observe in experiment, as this is the standard lore. But the data would not fit this picture, and the more and better data I gathered, the more conflicted the results became. At some point, it was clear that something was amiss.

At that point, I had two options. Either keep with the concepts of confinement and the Brout-Englert-Higgs effect as they have been since the 1960ies. Or to take the data seriously, assuming that these conceptions were wrong. It is probably signifying my difficulties that it took me more than a year to come to terms with the results. In the end, the decisive point was that, as a theoretician, I needed to take my theory seriously, no matter the results. There is no way around it. And it gave a prediction which did not fit my view of the experiments than necessarily either my view was incorrect or the theory. The latter seemed more improbable than the first, as it fits experiment very well. So, finally, I found an explanation, which was consistent. And this explanation accepted the curious mathematical statement from the 1970ies that confinement and the Brout-Englert-Higgs effect are qualitatively the same, but not quantitatively. And thus the conclusion was what we observe are not really the Higgs and the W/Z bosons, but rather some interesting composite objects, just like hadrons, which due to a quirk of the theory just behave almost as if they are the elementary particles.

This was still a very challenging thought to me. After all, this was quite contradictory to usual notions. Thus, it came as a very great relief to me that during a trip a couple months later someone pointed me to a few, almost forgotten by most, papers from the early 1980ies, which gave, for a completely different reason, the same answer. Together with my own observation, this made click, and everything started to fit together - the 1970ies curiosity, the standard notions, my data. That I published in the mid of 2012, even though this still lacked some more systematic stuff. But it required still to shift my thinking from agreement to really understanding. That came then in the years to follow.

The important click was to recognize that confinement and the Brout-Englert-Higgs effect are, just as pointed out in the 1970ies mathematically, really just two faces to the same underlying phenomena. On a very abstract level, essentially all particles which make up the standard model, are really just a means to an end. What we observe are objects which are described by them, but which they are not themselves. They emerge, just like hadrons emerge in the strong interaction, but with very different technical details. This is actually very deeply connected with the concept of gauge symmetry, but this becomes quickly technical. Of course, since this is fundamentally different from the usual way, this required confirmation. So we went, made predictions which could distinguish between the standard way of thinking and this way of thinking, and tested them. And it came out as we predicted. So, seems we are on the right track. And all details, all the if, how, and why, and all the technicalities and math you can find in the review.

To make now full circle to the starting point: That what happened during this decade in my mind was that the way I thought about how the physical theory I tried to describe, the standard model, changed. In the beginning I was thinking in terms of particles and their interactions. Now, very much motivated by gauge symmetry, and, not incidental, by its more deeper conceptual challenges, I think differently. I think no longer in terms of the elementary particles as entities themselves, but rather as auxiliary building blocks of actually experimentally accessible quantities. The standard 'small-ball' analogy went fully away, and there formed, well, hard to say, a new class of entities, which does not necessarily has any analogy. Perhaps the best analogy is that of, no, I really do not know how to phrase it. Perhaps at a later time I will come across something. Right now, it is more math than words.

This also transformed the way how I think about the original problem, confinement. I am curious, where this, and all the rest, will lead to. For now, the next step will be to go ahead from simulations, and see whether we can find some way how to test this actually in experiment. We have some ideas, but in the end, it may be that present experiments will not be sensitive enough. Stay tuned.

## March 28, 2018

### Marco Frasca - The Gauge Connection

Paper with a proof of confinement has been accepted

Recently, I wrote a paper together with Masud Chaichian (see here) containing a mathematical proof of confinement of a non-Abelian gauge theory based on Kugo-Ojima criterion. This paper underwent an extended review by several colleagues well before its submission. One of them has been Taichiro Kugo, one of the discoverers of the confinement criterion, that helped a lot to improve the paper and clarify some points. Then, after a review round of about two months, the paper has been accepted in Physics Letters B, one of the most important journals in particle physics.

This paper contains the exact beta function of a Yang-Mills theory. This confirms that confinement arises by the combination of the running coupling and the propagator. This idea was around in some papers in these latter years. It emerged as soon as people realized that the propagator by itself was not enough to grant confinement, after extended studies on the lattice.

It is interesting to point out that confinement is rooted in the BRST invariance and asymptotic freedom. The Kugo-Ojima confinement criterion permits to close the argument in a rigorous way yielding the exact beta funtion of the theory.

## March 20, 2018

### Marco Frasca - The Gauge Connection

Good news from Moriond

Some days ago, Rencontres of Moriond 2018 ended with the CERN presenting a wealth of results also about the Higgs particle. The direction that the two great experiments, ATLAS and CMS, took is that of improving the measurements on the Standard Model as no evidence has been seen so far of possible new particles. Also, the studies of the properties of the Higgs particle have been refined as promised and the news are really striking.

In a communicate to the public (see here), CERN finally acknowledge, for the first time, a significant discrepancy between data from CMS and Standard Model for the signal strengths in the Higgs decay channels. They claim a 17% difference. This is what I advocated for some years and I have published in reputable journals. I will discuss this below. I would like only to show you the CMS results in the figure below.

ATLAS, by its side, is seeing significant discrepancy in the ZZ channel ($2\sigma$) and a $1\sigma$ compatibility for the WW channel. Here are their results.

On the left the WW channel is shown and on the right there are the combined $\gamma\gamma$ and ZZ channels.

The reason of the discrepancy is due, as I have shown in some papers (see here, here and here), to the improper use of perturbation theory to evaluate the Higgs sector. The true propagator of the theory is a sum of Yukawa-like propagators with a harmonic oscillator spectrum. I solved exactly this sector of the Standard Model. So, when the full propagator is taken into account, the discrepancy is toward an increase of the signal strength. Is it worth a try?

This means that this is not physics beyond the Standard Model but, rather, the Standard Model in its full glory that is teaching something new to us about quantum field theory. Now, we are eager to see the improvements in the data to come with the new run of LHC starting now. In the summer conferences we will have reasons to be excited.

## March 17, 2018

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

Remembering Stephen Hawking

Like many physicists, I woke to some sad news early last Wednesday morning, and to a phoneful of requests from journalists for a soundbyte. In fact, although I bumped into Stephen at various conferences, I only had one significant meeting with him – he was intrigued by my research group’s discovery that Einstein once attempted a steady-state model of the universe. It was a slightly scary but very funny meeting during which his famous sense of humour was fully at play.

Yours truly talking steady-state cosmology with Stephen Hawking

I recalled the incident in a radio interview with RTE Radio 1 on Wednesday. As I say in the piece, the first words that appeared on Stephen’s screen were “I knew..” My heart sank as I assumed he was about to say “I knew about that manuscript“. But when I had recovered sufficiently to look again, what Stephen was actually saying was “I knew ..your father”. Phew! You can find the podcast here.

Hawking in conversation with my late father (LHS) and with Ernest Walton (RHS)

RTE TV had a very nice obituary on the Six One News, I have a cameo appearence a few minutes into the piece here.

In my view, few could question Hawking’s brilliant contributions to physics, or his outstanding contribution to the public awareness of science. His legacy also includes the presence of many brilliant young physicists at the University of Cambridge today. However, as I point out in a letter in today’s Irish Times, had Hawking lived in Ireland, he probably would have found it very difficult to acquire government funding for his work. Indeed, he would have found that research into the workings of the universe does not qualify as one of the “strategic research areas” identified by our national funding body, Science Foundation Ireland. I suspect the letter will provoke an angry from certain quarters, but it is tragically true.

Update

The above notwithstanding, it’s important not to overstate the importance of one scientist. Indeed, today’s Sunday Times contains a good example of the dangers of science history being written by journalists. Discussing Stephen’s 1974 work on black holes, Bryan Appleyard states  “The paper in effect launched the next four decades of cutting edge physics. Odd flowers with odd names bloomed in the garden of cosmic speculation – branes, worldsheets , supersymmetry …. and, strangest of all, the colossal tree of string theory”.

What? String theory, supersymmetry and brane theory are all modern theories of particle physics (the study of the world of the very small). While these theories were used to some extent by Stephen in his research in cosmology (the study of the very large), it is ludicrous to suggest that they were launched by his work.

## March 16, 2018

### Sean Carroll - Preposterous Universe

Stephen Hawking’s Scientific Legacy

Stephen Hawking died Wednesday morning, age 76. Plenty of memories and tributes have been written, including these by me:

I can also point to my Story Collider story from a few years ago, about how I turned down a job offer from Hawking, and eventually took lessons from his way of dealing with the world.

Of course Hawking has been mentioned on this blog many times.

When I started writing the above pieces (mostly yesterday, in a bit of a rush), I stumbled across this article I had written several years ago about Hawking’s scientific legacy. It was solicited by a magazine at a time when Hawking was very ill and people thought he would die relatively quickly — it wasn’t the only time people thought that, only to be proven wrong. I’m pretty sure the article was never printed, and I never got paid for it; so here it is!

(If you’re interested in a much better description of Hawking’s scientific legacy by someone who should know, see this article in The Guardian by Roger Penrose.)

Stephen Hawking’s Scientific Legacy

Stephen Hawking is the rare scientist who is also a celebrity and cultural phenomenon. But he is also the rare cultural phenomenon whose celebrity is entirely deserved. His contributions can be characterized very simply: Hawking contributed more to our understanding of gravity than any physicist since Albert Einstein.

“Gravity” is an important word here. For much of Hawking’s career, theoretical physicists as a community were more interested in particle physics and the other forces of nature — electromagnetism and the strong and weak nuclear forces. “Classical” gravity (ignoring the complications of quantum mechanics) had been figured out by Einstein in his theory of general relativity, and “quantum” gravity (creating a quantum version of general relativity) seemed too hard. By applying his prodigious intellect to the most well-known force of nature, Hawking was able to come up with several results that took the wider community completely by surprise.

By acclimation, Hawking’s most important result is the realization that black holes are not completely black — they give off radiation, just like ordinary objects. Before that famous paper, he proved important theorems about black holes and singularities, and afterward studied the universe as a whole. In each phase of his career, his contributions were central.

The Classical Period

While working on his Ph.D. thesis in Cambridge in the mid-1960’s, Hawking became interested in the question of the origin and ultimate fate of the universe. The right tool for investigating this problem is general relativity, Einstein’s theory of space, time, and gravity. According to general relativity, what we perceive as “gravity” is a reflection of the curvature of spacetime. By understanding how that curvature is created by matter and energy, we can predict how the universe evolves. This may be thought of as Hawking’s “classical” period, to contrast classical general relativity with his later investigations in quantum field theory and quantum gravity.

Around the same time, Roger Penrose at Oxford had proven a remarkable result: that according to general relativity, under very broad circumstances, space and time would crash in on themselves to form a singularity. If gravity is the curvature of spacetime, a singularity is a moment in time when that curvature becomes infinitely big. This theorem showed that singularities weren’t just curiosities; they are an important feature of general relativity.

Penrose’s result applied to black holes — regions of spacetime where the gravitational field is so strong that even light cannot escape. Inside a black hole, the singularity lurks in the future. Hawking took Penrose’s idea and turned it around, aiming at the past of our universe. He showed that, under similarly general circumstances, space must have come into existence at a singularity: the Big Bang. Modern cosmologists talk (confusingly) about both the Big Bang “model,” which is the very successful theory that describes the evolution of an expanding universe over billions of years, and also the Big Bang “singularity,” which we still don’t claim to understand.

Hawking then turned his own attention to black holes. Another interesting result by Penrose had shown that it’s possible to extract energy from a rotating black hole, essentially by bleeding off its spin until it’s no longer rotating. Hawking was able to demonstrate that, although you can extract energy, the area of the event horizon surrounding the black hole will always increase in any physical process. This “area theorem” was both important in its own right, and also evocative of a completely separate area of physics: thermodynamics, the study of heat.

Thermodynamics obeys a set of famous laws. For example, the first law tells us that energy is conserved, while the second law tells us that entropy — a measure of the disorderliness of the universe — never decreases for an isolated system. Working with James Bardeen and Brandon Carter, Hawking proposed a set of laws for “black hole mechanics,” in close analogy with thermodynamics. Just as in thermodynamics, the first law of black hole mechanics ensures that energy is conserved. The second law is Hawking’s area theorem, that the area of the event horizon never decreases. In other words, the area of the event horizon of a black hole is very analogous to the entropy of a thermodynamic system — they both tend to increase over time.

Black Hole Evaporation

Hawking and his collaborators were justly proud of the laws of black hole mechanics, but they viewed them as simply a formal analogy, not a literal connection between gravity and thermodynamics. In 1972, a graduate student at Princeton University named Jacob Bekenstein suggested that there was more to it than that. Bekenstein, on the basis of some ingenious thought experiments, suggested that the behavior of black holes isn’t simply like thermodynamics, it actually is thermodynamics. In particular, black holes have entropy.

Like many bold ideas, this one was met with resistance from experts — and at this point, Stephen Hawking was the world’s expert on black holes. Hawking was certainly skeptical, and for good reason. If black hole mechanics is really just a form of thermodynamics, that means black holes have a temperature. And objects that have a temperature emit radiation — the famous “black body radiation” that played a central role in the development of quantum mechanics. So if Bekenstein were right, it would seemingly imply that black holes weren’t really black (although Bekenstein himself didn’t quite go that far).

To address this problem seriously, you need to look beyond general relativity itself, since Einstein’s theory is purely “classical” — it doesn’t incorporate the insights of quantum mechanics. Hawking knew that Russian physicists Alexander Starobinsky and Yakov Zel’dovich had investigated quantum effects in the vicinity of black holes, and had predicted a phenomenon called “superradiance.” Just as Penrose had showed that you could extract energy from a spinning black hole, Starobinsky and Zel’dovich showed that rotating black holes could emit radiation spontaneously via quantum mechanics. Hawking himself was not an expert in the techniques of quantum field theory, which at the time were the province of particle physicists rather than general relativists. But he was a quick study, and threw himself into the difficult task of understanding the quantum aspects of black holes, so that he could find Bekenstein’s mistake.

Instead, he surprised himself, and in the process turned theoretical physics on its head. What Hawking eventually discovered was that Bekenstein was right — black holes do have entropy — and that the extraordinary implications of this idea were actually true — black holes are not completely black. These days we refer to the “Bekenstein-Hawking entropy” of black holes, which emit “Hawking radiation” at their “Hawking temperature.”

There is a nice hand-waving way of understanding Hawking radiation. Quantum mechanics says (among other things) that you can’t pin a system down to a definite classical state; there is always some intrinsic uncertainty in what you will see when you look at it. This is even true for empty space itself — when you look closely enough, what you thought was empty space is really alive with “virtual particles,” constantly popping in and out of existence. Hawking showed that, in the vicinity of a black hole, a pair of virtual particles can be split apart, one falling into the hole and the other escaping as radiation. Amazingly, the infalling particle has a negative energy as measured by an observer outside. The result is that the radiation gradually takes mass away from the black hole — it evaporates.

Hawking’s result had obvious and profound implications for how we think about black holes. Instead of being a cosmic dead end, where matter and energy disappear forever, they are dynamical objects that will eventually evaporate completely. But more importantly for theoretical physics, this discovery raised a question to which we still don’t know the answer: when matter falls into a black hole, and then the black hole radiates away, where does the information go?

If you take an encyclopedia and toss it into a fire, you might think the information contained inside is lost forever. But according to the laws of quantum mechanics, it isn’t really lost at all; if you were able to capture every bit of light and ash that emerged from the fire, in principle you could exactly reconstruct everything that went into it, even the print on the book pages. But black holes, if Hawking’s result is taken at face value, seem to destroy information, at least from the perspective of the outside world. This conundrum is the “black hole information loss puzzle,” and has been nagging at physicists for decades.

In recent years, progress in understanding quantum gravity (at a purely thought-experiment level) has convinced more people that the information really is preserved. In 1997 Hawking made a bet with American physicists Kip Thorne and John Preskill; Hawking and Thorne said that information was destroyed, Preskill said that somehow it was preserved. In 2007 Hawking conceded his end of the bet, admitting that black holes don’t destroy information. However, Thorne has not conceded for his part, and Preskill himself thinks the concession was premature. Black hole radiation and entropy continue to be central guiding principles in our search for a better understanding of quantum gravity.

Quantum Cosmology

Hawking’s work on black hole radiation relied on a mixture of quantum and classical ideas. In his model, the black hole itself was treated classically, according to the rules of general relativity; meanwhile, the virtual particles near the black hole were treated using the rules of quantum mechanics. The ultimate goal of many theoretical physicists is to construct a true theory of quantum gravity, in which spacetime itself would be part of the quantum system.

If there is one place where quantum mechanics and gravity both play a central role, it’s at the origin of the universe itself. And it’s to this question, unsurprisingly, that Hawking devoted the latter part of his career. In doing so, he established the agenda for physicists’ ambitious project of understanding where our universe came from.

In quantum mechanics, a system doesn’t have a position or velocity; its state is described by a “wave function,” which tells us the probability that we would measure a particular position or velocity if we were to observe the system. In 1983, Hawking and James Hartle published a paper entitled simply “Wave Function of the Universe.” They proposed a simple procedure from which — in principle! — the state of the entire universe could be calculated. We don’t know whether the Hartle-Hawking wave function is actually the correct description of the universe. Indeed, because we don’t actually have a full theory of quantum gravity, we don’t even know whether their procedure is sensible. But their paper showed that we could talk about the very beginning of the universe in a scientific way.

Studying the origin of the universe offers the prospect of connecting quantum gravity to observable features of the universe. Cosmologists believe that tiny variations in the density of matter from very early times gradually grew into the distribution of stars and galaxies we observe today. A complete theory of the origin of the universe might be able to predict these variations, and carrying out this program is a major occupation of physicists today. Hawking made a number of contributions to this program, both from his wave function of the universe and in the context of the “inflationary universe” model proposed by Alan Guth.

Simply talking about the origin of the universe is a provocative step. It raises the prospect that science might be able to provide a complete and self-contained description of reality — a prospect that stretches beyond science, into the realms of philosophy and theology. Hawking, always provocative, never shied away from these implications. He was fond of recalling a cosmology conference hosted by the Vatican, at which Pope John Paul II allegedly told the assembled scientists not to inquire into the origin of the universe, “because that was the moment of creation and therefore the work of God.” Admonitions of this sort didn’t slow Hawking down; he lived his life in a tireless pursuit of the most fundamental questions science could tackle.

### Ben Still - Neutrino Blog

Particle Physics Brick by Brick
It has been a very long time since I last posted and I apologise for that. I have been working the LEGO analogy, as described in the pentaquark series and elsewhere, into a book. The book is called Particle Physics Brick by Brick and the aim is to stretch the LEGO analogy to breaking point while covering as much of the standard model of particle physics as possible. I have had enormous fun writing it and I hope that you will enjoy it as much if you choose to buy it.

It has been available in the UK since September 2017 and you can buy it from Foyles / Waterstones / Blackwell's / AmazonUK where it is receiving ★★★★★ reviews

It is released in the US this Wednesday 21st March 2018 and you can buy it from all good book stores and Amazon.com

I just wanted to share a few reviews of the book as well because it makes me happy!

Spend a few hours perusing these pages and you'll be in a much better frame of mind to understand your place in the cosmos... The astronomically large objects of the universe are no easier to grasp than the atomically small particles of matter. That's where Ben Still comes in, carrying a box of Legos. A British physicist with a knack for explaining abstract concepts... He starts by matching the weird properties and interactions described by the Standard Model of particle physics with the perfectly ordinary blocks of a collection of Legos. Quarks and leptons, gluons and charms are assigned to various colors and combinations of plastic bricks. Once you've got that system in mind, hang on: Still races off to illustrate the Big Bang, the birth of stars, electromagnetism and all matter of fantastical-sounding phenomenon, like mesons and beta decay. "Given enough plastic bricks, the rules in this book and enough time," Still concludes, "one might imagine that a plastic Universe could be built by us, brick by brick." Remember that the next time you accidentally step on one barefoot.--Ron Charles, The Washington Post

Complex topics explained simply An excellent book. I am Head of Physics at a school and have just ordered 60 copies of this for our L6th students for summer reading before studying the topic on particle physics early next year. Highly recommended. - Ben ★★★★★ AmazonUK

It's beautifully illustrated and very eloquently explains the fundamentals of particle ...
This is a gem of a pop science book. It's beautifully illustrated and very eloquently explains the fundamentals of particle physics without hitting you over the head with quantum field theory and Lagrangian dynamics. The author has done an exceptional job. This is a must have for all students and academics of both physics and applied maths! - Jamie ★★★★★ AmazonUK

## March 02, 2018

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

Snowbound academics are better academics

Like most people in Ireland, I am working at home today. We got quite a dump of snow in the last two days, and there is no question of going anywhere until the roads clear. Worse, our college closed quite abruptly and I was caught on the hop – there are a lot of things (flash drives, books and papers) sitting smugly in my office that I need for my usual research.

The college on Monday evening

That said, I must admit I’m finding it all quite refreshing. For the first time in years, I have time to read interesting things in my daily email; all those postings from academic listings that I never seem to get time to read normally. I’m enjoying it so much, I wonder how much stuff I miss the rest of the time.

The view from my window as I write this

This morning, I thoroughly enjoyed a paper by Nicholas Campion on the representation of astronomy and cosmology in the works of William Shakespeare. I’ve often wondered about this as Shakespeare lived long enough to know of Galileo’s ground-breaking astronomical observations. However, anyone expecting coded references to new ideas about the universe in Shakespeare’s sonnets and plays will be disappointed; apparently he mainly sticks to classical ideas, with a few vague references to the changing order.

I’m also reading about early attempts to measure the parallax of light from a comet, especially by the great Danish astronomer Tycho de Brahe. This paper comes courtesy of the History of Astronomy Discussion Group listings, a really useful resource for anyone interested in the history of astronomy.

While I’m reading all this, I’m also trying to keep abreast of a thoroughly modern debate taking place worldwide, concerning the veracity of an exciting new result in cosmology on the formation of the first stars. It seems a group studying the cosmic microwave background think they have found evidence of a signal representing the absorption of radiation from the first stars. This is exciting enough if correct, but the dramatic part is that the signal is much larger than expected, and one explanation is that this effect may be due to the presence of Dark Matter.

If true, the result would be a major step in our understanding of the formation of stars,  plus a major step in the demonstration of the existence of Dark Matter. However, it’s early days – there are many possible sources of a spurious signal and signals that are larger than expected have a poor history in modern physics! There is a nice article on this in The Guardian, and you can see some of the debate on Peter Coles’s blog In the Dark.  Right or wrong, it’s a good example of how scientific discovery works – if the team can show they have taken all possible spurious results into account, and if other groups find the same result, skepticism will soon be converted into excited acceptance.

All in all, a great day so far. My only concern is that this is the way academia should be – with our day-to-day commitments in teaching and research, it’s easy to forget there is a larger academic world out there.

Update

Of course, the best part is the walk into the village when it finally stops chucking down. can’t believe my local pub is open!

Dunmore East in the snow today