Insights Why Supersymmetry? Because of Deligne's theorem - Comments

[kodama]

I'm currently doing my PhD in theoretical particle physics. I understand SUSY, the Poincare Group and Wigner's Classification quite good. I've read the article twice. However I have no clue what the author is talking about.

To me it reads like the usual SUSY propaganda: SUSY must be correct, because otherwise string theory is in deep trouble. Thus let's find some good sounding reasons why SUSY is inevitable.

This article seems motivated by the current doomsday mood in the HEP community. Everyone was certain that SUSY shows up at the LHC, just as everyone was certain that SUSY shows up at LEP or the Tevatron. (And sure, the 100 TeV collider certainly will find SUSY.) Howecer, there is no experimental evidence for anything beyond the standard model and certainly no signal that hints towards SUSY particles. The fact that the LHC did not find any SUSY particles is a big problem for SUSY fans, because now one main motivation is no longer valid (SUSY as a solution of the naturalness problem).

Therefore, SUSY isn't very attractive anymore. There are four main motivations for SUSY:

Solving the naturalness problem (Higgs mass problem)
Unfication of the three standard model forces. However this argument is rather weak, because any BSM theory with as many free paramters as SUSY can be easily fitted such that the couplings unify. In addition, it's quite unlikely that a big unified symmetry (SO(10), E6) breaks directly to SU(3)xSU(2)xU(1). Instead an intermediate symmetry group between the unifcation and the standard model group, like the Pati-Salam group possibly exists. If this ist the case the couplings ALWAYS unify with SUSY or without.
Solving the Dark Matter problem. This argument is rather weak, too. Any expansion of the standard model with additional particles contains a dark matter candidate if we impose an additional discrete symmtry to guarantee its stability, which is what SUSY does.
The Coleman-Mandula-Theorem and the argument that SUSY is the only possibility to mix spacetime and internal symmetries. I've also problems with this argument, but this comment is already too long. In short, I don't think that SUSY helps to understand why fermions and bosons behave so differently (which is one of the biggest mysteries in modern physics), because this difference is simply the assumption at the start of SUSY. Thus I don't see which theoretical problem SUSY solves or why the proposed "unification" of spacetime and internal symmetries helps in any way.

Now the first one is no longer valid. The second and third are very weak arguments anyway. Thus it is not suprising that many SUSY researches are stopping to work on SUSY topics now. However there is a group of researches that can not stop and thus needs to find new motivation for SUSY: string theorists.

This is how we end with an article like this. Lots of highbrow mathematics and complicated wording, which impresses students and laymans and leaves the impression that SUSY is inevitable.

No. The article presents a fact that was discovered by somebody with no interest in supersymmetry, either way.

Most of what you quote are standard arguments for unbroken low-energy susy. As explained in the article right at the beginning, this is not what it is about. Remains the Coleman-Mandula theorem, on which the article comments in some detail towards the end.

Try to read it. Try to read it without ideology. It is an exposition of a mathematical theorem, which you may try to understand and accept, but which does not go away by becoming angry at it.

um his argument was on *broken* low energy SUSY. not unbroken. and in your account SUSY still has to be broken though obviously at a higher energy scale than what LHC can find.

mathematician Deligne may not have had a personal interest in supersymmetry/string theory, but it seems clear that strings/susy is your own personal belief and research interest and by you i mean Urs schreiber.

peter woit's assessement

http://www.math.columbia.edu/~woit/wordpress/?p=8708#comments

Urs,
You are trying to derive from an extremely general abstract theorem (that Tannaka duality works for not just groups but also Z2-graded groups) an argument for a very specific supergroup, a rather ugly one with no experimental evidence at all for it. I just don’t see any argument at all for this.

“All groups” covers almost all of mathematics, and adding in Z2-graded groups makes this even more general. I’m a big fan of the idea that quantum mechanics is fundamentally representation theory, and (see the book I’ve been writing) I think there’s a huge amount to say about how highly non-trivial and specific basic structures in representation theory govern quantum theory. But, you can’t get something from nothing: an extremely general piece of abstraction applying to almost the entire mathematical universe cannot possibly do the job of distinguishing the very specific mathematical structure that seems to govern the physical universe.

and

Peter Woit says:
August 29, 2016 at 10:27 am
John,
I think we agree about strategy: step back and look for new mathematical insights that may later find applications in fundamental physics. Even if you don’t get what you want for physics, you’ll learn more about deep mathematics, which is all to the good. And sure, Z2-graded mathematics may very well be part of those insights. Now that I’m wrapping up work on the book, I’m looking forward to going back to doing precisely that, thinking about Dirac cohomology.

My problem with Urs is that while he’s often doing this sort of thing, at the same time he finds it necessary to try to use this to defend the central failed research program that has dominated (and done a huge amount of damage to) theoretical physics for over 30 years. His argument starting with Z2-graded Tannaka duality ends up with the specific endpoint of an argument for supergravity, in ten dimensions (whatever you want to call the local supergroup there, that’s the one I’m referring to). I don’t think there’s a serious argument there. You can’t get to that kind of specific theory from general ideas about the relation of QM, representation theory and Tannaka duality. When you try and do it, you’re just adding in lots of unexamined assumptions and eliding distinctions that are exactly the ones you need to be looking at to figure out where this train of inference goes wrong.

Defending 10d superstring theory and supergravity as the fundamental theory while arguing that any possible actual experiments are irrelevant is very dangerous, the “Not Ever Wrong” danger I’m trying to point out. Bringing very abstract not relevant mathematical statements into help do this is a really bad idea. I think in this year we’re going to finally see the collapse of any hope that supersymmetric extensions of the standard model will ever see a test or get experimental support. I hope the community reacts to this by challenging the assumptions that led to enthusiasm for these models, not by permanently seeking refuge in excuses (“only visible at high energy”) and dubious invocations of abstract mathematics.

 

[Urs Schreiber]

Kodama, If you have any technical questions or comments, I'd be happy to discuss them.

But please try to omit the believes and the agenda and the gossip. Try to stick to the objective facts that we are trying to discuss here. There is some really interesting stuff here, and it would be sad to drown it in noise.

The material you quote is subject to various misunderstandings. Since you are lazy and just copy-and-pasting stuff other people said elsewhere, I don't feel motivated to guess which questions you might actually have. Let me know which points you find unclear, and I'll try to help.

 

[ohwilleke]

What precisely must a theory do to qualify as supersymmetric within the meaning of the theorem?

For example, must there be symmetry between the Fermionic and Bosonic sector of fundamental particles generally, or must there actually be a one to one correspondence such that each fundamental fermion have precisely one bosonic counterpart and visa versa?

 

[Urs Schreiber]

There is a comment on that at the end of the article, where it says

"Notice here that a super-group is understood to be a group that may contain odd-graded components. So also an ordinary group is a super-group in this sense. The statement does not say that spacetime symmetry groups need to have odd supergraded components (that would evidently be false). But it says that the largest possible class of those groups that are sensible as local spacetime symmetry groups is precisely the class of possibly-super groups. Not more. Not less."

But let me expand on this point more:

The theorem says that the most general admissible symmetry groups are algebraic super-groups (as opposed to more exotic things like quantum groups etc.) An algebraic group is equivalently a commutative Hopf algebra. An algebraic super-group is equivalently a super-commutative Hopf algebra, hence a Hopf algebra that may be non-commutative, but only in the mild sense that it carries a Z/2-grading and commuting two odd-graded elements past each other introduces a minus sign. What is excluded are Hopf algebras that are more non-commutative than this (e.g. quantum groups) as well as yet more exotic situations.

It is the odd-graded elements in a super-commutative Hopf algebra that correspond to super-group elements that mix bosons and fermions.

Now, as highlighted at the end of the article, ordinary commutative Hopf algebras, all whose elements are in even degree, are included within super-Hopf algebras. These do not mix bosons and fermions.

The theorem does not say that the spacetime symmetry group necessarily needs to mix bosons and fermions, and how much. It only says that there is guaranteed to be a kind of spacetime symmtry group for every consistent collection of elementary particles, and the kind of groups arising this way are precisely the super-groups (as opposed to more exotic things like quantum groups etc.)

 

[ohwilleke]

So, would I be correct in saying that the class of models consistent with this theorem is quite a bit broader than the class of models conventionally described as supersymmetric?

 

[Urs Schreiber]

The theorem itself speaks about super-groups in the general sense, which includes the super-group extensions of the Poincare group (physicist's supersymmetry) as well as more general super-groups. The theorem itself does not know the Poincare group, it only knows that spacetime symmetry needs to be some possibly-super-group. But given that we know that spacetime symmetry looks at least approximately (at low energy) like Poincare, together this means that all that can happen at higher energy is that some super-group extension of Poincare becomes visible. And the super-group extension of Poincare, that's what's conventionally called super-symmetry in physics.

 

[Haelfix]

Out of curiosity, how many loopholes are there to this theorem? For instance, the Coleman-Mandula theorem offered a rather large amount of interesting outs, and just glancing at the statement of the theorem, it looks like some of the same sorts of tricks can be used.

For instance, Coleman-Mandula/Haag-Lopuszanski fails when you consider an infinite amount of particle species (like theories with an infinite tower of higher spin states)...

 

[Urs Schreiber]

There is a set theoretic size bound for the theorem to work (that regularity condition that I mentioned ), but it is very mild. I think it is hard to construct categories that violate this size bound, and the example that do are contrived and won't show up in mathematical practice, much less in physics. This is a point that I should eventually expand on.

 

[Jimster41]

"First of all, the only thing we need to believe about physics, for it to give us information, is an utmost minimum: that particle species transform linearly under spacetime symmetry groups." - Reference https://www.physicsforums.com/insights/supersymmetry-delignes-theorem/

Does it make sense to ask what Background(s) support the existence of regular groups - but don't start with any?

"The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not exist any faithful functor from hTop to Set was first proven by Peter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalence-classes of functors" also fails to be concretizable." - https://en.wikipedia.org/wiki/Concrete_category

Doesn't this bear on the question of whether or not space-time is likely to be discrete or continuous?

 

[Urs Schreiber]

Does it make sense to ask what Background(s) support the existence of regular groups - but don't start with any?

Not entirely sure what you mean to ask here, but I'll highlight again that there is an utmost minimum of assumption that goes into the argument given in the entry above. All it needs is that locally the collection of particle species satisfies the most minimalistic conditions (such as that the space of interaction vertices is a linear space over a field of characteristic zero). No assumption on "backgrounds" enters. And crucially, no assumption on groups enters. The statement about the spacetime symmetry groups is all a consequence of the theorem (that appropriate groups exist at all, and that they span the space of algebraic supergroups).

Doesn't this bear on the question of whether or not space-time is likely to be discrete or continuous?

No. First of all, the classical homotopy category is well known to be pathological in many ways, and the fact that it is not concrete is absolutely no cause of worry or concern, it only serves as a counterexample to concrete categories that people like to cite. If you are looking into actual geometry (discrete or continuous or whatsoever) then the classical homotopy category is not the place to look. It's not about geometry, but about abstract homotopy theory. More importantly however, apart from the word "category" appearing both in the above entry and in the blurb on the homotopy category that you cite, there is no relation between the two.

 

[DrDu]

Technically what I mean are the spaces of "intetwiners" between representations. In physics these are the possible spaces of interaction vertices.

For instance the space of interaction vertices for two spinors merging to become a vector boson include the linear maps which in components are given by the Gamma-matrices, as familiar from QCD. But there is an arbitrary prefactor in front of the Gamma-matrix, the "coupling constant", and hence the space of interaction vertices is in fact a vector space.

First, let me say that this is a very nice and interesting insights article!

If I remember correctly, e.g. Galilean relativity fits well into the Klein schema but there are more general kinds of interactions possible than in the Poincare setting, namely the ones where particles interact via a potential. Are these representations compartible with the prerequisites of Delignes theorem?

 

[Urs Schreiber]

First, let me say that this is a very nice and interesting insights article!

Thanks. Glad you liked it.

If I remember correctly, e.g. Galilean relativity fits well into the Klein schema but there are more general kinds of interactions possible than in the Poincare setting, namely the ones where particles interact via a potential. Are these representations compartible with the prerequisites of Delignes theorem?

Where the theorem speaks about groups, the only condition is that these are affine algebraic. So all the usual matrix groups fit in.

On the other hand, to make the theorem say something about physics, we are to think of these groups as spacetime symmetry groups at high energy, equivalently at small scales. That makes the Galilean group be an odd choice.

 

[DrDu]

What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.

 

[Urs Schreiber]

What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.

The theorem is about tensor categories, whose morphisms in physics translate to possible interactions between particle species. It doesn't say anything about potentials.

 

[john baez]

In short, Deligne's theorem applies perfectly well to the tensor category of representations of the Galilean group: it says this tensor category can be seen as consisting of representations of a group. But that's not very surprising!