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

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1. Aug 21, 2016

### Urs Schreiber

Last edited by a moderator: Aug 21, 2016
2. Aug 21, 2016

### klotza

Is Deligne's theorem here the same one, or related to, as the one that is used to help solve the twin prime conjecture?

3. Aug 21, 2016

### strangerep

Heh, well, in ancient history (when s.p.r. was a great place and you were a moderator) you once rejected a post of mine because I suggested that spacetime itself is not fundamental. I'm glad to see you've (apparently?) changed your mind.

IIUC, the word "exactly" is not correct for massless particles -- one must manually impose a constraint that the representation should be trivial wrt the continuous spin degrees of freedom (i.e., the 2 translation-like generators in the E(2) little group for massless representations). Weinberg vol 1 covers this.

Separately, I have a question about Klein/Cartan geometry. Are there already any extensions of that framework for the case where G/H is a semigroup? I'm thinking here about how one might embed temporal casuality into physical theories in a more fundamental way, rather than being imposed by hand as is currently the case in GR and QFT.

Cheers.

4. Aug 21, 2016

### Urs Schreiber

What was used in the discussion of the twin prime conjecture is Deligne's theorem on extending the Weil bound on Kloosterman sums. This is unrelated to the theorem on Tannakian reconstruction of tensor categories that the above entry is referring to.
Pierre Deligne proved many important theorems.

5. Aug 22, 2016

### Urs Schreiber

Regarding the ancient history: I don't remember the contribution you are referring to, maybe you could remind me.

Regarding the quibble: True, I have swept some technical fine print under the rug, in order to keep the discussion informal. Also, either way this fine print does not affect the point of the article.

Regarding Cartan geometry for semigroups: I haven't seen this discussed anywhere. It seems plausible that one could generalize the definition to that case in a fairly straightforward way, but I haven't seen it considered.

6. Aug 22, 2016

### strangerep

Oh, I didn't keep a copy. At the time, it was all too hard to convince anyone that symmetries are more fundamental than spacetime. Nowadays, I sense that it's a more respectable point of view.

What precisely do you mean by "spaces of particle interaction vertices"? In the context of ordinary QFT, I imagine tensoring together the Fock spaces of the various elementary fields so that (e.g., in QED) one can express interaction terms like $\bar \psi \gamma_\mu A^\mu \psi$. But such Fock-like spaces are known to be incapable of accommodating nontrivial interacting QFTs (according to Haag's thm, etc). So perhaps you mean something else?

Last edited: Aug 22, 2016
7. Aug 22, 2016

### atyy

Nice article!

8. Aug 22, 2016

### MathematicalPhysicist

s.p.r? is that a usenet group?

9. Aug 22, 2016

### strangerep

sci.physics.research

10. Aug 22, 2016

### MathematicalPhysicist

Are researchers in physics still using these usenet groups?

I think that nowadays with stackexchange and PF that why would anyone still use those primitive forums.
I know that they still exist.

11. Aug 22, 2016

### Demystifier

Suppose that the Standard Model, as we know it, is the final theory of "everything". Since it is not supersymmetric, it must violate some assumptions of the Deligne's theorem. My question is: what these assumptions (violated by the Standard Model) are?

12. Aug 22, 2016

### MathematicalPhysicist

How can this possibly be if it doesn't include gravity? (do you refer to SM of particle physics?).

13. Aug 22, 2016

### unknown1111

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.

14. Aug 22, 2016

### john baez

No. Deligne's theorem says, very very roughly, that under certain conditions particles must have some super-group of symmetries. However, for the purposes of this theorem, an ordinary group counts as a special case of a super-group, namely one that has no transformations mixing fermions and bosons. So a non-supersymmetric theory, like the Standard Model, is allowed. Urs explained it this:

15. Aug 22, 2016

### Urs Schreiber

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.

16. Aug 22, 2016

### Urs Schreiber

The formatting here in the comment section tends to come out differently from what the people editing a comment expect. I think in the message by "Mathematical Physicist" above in fact everything except the last line is meant as a blockquote from a previous comment, the only line that "Mathematical Physicist" meant to add is

"s.p.r? is that a usenet group?"

17. Aug 22, 2016

### Urs Schreiber

No, the entry comments on this point in the paragraph starting with the sentence:

"Notice here that a super-group is understood to be a group that _may_ contain odd-graded components." But it need not.

A super-group is a group in super-geometry. It's underlying space may have add-graded coordinates, but it need not. In this terminology, an ordinary group is also a super-group, just one where the super-odd piece happens to be trivial.

It's all explained in the article, but since you missed it, I'll say it again: the theorem of course does not say that ordinary groups are ruled out, that would clearly be wrong. Instead the force of the theorem is to say that the largest class of admissible groups is that of super-groups (i.e. ordinary and possibly super groups), instead of, say, the even larger class of non-commutative groups or what not.

18. Aug 22, 2016

### Urs Schreiber

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.

19. Aug 22, 2016

### A. Neumaier

Nice article!

You referred to https://ncatlab.org/nlab/show/unitary+representation+of+the+Poincaré+group where you could delete the remark by John Baez.

Indeed, Wigner had classified all irreducible unitary representations of the Poincare group, including the unphysical ones. The physical ones are almost characterized by causality requirements, but to exclude zero mass continuous spin (which is causal but apparently not realized in Nature) they should rather be characterized by the requirement that one can create from them a free Wightman field theory.

20. Aug 22, 2016

### Urs Schreiber

Thanks, Arnold. That nLab entry is waiting for somebody to take a little care of it. It was started by John and/or people discussing with him long time back,but the editing was abandoned before a stable version was reached. Might you have 10 minutes to spare on this? It would be greatly appreciated! Just hit "edit" at the bottom of the entry. The syntax is simple and should be self-explanatory.