Why Supersymmetry? Because of Deligne's theorem - Comments

[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!