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quick entry Haag–Lopuszanski–Sohnius theorem
Would anyone have a pdf-copy of “All possible generators of supersymmetries of the S-matrix” for me? I am currently not in a network through which I’d have subscription to the journal.
My library only gives access post-1995.
Ah, I see, maybe that’s the same reason for me.
I’d be happy already to see a review that goes into the details of what exactly is assumed and what is shown. This article has an immense ratio of number of citations over number of citations that are more than a hat tip.
I can check.
Well, it seems the Adelaide Uni library does have access ;-)
Thanks David!
So I was trying to sort out the following:
Consider super Lie alegbras whose bosonic part is precisely the ordinary Poincaré algebra. Add a direct summand vector space $S$ in odd degree. Then what is the most general possible extension of a bracket to a superbracket?
It is easy to see that a sufficient condition is that $S$ is a $\mathfrak{so}$-rep that is equipped with a $\mathfrak{so}$-equivariant symmetric bilinear pairing to the translation subalgebra.
What is more subtle to see is that or whether this is necessary, i.e. to exclude the case that the pairing on $S$ lands possibly also in $\mathfrak{so}$.
In Haag–Lopuszanski–Sohnius this case is ruled out by adding further assumptions, motivated from the behaviour of S-matrices for field theories with mass gap. One such assumption is that $P_a P^a$ is a Casimir operator still for the super-extension (i.e. an element in the center of the universal enveloping algebra). They have further assumptions on top of that.
On the other hand, they allow further central bosonic elements. I am interested here in the case where the bosonic algebra is strictly ordinary Poincaré. And I would like a purely mathematical statement, not falling back to input from scattering theory.
What may be said?
I have forwarded this question to MO, here.
Discussing the issue with John Huerta, we re-discovered/re-remembered an example of a super-extension of the Poincaré Lie algebra which is not of the standard super-Poincaré algebra form. I have added this example at geometry of physics – supersymmetry here and also at supersymmetry here.
Fixed some typos.
So you think this exotic extension isn’t physically relevant?
Fixed some typos.
Thanks! I have copied the fixes over also to the big entry.
So you think this exotic extension isn’t physically relevant?
The question was to which extent the concept of super-extensions of ordinary symmetry algebras is sufficient to single out Poincaré-super-algebras, hence spinorial “supersymmetry” as it is understood. We see that it is not sufficient. Hence the traditional approach cannot derive its conclusion from first principles.
Good, so your approach comes to the rescue.
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