Non-Relativistic SUSY: Group Theory Motivation

[haushofer]

Hi, I have a question about non-relativistic SUSY, see e.g. "non-relativistic SUSY" by Clark and Love.

The supersymmetric Galilei algebra with central extension M can easily be obtained from the N=1 Super Poincaré algebra by an Inonu-Wigner contraction. In this proces, SUSY and spacetime translations are decoupled! The characteristic commutator of rel. SUSY is schematically (using Weyl spinors)

$$\{ Q, \bar{Q} \} = P$$

This can be motivated by the fact that Q, being a Weyl spinor, is in the (1/2,0) rep. of the Lorentz algebra, and Q-bar is in the (0,1/2) rep. such that the commutator must be in the (1/2,1/2) rep. which is the vector representation. This lead you to use $P_{\mu}$ on the right hand side of the commutator.

Now, non-relativistically one obtains the commutator

$$\{ Q, \bar{Q} \} = M$$

with M being the central extension playing the role of mass, and Q only transforming under SO(3) rotations. SUSY becomes an "internal symmetry", and perhaps calling it "SUSY" is somewhat of a misnomer.

My question is: how can I again use a group-theoretical argument to motivate that this is what you expect, as in the rel. case? Instead of a vector one now seems to get a scalar on the RHS of the commutator, but I can't see how to motivate this.

 

[haushofer]

So, naively I would say that we are talking about spinors sitting in the fundamental rep. of SU(2), namely the $2$, and that

$$2 \otimes 2 = 1_A \oplus 3_S$$

The non.rel. SUSY anticommutator is the symmetric product, but then I don't get the singlet but the $3$! What is going wrong in my reasoning?

-edit: strictly speaking Q sits in the $2$ and $\bar{Q}$ in the $\bar{2}$, but I thought these two reps are identical. Perhaps here something subtle is going on?

 

[haushofer]

Perhaps this topic fits better in another subforum?