non-relativistic SUSY

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)

[tex]
\{ Q, \bar{Q} \} = P
[/tex]

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 [itex]P_{\mu}[/itex] on the right hand side of the commutator.

Now, non-relativistically one obtains the commutator

[tex]
\{ Q, \bar{Q} \} = M
[/tex]

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 [itex]2[/itex], and that

[tex]
2 \otimes 2 = 1_A \oplus 3_S
[/tex]

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

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

haushofer

Perhaps this topic fits better in another subforum?