- #1

haushofer

Science Advisor

- 2,414

- 817

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.