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Non-relativistic SUSY

  1. Jun 26, 2012 #1

    haushofer

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    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.
     
  2. jcsd
  3. Jun 26, 2012 #2

    haushofer

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    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?
     
    Last edited: Jun 26, 2012
  4. Jun 28, 2012 #3

    haushofer

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    Perhaps this topic fits better in another subforum?
     
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