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Reps of the SUSY algebra: raising and lowering operators

  1. Jun 3, 2015 #1
    I'm reading Alvarez-Gaume review on Seiberg-Witten theory: http://arxiv.org/abs/hep-th/9701069.
    Around page 23 you can find the following claim:

    "This is a Clifford algebra with 2N generators and has a 2N-dimensional representation. From the point of view of the angular momentum algebra, [itex]a^I[/itex] is a rising operator and [itex](a^I)^\dagger[/itex] is a lowering operator for the helicity of massless states"

    The definitions of [itex]a^I[/itex] and [itex](a^I)^\dagger[/itex] are given a few lines above. How to see that the first raise helicity while the second lowers it?
  2. jcsd
  3. Jun 3, 2015 #2


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    It's just because of the convention that the spinors with the ##\alpha## index are taken to be of positive helicity, while those with the ##\dot{\alpha}## have negative helicity. You can probably show this explicitly from the relation

    $$[M_{\mu\nu},\bar{Q}_{\dot{\alpha}}] = \frac{1}{2} {(\bar{\sigma}_{\mu\nu})_{\dot{\alpha}}}^{\dot{\beta}} \bar{Q}_{\dot{\beta}}.$$

    Apply this to ##J_3\propto M_{12}## and follow the minus sign in the definition of ##\bar{\sigma}^\mu=(1,-\sigma^i)##.
  4. Feb 15, 2017 #3


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    Fix: 2N-dimensional
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