Reps of the SUSY algebra: raising and lowering operators

[hyperkahler]

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, a^I is a rising operator and (a^I)^\dagger is a lowering operator for the helicity of massless states"

The definitions of a^I and (a^I)^\dagger are given a few lines above. How to see that the first raise helicity while the second lowers it?

 

[fzero]

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 $\bar{\sigma}^\mu=(1,-\sigma^i)$.

 

[arivero]

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, a^I is a rising operator and (a^I)^\dagger is a lowering operator for the helicity of massless states"

The definitions of a^I and (a^I)^\dagger are given a few lines above. How to see that the first raise helicity while the second lowers it?

Fix: 2^N-dimensional