# Polchinski "String theory" (B.2.8)

The $$\mathcal N = 1$$ algebra $$$$\{Q_\alpha,\overline Q_\beta\} = -2P_\mu\Gamma^\mu_{\alpha\beta}$$$$ is in a frame where $$k_1 = k_0$$ $$$$\{Q_\alpha, Q_\beta^\dagger\} = 2k^0(1 + 2S_0)_{\alpha\beta}$$$$ or in $$\bf s$$ basis, $$$$\{Q_{s_0',s_1'},Q^\dagger_{s_0,s_1}\} = 4k^0\delta_{s_0,1/2}\delta_{\bf s'\bf s}.$$$$

The matrix elements of $$Q_{-1/2,s_1}$$ vanish. The remaining operator gives the raising and lowering operator: $$$$b = (4k^0)^{-1/2}Q_{1/2,-1/2},\;\; b^\dagger = (4k^0)^{-1/2}Q_{1/2,1/2},\;\; \{b,b^\dagger\} = 1,\;\; b^2 = (b^\dagger)^2 = 0$$$$

We construct the representation by starting from $$|\lambda\rangle$$ {\it s.t.}, $$$$S_1|\lambda\rangle = \lambda|\lambda\rangle,\;\; b|\lambda\rangle = 0.$$$$ By operating $$b^\dagger$$, $$$$b^\dagger|\lambda\rangle = \Big|\lambda + \frac12\Big\rangle,\;\; S_1\Big|\lambda + \frac12\Big\rangle = \Big(\lambda + \frac12\Big) \Big|\lambda + \frac12\Big\rangle.$$$$

My question is why the state $$b^\dagger|\lambda\rangle$$ is $$|\lambda + 1/2\rangle$$?

P.S. I think that in order to confirm it we need to calculate the commutator $$[b^\dagger,S_1]$$ and it is equal to $$1/2$$.

You need to use the super Poncaré algebra, that besides the commutator $$[q,q]\sim p$$ there is also the commutators $$[M,p]\sim p$$, $$[M,q]\sim q$$, and $$[M,M]\sim M$$. Here $$q$$ are the supercharges, $$p$$ are the momentum components and $$M$$ the Lorentz generators.

Turns out that using Jacobi identity

$$[M,[q,q]]=[[M,q],q]+[q,[M,q]]$$

and that $$[q,q]\sim p$$ we learn that $$q$$ must be a spinor, so it has charge $$\pm \frac{1}{2}$$ under $$S_0, \dots, S_{\frac{d}{2}}$$ Cartan subalgebra of the Lorentz generators

$$[S,q]\sim \frac{1}{2} q$$

I did not put indices up to here since all I said is valid to any dimension and notation. For your case where we have $$d=1+3$$, we can work with the $$SL(2,\mathbb{C})$$ notation

$$\alpha=(1\equiv ++,\,2\equiv --)\qquad \dot\alpha=(\dot 1\equiv +-,\,\dot 2\equiv -+)$$

where the first sign is the charge under $$S_0$$ and the second is the charge under $$S_1$$. The momentum is given by $$p_{\alpha\dot\alpha}$$ and the supercharges are $$q_{\alpha}$$ and $$\bar q_{\dot\alpha}$$ such that

$$[q_{\alpha},\bar q_{\dot\alpha}]=2 p_{\alpha\dot\alpha}$$

and we have

$$[S_0,q_{\pm \,\star}] = \pm\frac{1}{2} q_{\pm \,\star},\qquad [S_1,q_{\star\, \pm}] = \pm\frac{1}{2} q_{\star \,\pm}$$

that follows from the Jacobi identity of

$$[S_i,2p_{\alpha\dot\alpha}]=[S_{i}, [q_{\alpha},\bar q_{\dot\alpha}]]=[[S_i,q_{\alpha}],\bar q_{\dot\alpha}]+[q_{\alpha},[S_i,\bar q_{\dot\alpha}]]$$