= Solution
Use $\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$, $\sigma^\mu=(I,\boldsymbol\sigma)$ and $\bar\sigma^\mu=(I,-\boldsymbol\sigma)$, where $\boldsymbol\sigma$ are the <Pauli matrices>. The translation and <Lorentz generators> obey the <Poincare algebra>
$$
\begin{aligned}
[P_\mu,P_\nu]&=0,\\
[M_{\mu\nu},P_\rho]&=i(\eta_{\nu\rho}P_\mu-\eta_{\mu\rho}P_\nu),\\
[M_{\mu\nu},M_{\rho\sigma}]&=i(\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\nu\sigma}M_{\mu\rho}+\eta_{\mu\sigma}M_{\nu\rho}).
\end{aligned}
$$
One can derive the additional relations by realizing the <supercharges> as differential operators on <superspace>. With <left Grassmann derivatives> and $P_\mu=i\partial_\mu$, take
$$
Q_\alpha=\partial_{\theta^\alpha}-i\sigma^\mu_{\alpha\dot\beta}\bar\theta^{\dot\beta}\partial_\mu,
\qquad
\bar Q_{\dot\beta}=-\partial_{\bar\theta^{\dot\beta}}+i\theta^\alpha\sigma^\mu_{\alpha\dot\beta}\partial_\mu.
$$
Using $\{\partial_{\theta^\alpha},\theta^\beta\}=\delta_\alpha{}^\beta$, the two mixed derivative-coordinate terms each give $i\sigma^\mu_{\alpha\dot\beta}\partial_\mu$. The remaining terms cancel by Grassmann anticommutation. Equal-chirality terms cancel in the same way. Hence
$$
\boxed{\{Q_\alpha,\bar Q_{\dot\beta}\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu,\qquad\{Q_\alpha,Q_\beta\}=\{\bar Q_{\dot\alpha},\bar Q_{\dot\beta}\}=0.}
$$
All coefficients are independent of $x$, so direct differentiation gives $[P_\mu,Q_\alpha]=[P_\mu,\bar Q_{\dot\alpha}]=0$. This also shows that <supercharges commute with translations> and preserve the four-momentum of a state.
For the <Lorentz generators>, define the left-handed spinor matrices
$$
S_{\mu\nu}=\frac{i}{4}(\sigma_\mu\bar\sigma_\nu-\sigma_\nu\bar\sigma_\mu).
$$
The identity $\sigma^\mu\bar\sigma^\nu+\sigma^\nu\bar\sigma^\mu=2\eta^{\mu\nu}I$ gives their Lorentz commutation relations. Acting on the spinor superspace coordinates and their dual derivatives then gives, in a column-component convention,
$$
\boxed{[M_{\mu\nu},Q_\alpha]=-(S_{\mu\nu})_\alpha{}^\beta Q_\beta,\qquad
[M_{\mu\nu},\bar Q_{\dot\alpha}]=(S_{\mu\nu}^{*})_{\dot\alpha}{}^{\dot\beta}\bar Q_{\dot\beta}.}
$$
Here dotted index values are identified with the conjugate undotted values, and $\bar Q_{\dot\alpha}=Q_\alpha^\dagger$. The second formula also follows directly by taking the adjoint of the first, using Hermitian $M_{\mu\nu}$. For example, $S_{12}=\sigma^3/2$, so $[J_3,Q_1]=-Q_1/2$ and $[J_3,Q_2]=Q_2/2$ for $J_3=M_{12}$. This explicit component convention avoids reversing signs by mixing state matrices with operator-component matrices. Covariance of the mixed <anticommutator> follows by applying these matrices to $\sigma^\rho P_\rho$.
These are the minimal <Super-Poincaré algebra> relations. In the ordinary particle algebra a possible scalar term in $\{Q_\alpha,Q_\beta\}$ would be proportional to $\epsilon_{\alpha\beta}$, but an anticommutator is symmetric in its two labels; with only one supersymmetry such a central term vanishes. Extended algebras can instead use an antisymmetric internal index tensor. Tensor charges associated with extended objects are not part of this minimal particle algebra.
For a positive-energy <massless supermultiplet>, choose $p^\mu=(E,0,0,E)$ with $E>0$, so $p_\mu=(E,0,0,-E)$ and
$$
\{Q_\alpha,Q_\beta^\dagger\}=2\sigma^\mu_{\alpha\dot\beta}p_\mu
=\begin{pmatrix}0&0\\0&4E\end{pmatrix}.
$$
Positivity of the Hilbert-space norm implies that both $Q_1$ and $Q_1^\dagger$ annihilate every state: their squared norms sum to zero. Set
$$
a=\frac{Q_2}{\sqrt{4E}},\qquad a^\dagger=\frac{Q_2^\dagger}{\sqrt{4E}},\qquad
\{a,a^\dagger\}=1,\quad a^2=(a^\dagger)^2=0.
$$
For momentum along the positive third axis, <helicity> is $h=J_3$. The Lorentz commutators give $[h,a]=a/2$ and $[h,a^\dagger]=-a^\dagger/2$. Choose a highest-helicity <Clifford vacuum> $|\lambda\rangle$ with $a|\lambda\rangle=0$. Its partner $a^\dagger|\lambda\rangle$ has unit norm if the first state does and has helicity $\lambda-1/2$. No further state is generated because $(a^\dagger)^2=0$. Therefore
$$
\boxed{\text{an irreducible finite-helicity massless }\mathcal N=1\text{ multiplet has helicities }(\lambda,\lambda-\tfrac12).}
$$
The two states have opposite <fermion parity>. A <CPT completion of a supermultiplet> adds helicities $(-\lambda,-\lambda+1/2)$ and conjugate internal charges when necessary. Taking $\lambda=1/2$ produces a complex scalar and a <Weyl spinor> after CPT completion; taking $\lambda=1$ produces the massless <vector multiplet>. This construction assumes ordinary finite-helicity particle representations, rather than continuous-spin representations.
Back to article page
