Super-Poincaré algebra and representation multiplets

Throughout the four-dimensional lessons use $\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$, $\sigma^\mu=(I,\boldsymbol\sigma)$, and $\bar\sigma^\mu=(I,-\boldsymbol\sigma)$. Undotted indices label left-handed Weyl spinors and dotted indices their conjugates. Choose $\epsilon^{12}=1$ and $\epsilon_{12}=-1$, with identical dotted conventions, raising by $\psi^\alpha=\epsilon^{\alpha\beta}\psi_\beta$. Contract left spinors as $\psi\chi=\psi^\alpha\chi_\alpha$ and right spinors as $\bar\psi\bar\chi=\bar\psi_{\dot\alpha}\bar\chi^{\dot\alpha}$. These definitions, rather than remembered component signs, determine every contraction below.

Classical fermionic fields and supersymmetry parameters are Grassmann odd. Exchanging two such objects introduces a minus sign, and Hermitian conjugation reverses product order. With our antisymmetric index contraction, the contracted bilinear obeys $\psi\chi=\chi\psi$ even though the uncontracted components anticommute. Supercharges instead act as odd Hilbert-space operators. Their graded brackets are anticommutators with one another and ordinary commutators with momentum. The adjoint of Q is the corresponding dotted charge, and all norm-positivity arguments assume a physical positive-inner-product Hilbert space.

For a state of positive mass M in its rest frame, the mixed anticommutator becomes twice M times the identity spinor matrix. Divide the two independent charges by square root of twice M to obtain normalized fermionic annihilation operators. Acting with their adjoints on a Clifford vacuum of spin j gives occupation numbers zero or one for each mode. The total state count is four times the vacuum spin degeneracy. This is a representation statement at fixed momentum, not a count of independent off-shell field components.

For a spin-zero Clifford vacuum, the zero- and two-occupation levels are scalar states, while the one-occupation level is a spin-one-half doublet. There are two bosonic and two fermionic states. In a massless frame one charge combination instead has zero norm and acts trivially, leaving a smaller helicity construction. CPT completion can require adding the conjugate helicities or charge sector. Consequently, a minimal helicity ladder and a complete physical chiral or vector multiplet should not be confused when comparing tables of state counts.

Extended supersymmetry can contain a central charge. For N equals two choose $\{Q^I_\alpha,Q^J_\beta\}=2\epsilon_{\alpha\beta}\epsilon^{IJ}Z$, while the mixed bracket remains twice momentum times delta I J. The internal antisymmetric symbol also has $\epsilon^{12}=1$. A phase rotation makes the relevant central charge real and nonnegative for diagonalizing the rest-frame norm matrix. Its eigenvalues are $2(M+|Z|)$ and $2(M-|Z|)$, each with multiplicity two. Positivity therefore requires M at least the magnitude of Z in this declared normalization. Different central-charge normalizations change the displayed numerical bound.

At saturation, half the charge combinations have zero norm and annihilate the state. The resulting BPS representation is shorter than a generic massive representation. For spin-zero Clifford vacuum its size falls from sixteen states to four, before any additional CPT completion. Shortening protects specific mass-charge relations under appropriate dynamical assumptions, but does not make every state in an interacting theory BPS. Nor does representation theory establish that a proposed charged state actually exists or remains stable across all parameter regions. Those are further questions about the theory's spectrum.