Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 307 / 1

The Coleman–Mandula theorem says that, under its assumptions on a nontrivial analytic relativistic S-matrix and the particle spectrum, every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra. Supersymmetry becomes possible by weakening the assumption that the symmetry algebra is an ordinary Lie algebra: a Z2-graded Lie superalgebra admits odd generators whose bracket is an anticommutator. The Haag–Łopuszański–Sohnius theorem then classifies the allowed extension and leads to the Super-Poincaré algebra.

An internal generator $T_a$ is a Lorentz scalar. The Coleman–Mandula theorem and the graded extension allow it to act nontrivially on supercharges only as an R-symmetry. Since R-symmetries are excluded,

Translation invariance of a conserved global supercharge and the graded Jacobi identities giveLorentz covariance requires the supercharges to transform as Weyl spinors,with the complex-conjugate dotted-spinor relation for ${\bar{Q}}_{\dot{\alpha }}$, up to the sign convention used for the action of generators.

The anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}$ transforms as a Lorentz vector because $(1/2,0)\otimes (0,1/2)=(1/2,1/2)$. The only translation generator with that transformation law is $P_{\mu }$, and a normalization of $Q$ fixesAn equal-chirality anticommutator could only contain an antisymmetric spinor contraction times a central charge, but the anticommutator is symmetric under exchange of the complete supercharges. For one supercharge and no central extension this forcesTogether with the stated Poincare brackets, these are the four-dimensional $\mathcal{N}=1$ Super-Poincare relations.

Apply parity to the mixed anticommutator. The transformed left-hand side isbecause $|{\eta }_P|=1$. The Pauli-matrix identity represented by this product leaves the temporal matrix unchanged and reverses the three spatial matrices, so it equalsBut a parity transformation acts on momentum as $P^0\mapsto P^0$ and $P^i\mapsto -P^i$. This is exactly the parity transform ofso $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$ is consistent with the stated transformation law. The arbitrary intrinsic phase cancels.

Fermion parity is defined byIt anticommutes with $Q_{\alpha }$ and ${\bar{Q}}_{\dot{\alpha }}$. At fixed nonzero energy and momentum, choose a supercharge combination $S$ for which $\{S,S^{†}\}=cI$ with $c>0$. Taking the finite-dimensional trace over one supermultiplet givesbecause cyclicity of the trace and anticommutation of $(-1{)}^F$ with $S$ make the two terms cancel. ThereforeThis is boson-fermion degeneracy in a supermultiplet.

If explicit or soft supersymmetry breaking terms are added, the supercharge is no longer a conserved symmetry of the full Hamiltonian and states need not form representations of the supersymmetry algebra at equal energy. The positive anticommutator cannot be replaced by a constant on a purported multiplet, so the supertrace proof and mass degeneracy fail.