Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 54 / 2 / ii / Solution

Sum the diagonal mixed anticommutators in the Super-Poincaré algebra. Their spatial terms have zero Pauli trace, givingFor every normalized state $|u⟩$ in the common operator domain,This proves energy positivity in global supersymmetry. The zero of energy is fixed by this algebra, rather than by an arbitrary independent subtraction.

A supersymmetric vacuum is invariant under every supercharge and its adjoint. Its energy therefore vanishes. Conversely, a normalized vacuum of zero energy makes every nonnegative norm on the right zero, so all supercharges annihilate it. ConsequentlyFor an infinite spatial volume the corresponding statement is about vacuum energy density, obtainable by first regulating the volume. It concerns rigid supersymmetry; the gravitational scalar potential in supergravity has additional terms and is not generally nonnegative.