Supersymmetric field equations from momentum space

TL;DR

This paper develops a geometric and Fourier-analytic framework to derive supersymmetric field equations from super Poincaré group representations, connecting them to known Wess-Zumino equations.

Contribution

It introduces a super Fourier transform approach to realize super Poincaré representations as constrained superfunctions, deriving supersymmetric equations in ordinary fields.

Findings

Realized super Poincaré representations via super Fourier transform.

Derived supersymmetric equations in non-Grassmannian fields.

Connected these equations to Wess-Zumino superfield equations.

Abstract

It is known that every irreducible unitary representation of positive energy of the Poincar\'e group can be realized as a subspace of tensor fields on Minkowski spacetime subjected to suitable partial differential equations. We first describe geometrically the general mechanism that produces, via Fourier transform, the invariant differential operators corresponding to those representations. Then, using a super-version of the Fourier transform, we show explicitly how a massive irreducible unitary representation of the super Poincar\'e group in dimension $(4|4)$ can be realized as a linear sub-supermanifold of suitably constrained superfunctions. In this way, we obtain supersymmetric equations in terms of ordinary (non-Grassmannian) fields. Finally, using the functor of points, we show how our equations can be related in a natural way to the Wess-Zumino equations for massive chiral superfields.