Physical States in Canonically Quantized Supergravity

TL;DR

This paper investigates the canonical quantization of N=1 supergravity, demonstrating that physical states likely involve infinite Grassmann numbers, with no finite-order fermionic solutions existing.

Contribution

It shows that solutions with finite fermion order do not exist in supergravity, implying physical states have infinite Grassmann number, confirmed through analysis of the gravitino field.

Findings

Finite fermion order states are absent in supergravity.

Physical states involve infinite Grassmann numbers.

Confirmed with free spin-3/2 field analysis.

Abstract

We discuss the canonical quantization of $N=1$ supergravity in the functional Schrodinger representation. Although the form of the supersymmetry constraints suggests that there are solutions of definite order $n$ in the fermion fields, we show that there are no such states for any finite $n$. For $n=0$, a simple scaling argument definitively excludes the purely bosonic states discussed by D'Eath. For $n>0$, the argument is based on a mode expansion of the gravitino field on the quantization 3-surface. It is thus suggested that physical states in supergravity have infinite Grassmann number. This is confirmed for the free spin-3/2 field, for which we find that states satisfying the gauge constraints contain an infinite product of fermion mode operators.