Quantization of a Locally Supersymmetric Friedmann Model with Supermatter

Abstract

The general theory of N = 1 supergravity with supermatter is studied using a canonical approach. The supersymmetry and gauge constraint generators are found. The framework is applied to the study of a Friedmann minisuperspace model. We consider a Friedmann k = + 1 geometry and a family of spin-0 as well as spin-1 gauge fields together with their odd (anti-commuting) spin-1/2 partners. The quantum supersymmetry constraints give rise to a set of first-order coupled partial differential equations for the components of the wave function. As an intermediate stage in this project, we put both the spin-1 field and its fermionic partner equal to zero. The physical states of our simplified model correspond effectively to those of a mini-superspace quantum cosmological model possessing N=4 local supersymmetry coupled to complex scalars with spin-1/2 partners. The different supermatter models are given by specifying a K\"ahler metric for the scalars; the allowed quantum states then depend on the K\"ahler geometries. For the cases of spherically symmetric and flat K\"ahler geometries we find the general solution for the quantum state with a very simple form. However, although they allow a Hartle-Hawking state, they do not allow a wormhole state.