Quantum Spin Dynamics (QSD) II

TL;DR

This paper advances the mathematical understanding of the Wheeler-DeWitt operator in quantum gravity by deriving its kernel, defining symmetric versions, and establishing self-adjoint extensions, thereby contributing to the foundational structure of quantum gravity.

Contribution

It provides a complete analysis of the Wheeler-DeWitt operator, including kernel derivation, inner product definition, and self-adjoint extension methods for Lorentzian quantum gravity.

Findings

Derived the complete kernel of the non-symmetric Wheeler-DeWitt operator

Defined a symmetric Wheeler-DeWitt operator and proved self-adjoint extensions

Analyzed the Wick rotation and its implications for quantum gravity

Abstract

We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then define a symmetric version of the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator. Finally we comment on the status of the Wick rotation transform in the light of the present results.