Reality conditions inducing transforms for quantum gauge field theory and quantum gravity

TL;DR

This paper introduces a canonical method to incorporate reality conditions in quantum gauge theories and quantum gravity, simplifying the Hamiltonian and constraints through a unique holomorphic representation derived via a Wick rotation transform.

Contribution

It provides a canonical algorithm for constructing the unique holomorphic representation that enforces reality conditions and simplifies the Hamiltonian in quantum gauge theories and gravity.

Findings

Derived a canonical algorithm for the holomorphic representation

Explicitly computed the representation for quantum gravity

Introduced a Wick rotation transform to implement reality conditions

Abstract

For various theories, in particular gauge field theories, the algebraic form of the Hamiltonian simplifies considerably if one writes it in terms of certain complex variables. Also general relativity when written in the new canonical variables introduced by Ashtekar belongs to that category, the Hamiltonian being replaced by the so-called scalar (or Wheeler-DeWitt) constraint. In order to ensure that one is dealing with the correct physical theory one has to impose certain reality conditions on the classical phase space which generally are algebraically quite complicated and render the task of finding an appropriate inner product into a difficult one. This article shows, for a general theory, that if we prescribe first a {\em canonical} complexification and second a $^*$ representation of the canonical commutation relations in which the real connection is diagonal, then there is only one choice of a holomorphic representation which incorporates the correct reality conditions {\em and} keeps the Hamiltonian (constraint) algebraically simple ! We derive a canonical algorithm to obtain this holomorphic representation and in particular explicitly compute it for quantum gravity in terms of a {\em Wick rotation transform}.