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

TL;DR

This work introduces a canonical, broadly applicable algorithm to construct holomorphic (complex) representations for constrained quantum theories by implementing the correct reality conditions through a canonical complexification and a Wick rotation generated by a Complexifier $C$. Central to the approach is the generalized coherent state transform, realized as $ ext{U}_t= ext{K} ext{W}_t$, which unitarily maps the real representation to a holomorphic one in which the constraints are algebraically simpler, and an isometry is established via a carefully defined measure $ u_t$. The paper provides explicit formulations for type 1, type 2 gauge field transforms and, crucially, derives the gravity-specific Wick rotation with $A^{ m C}=oldsymbol{ m \Gamma}-iK$ and $P^{ m C}=iP$, generated by $C= extstylerac{1}{ m ext{ } ext} ilde{C}=rac{1}{ ext{k}}\,tint_ extSigma K_a^i P^a_i$, leading to a polynomial, Euclidean-like $H_{ m C}$ and a route to solving the theory non-perturbatively. The framework links the complexified theory to a real-variables formulation and outlines strategies to extract physical spectra, with insights into regulator choices and potential connections to Euclidean gravity and topological constructs such as Chern-Simons theory. Together, these results aim to render non-perturbative canonical quantum gravity more tractable by providing a rigorous, unitary bridge between real and holomorphic quantum representations while implementing reality conditions transparently.

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}.