A gravitationally induced decoherence model using Ashtekar variables

TL;DR

This paper develops a relativistic model of gravitationally induced decoherence using Ashtekar variables, gauge invariance, and phase space quantization, providing a detailed derivation of the master equation with thermal gravitational effects.

Contribution

It introduces a gauge invariant formulation of gravitational decoherence with Ashtekar variables and extends the observable map with a dual map, broadening existing models.

Findings

Derived a relativistic decoherence model using Ashtekar variables.

Formulated the model at gauge invariant level with geometrical clocks.

Obtained a master equation involving thermal Wightman functions and residual time dependence.

Abstract

We consider the coupling of a scalar field to linearised gravity and derive a relativistic gravitationally induced decoherence model using Ashtekar variables. The model is formulated at the gauge invariant level using suitable geometrical clocks in the relational formalism, broadening existing gauge invariant formulations of decoherence models. For the construction of the Dirac observables we extend the known observable map by a kind of dual map where the role of clocks and constraints is interchanged. We also discuss a second choice of geometrical clocks existing in the ADM literature. Then we apply a reduced phase space quantisation on Fock space and derive the final master equation choosing a Gibbs state for the gravitational environment and using the projection operator technique. The resulting master equation is not automatically of Lindblad type, a starting point sometimes assumed for phenomenological models, but still involves a residual time dependence at the level of the effective operators in the master equation due to the form of the correlation functions that we express in terms of thermal Wightman functions. Furthermore, we discuss why in the model analysed here the application of a second Markov approximation in order to obtain a set of time independent effective system operators is less straightforward than in some of the quantum mechanical models.