Role of Quantum Entropy and Establishment of H-Theorems in the Presence of Graviton Sinks for Manifestly-Covariant Quantum Gravity.

Massimo Tessarotto,Claudio Cremaschini

doi:10.3390/e21040418

Abstract

Based on the introduction of a suitable quantum functional, identified here with the Boltzmann–Shannon entropy, entropic properties of the quantum gravitational field are investigated in the framework of manifestly-covariant quantum gravity theory. In particular, focus is given to gravitational quantum states in a background de Sitter space-time, with the addition of possible quantum non-unitarity effects modeled in terms of an effective quantum graviton sink localized near the de Sitter event horizon. The theory of manifestly-covariant quantum gravity developed accordingly is shown to retain its emergent-gravity features, which are recovered when the generalized-Lagrangian-path formalism is adopted, yielding a stochastic trajectory-based representation of the quantum wave equation. This permits the analytic determination of the quantum probability density function associated with the quantum gravity state, represented in terms of a generally dynamically-evolving shifted Gaussian function. As an application, the study of the entropic properties of quantum gravity is developed and the conditions for the existence of a local H-theorem or, alternatively, of a constant H-theorem are established.

Highlights

The understanding of the abstract, i.e., geometrical, structure of space-time poses challenging mathematical questions that have to be established on rigorous grounds
CQG-theory is characterized by the fact that all the dynamical variables and quantum operators that characterize the theory are expressed in terms of continuum field variables that behave as four-tensors with respect to the local point transformations (LPT)-group (see Equation (4)), i.e., the group of point transformations which preserve the differential manifold structure of space-time
The entropic properties of the gravitational field, and more generally the thermodynamic ones, investigated in previous literature have been traditionally associated with the mathematical properties of classical black hole solutions, i.e., in the framework of general relativity

Summary

Introduction

The understanding of the abstract, i.e., geometrical, structure of space-time poses challenging mathematical questions that have to be established on rigorous grounds. CQG-theory is characterized by the fact that all the dynamical variables and quantum operators that characterize the theory are expressed in terms of continuum field variables that behave as four-tensors with respect to the LPT-group (see Equation (4)), i.e., the group of point transformations which preserve the differential manifold structure of space-time. Given these premises, the present paper is part of a research effort about the theoretical foundations and principles of classical and quantum gravity. Sufficiently close to the de Sitter event horizon, if localized quantum sinks are present, the BS entropy generally exhibits an irreversible proper-time monotonically-increasing behavior as corresponds to the occurrence of a strictly-positive entropy production rate

Historical Setting and Background

Covariant Quantum Gravity Theory

GLP Representation of Non-Unitary CQG-Theory

CQG Boltzmann–Shannon Entropy

Entropic Properties and the Search of H-Theorems for Quantum Gravity

Conclusions