Quantum-wave equation and Heisenberg inequalities of covariant quantum gravity

Abstract

Key aspects of the manifestly-covariant theory of quantum gravity (Cremaschini and Tessarotto 2015-2017) are investigated. These refer, first, to the establishment of the 4-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG) wave equation, which advances the quantum state $\psi$ associated with a prescribed background space-time. In this paper, the CQG-wave equation is proved to follow at once by means of a Hamilton-Jacobi quantization of the classical variational tensor field $g\equiv \left\{g_{\mu \nu }\right\}$ and its conjugate momentum, referred to as (canonical) $g-$quantization. The same equation is also shown to be variational and to follow from a synchronous variational principle identified here with the quantum Hamilton variational principle. The corresponding quantum hydrodynamic equations are then obtained upon introducing the Madelung representation for $\psi$, which provide an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state $\psi$ is proved to fulfill generalized Heisenberg inequalities, relating the statistical measurement errors of quantum observables. These are shown to be represented in terms of the standard deviations of the matric tensor $g\equiv \left\{g_{\mu \nu }\right\}$ and its quantum conjugate momentum operator.

arXiv commentEntropy Journal Special Issue: Advances in Relativistic Statistical Mechanics

References in corpus (16)

• Canonical Structure of Classical Field Theory in the Polymomentum Phase Space
• Relativistic magnetohydrodynamics in dynamical spacetimes: A new AMR implementation
• Exact Solutions with Noncommutative Symmetries in Einstein and Gauge Gravity
• The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory
• Covariant Hamiltonian Field Theory
• Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries
• Off-Diagonal Deformations of Kerr Black Holes in Einstein and Modified Massive Gravity and Higher Dimensions
• Hamiltonian approach to GR - Part 1: covariant theory of classical gravity
• Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity
• Synchronous Lagrangian variational principles in General Relativity
• Hamiltonian formulation for the classical EM radiation-reaction problem: application to the kinetic theory for relativistic collisionless plasmas
• On Relativistic Generalization of Perelman's W-entropy and Statistical Thermodynamic Description of Gravitational Fields
• Exact solution of the EM radiation-reaction problem for classical finite-size and Lorentzian charged particles
• Manifest Covariant Hamiltonian Theory of General Relativity
• p-Mechanics as a Physical Theory. An Introduction
• Inverse kinetic theory for quantum hydrodynamic equations

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