Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity

Abstract

Key aspects of the manifestly-covariant theory of quantum gravity Tessarotto 2015-2017) are investigated. These refer, first, to the establishment of the four-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG) wave equation, which advances the quantum state ψ 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 ≡ g µν 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 ψ, which provides an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state ψ is proven 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 metric tensor g ≡ g µν and its quantum conjugate momentum operator.

Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory

“…However, just as in the case of the Schroedinger equation (see related discussion in [22]), a basic difficulty of such an interpretations lies in the uniqueness feature, and consequently the intrinsic deterministic character, of each LP. Such a property, in fact, appears potentially in contradiction with the notion of quantum measurement holding in the context of CQG-theory and the validity of Heisenberg inequalities [11].…”

“…Elementary algebra [10,11] then shows that, based on the quantum-wave Equation (16), the same quantum fluid fields necessarily fulfill the corresponding set of Eulerian CQG-quantum hydrodynamic equations. In the Eulerian representation, upon identifying again d ds with the total covariant s−derivative operator in Equation (14), these are realized respectively by the continuity and quantum Hamilton-Jacobi equations:…”

Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity

“…The theory of manifestly-covariant quantum gravity (CQG-theory) recently proposed in a series of papers (see Refs. [1][2][3][4][5][6]) provides a possible new self-consistent route to Quantum Gravity and the cosmological interpretation of quantum vacuum. This refers specifically to the quantum prescription of the cosmological constant and the longstanding question whether or not it can be ascribed exclusively to suitable vacuum fluctuations arising at the quantum level.…”

“…CQG-theory realizes, as such, a first-quantization picture of space-time which embodies simultaneously all the required fundamental principles (for an extended related discussion we refer again to Refs. [1][2][3][4][5][6]). One of its notable features is realized, first of all, by the distinction between a continuum classical background metric tensor g ≡ { g µν }, yielding the geometric properties of the space-time, and the quantum gravitational field g µν which dynamically evolves over g according to a defined quantum wave equation (CQG-wave equation) [4].…”

“…• The statistical interpretation of the CQG-wave equation in terms of corresponding quantum hydrodynamic equations [5].…”