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Article

Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity

1
Institute of Physics and Research Center for Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava, CZ-74601 Opava, Czech Republic
2
Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy
3
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, CZ-74601 Opava, Czech Republic
*
Author to whom correspondence should be addressed.
Entropy 2017, 19(7), 339; https://doi.org/10.3390/e19070339
Received: 18 June 2017 / Revised: 30 June 2017 / Accepted: 2 July 2017 / Published: 6 July 2017
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
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 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 μ ν 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 μ ν and its quantum conjugate momentum operator. View Full-Text
Keywords: covariant quantum gravity; Hamilton–Jacobi quantization; quantum-wave equation; quantum hydrodynamic equations; Heisenberg inequalities covariant quantum gravity; Hamilton–Jacobi quantization; quantum-wave equation; quantum hydrodynamic equations; Heisenberg inequalities
MDPI and ACS Style

Cremaschini, C.; Tessarotto, M. Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity. Entropy 2017, 19, 339. https://doi.org/10.3390/e19070339

AMA Style

Cremaschini C, Tessarotto M. Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity. Entropy. 2017; 19(7):339. https://doi.org/10.3390/e19070339

Chicago/Turabian Style

Cremaschini, Claudio, and Massimo Tessarotto. 2017. "Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity" Entropy 19, no. 7: 339. https://doi.org/10.3390/e19070339

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