Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity
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
Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, CZ-74601 Opava, Czech Republic
Author to whom correspondence should be addressed.
Received: 18 June 2017 / Revised: 30 June 2017 / Accepted: 2 July 2017 / Published: 6 July 2017
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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
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
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
, which provides an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state
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
and its quantum conjugate momentum operator.
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Cremaschini, C.; Tessarotto, M. Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity. Entropy 2017, 19, 339.
Cremaschini C, Tessarotto M. Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity. Entropy. 2017; 19(7):339.
Cremaschini, Claudio; Tessarotto, Massimo. 2017. "Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity." Entropy 19, no. 7: 339.
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