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arXiv · 1707.02399

Quantum-wave equation and Heisenberg inequalities of covariant quantum gravity

Claudio Cremaschini·Massimo Tessarotto
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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.

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Claudio Cremaschini, Massimo Tessarotto. 2017-07-08. Quantum-wave equation and Heisenberg inequalities of covariant quantum gravity. https://doi.org/10.3390/e19070339

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