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Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity

1
Institute of Physics and Research Center for Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám.13, 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, Bezručovo nám.13, CZ-74601 Opava, Czech Republic
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(4), 592; https://doi.org/10.3390/sym11040592
Received: 17 February 2019 / Revised: 18 April 2019 / Accepted: 19 April 2019 / Published: 24 April 2019
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Abstract

The axiomatic geometric structure which lays at the basis of Covariant Classical and Quantum Gravity Theory is investigated. This refers specifically to fundamental aspects of the manifestly-covariant Hamiltonian representation of General Relativity which has recently been developed in the framework of a synchronous deDonder–Weyl variational formulation (2015–2019). In such a setting, the canonical variables defining the canonical state acquire different tensorial orders, with the momentum conjugate to the field variable g μ ν being realized by the third-order 4-tensor Π μ ν α . It is shown that this generates a corresponding Hamilton–Jacobi theory in which the Hamilton principal function is a 4-tensor S α . However, in order to express the Hamilton equations as evolution equations and apply standard quantization methods, the canonical variables must have the same tensorial dimension. This can be achieved by projection of the canonical momentum field along prescribed tensorial directions associated with geodesic trajectories defined with respect to the background space-time for either classical test particles or raylights. It is proved that this permits to recover a Hamilton principal function in the appropriate form of 4-scalar type. The corresponding Hamilton–Jacobi wave theory is studied and implications for the manifestly-covariant quantum gravity theory are discussed. This concerns in particular the possibility of achieving at quantum level physical solutions describing massive or massless quanta of the gravitational field. View Full-Text
Keywords: covariant quantum gravity; Hamilton equations; Hamilton–Jacobi theory; wave theory; massive/massless gravitons covariant quantum gravity; Hamilton equations; Hamilton–Jacobi theory; wave theory; massive/massless gravitons
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Cremaschini, C.; Tessarotto, M. Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity. Symmetry 2019, 11, 592.

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