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

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## Abstract

**:**

## 1. Introduction

- manifestly covariant, i.e., it can be set in 4-tensor form;
- unconstrained, i.e., the same Hamiltonian system can be expressed in terms of an independent set of canonical variables;
- variational. Indeed, as typical of classical Hamiltonian systems occurring in classical mechanics, one can show that also the new abstract Hamiltonian system can be determined via a suitable path-integral variational principle.

- The space-time transformation properties with respect to the group of local point transformations, i.e., coordinate transformations, and the consistency of current realizations adopted for classical and quantum gravity theories with respect to the principle of manifest covariance [13,14,15,16,17]. The issue pertains both to the identification of the classical Hamiltonian and Hamilton–Jacobi structures of General Relativity, as well as to a corresponding prescription of the physical postulates at the basis of a quantum mechanical description of space-time and canonical quantization in terms of continuum or discrete space-time configurations [18,19,20].
- The symmetry properties of space-time related to the emergent gravity phenomenon, whereby certain physical observables/characteristics of classical General Relativity follow from quantum gravity theory. These concern both the prescription of the local-coordinate value of the space-time metric tensor, via a suitable quantum expectation value, as well as the establishment of the very functional form of the General Relativity field equations [22,24].

## 2. Extended Hamiltonian Formulation

**Property**

**1.**

**Proof.**

**Property**

**2.**

**Proof.**

## 3. Canonical Transformations and Extended Hamilton–Jacobi Theory

## 4. Search of a Reduced Hamiltonian Theory

## 5. Reduced Hamilton–Jacobi Theory

## 6. The Projection Operator

## 7. Hamilton–Jacobi Waves

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Extended Canonical Equation in 4-Vector Poisson Brackets Notation

## Appendix B. Admissible Form of Hamilton Principal Function

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Cremaschini, C.; Tessarotto, M.
Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity. *Symmetry* **2019**, *11*, 592.
https://doi.org/10.3390/sym11040592

**AMA Style**

Cremaschini C, Tessarotto M.
Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity. *Symmetry*. 2019; 11(4):592.
https://doi.org/10.3390/sym11040592

**Chicago/Turabian Style**

Cremaschini, Claudio, and Massimo Tessarotto.
2019. "Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity" *Symmetry* 11, no. 4: 592.
https://doi.org/10.3390/sym11040592