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

^{1}

^{2}

^{3}

^{*}

## 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

## References

- De Donder, T. Théorie Invariantive Du Calcul des Variations; Gaultier-Villars & Cia.: Paris, France, 1930. [Google Scholar]
- Weyl, H. Geodesic Fields in the Calculus of Variation for Multiple Integrals. Ann. Math.
**1935**, 36, 607. [Google Scholar] [CrossRef] - Saunders, D.J. The Geometry of Jet Bundles; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Sardanashvily, G. Generalized Hamiltonian Formalism for Field Theory; World Scientific Publishing: Singapore, 1995. [Google Scholar]
- Echeverría-Enríquez, A.; Muñoz-Lecanda, M.C.; Román-Roy, N. Geometry of Lagrangian first-order classical field theories. Fortschr. Phys.
**1996**, 44, 235. [Google Scholar] [CrossRef] - Kanatchikov, I.V. Canonical structure of classical field theory in the polymomentum phase space. Rep. Math. Phys.
**1998**, 41, 49. [Google Scholar] [CrossRef] - Forger, M.; Paufler, C.; Romer, H. The Poisson bracket for Poisson forms in multisymplectic field theory. Rev. Math. Phys.
**2003**, 15, 705. [Google Scholar] [CrossRef] - Kisil, V.V. p-Mechanics as a physical theory: An introduction. J. Phys. A Math. Gen.
**2004**, 37, 183. [Google Scholar] [CrossRef] - Struckmeier, J.; Redelbach, A. Covariant Hamiltonian field theory. Int. J. Mod. Phys. E
**2008**, 17, 435. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Synchronous Lagrangian variational principles in General Relativity. Eur. Phys. J. Plus
**2015**, 130, 123. [Google Scholar] [CrossRef] - Dirac, P.A.M. Generalized Hamiltonian Dynamics. Can. J. Math.
**1950**, 2, 129. [Google Scholar] [CrossRef] - Sundermeyer, K. Constrained Dynamics; Lecture Notes in Physics; Springer: Berlin, Germany, 1982. [Google Scholar]
- Einstein, A. The Meaning of Relativity; Princeton University Press: Princeton, NJ, USA, 2004. [Google Scholar]
- Landau, L.D.; Lifschitz, E.M. Field Theory, Theoretical Physics; Addison-Wesley: New York, NY, USA, 1957; Volume 2. [Google Scholar]
- Misner, C.W.; Thorne, K.S.; Wheeler, J.A. Gravitation, 1st ed.; W.H. Freeman: Princeton, NJ, USA, 1973. [Google Scholar]
- Tessarotto, M.; Cremaschini, C. Theory of Nonlocal Point Transformations in General Relativity. Adv. Math. Phys.
**2016**, 2016, 9619326. [Google Scholar] [CrossRef] - Jordi, G.; Narciso, R.R. Multisymplectic unified formalism for Einstein-Hilbert gravity. J. Math. Phys.
**2018**, 59, 032502. [Google Scholar][Green Version] - Cremaschini, C.; Tessarotto, M. Manifest covariant Hamiltonian theory of General Relativity. Appl. Phys. Res.
**2016**, 8, 2. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Hamiltonian approach to GR—Part 1: Covariant theory of classical gravity. Eur. Phys. J. C
**2017**, 77, 329. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Hamiltonian approach to GR—Part 2: Covariant theory of quantum gravity. Eur. Phys. J. C
**2017**, 77, 330. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Quantum-wave equation and Heisenberg inequalities of covariant quantum gravity. Entropy
**2017**, 19, 339. [Google Scholar] [CrossRef] - Tessarotto, M.; Cremaschini, C. Generalized Lagrangian path approach to manifestly-covariant quantum gravity theory. Entropy
**2018**, 20, 205. [Google Scholar] [CrossRef] - Tessarotto, M.; Cremaschini, C. Role of quantum entropy and establishment of H-theorems in the presence of graviton sinks for manifestly-covariant quantum gravity. Entropy
**2019**, 21, 418. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Space-time second-quantization effects and the quantum origin of cosmological constant in covariant quantum gravity. Symmetry
**2018**, 10, 287. [Google Scholar] [CrossRef] - Arnowitt, R.; Deser, S.; Misner, C.W. Gravitation: An Introduction to Current Research; Witten, L., Ed.; Wiley: New York, NY, USA, 1962. [Google Scholar]
- Etienne, Z.B.; Liu, Y.T.; Shapiro, S.L. Relativistic magnetohydrodynamics in dynamical spacetimes: A new AMR implementation. Phys. Rev. D
**2010**, 82, 084031. [Google Scholar] [CrossRef] - Alcubierre, M. Introduction to 3+1 Numerical Relativity; Oxford University Press: Oxford, UK, 2008. [Google Scholar]
- Gheorghiu, T.; Vacaru, O.; Vacaru, S. Off-diagonal deformations of kerr black holes in Einstein and modified massive gravity and higher dimensions. Eur. Phys. J. C
**2014**, 74, 3152. [Google Scholar] [CrossRef] - Messiah, A. Quantum Mechanics; Dover Pubs: New York, NY, USA, 1999. [Google Scholar]
- Cremaschini, C.; Tessarotto, M. Quantum theory of extended particle dynamics in the presence of EM radiation-reaction. Eur. Phys. J. Plus
**2015**, 130, 166. [Google Scholar] [CrossRef] - Tessarotto, M.; Cremaschini, C. Generalized Lagrangian-path representation of non-relativistic quantum mechanics. Found. Phys.
**2016**, 46, 1022–1061. [Google Scholar] [CrossRef] - Goldstein, H. Classical Mechanics, 2nd ed.; Addison-Wesley: Reading, MA, USA, 1980. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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