# Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Goal #1: Explicit solutions of the CQG-quantum hydrodynamic equations satisfying suitable physical requirements.
- Goal #2: The “emergent” character of the classical background space-time metric tensor $\widehat{g}\left(r\right)$, to be determined in terms of quantum theory. Accordingly, the background metric tensor $\widehat{g}\left(r\right)$ should be identified with a suitably-defined quantum expectation value of the quantum state, i.e., weighted in terms of the corresponding quantum probability density (PDF).
- Goal #3: The existence of either stationary or, more generally, non-stationary solutions with respect to the proper-time s, i.e., explicitly dependent on s, for the quantum state $\psi $ expressed via the Madelung representation (see Equation (7)).
- Goal #4: The search of Gaussian-like or Gaussian realizations for the quantum PDF $\rho $.
- Goal #5: The search of separable solutions of the quantum Hamilton-Jacobi (H-J) equation in terms of the quantum phase-function ${S}^{\left(q\right)}$ and the investigation of their qualitative properties and in particular their asymptotic behavior for $s\to +\infty $.

- Requirement #1: the quantum wave-function $\psi \left(s\right)$ is dynamically consistent, namely for which the PDF $\rho (g,r,s)\equiv {\left|\psi (g,r,s)\right|}^{2}$ associated with the quantum wave-function $\psi (g,r,s)$ is globally prescribed and summable in the quantum configuration space ${U}_{g}$ in such a way that the corresponding probability ${\left|\psi \right|}^{2}d\left(g\right)$ is similarly globally conserved for arbitrary subsets of the quantum configuration space ${U}_{g}$. As discussed below a prerequisite for meeting such a requirement is the validity of suitable Heisenberg inequalities earlier determined in [11].
- Requirement #2:$\psi \left(s\right)$ exhibits the explicit dependence in terms of a stochastic observable, so to yield a so-called Stochastic-Variable Approach to quantum theory [22,33,34,35]. In the context of CQG-theory this should be generally identified with a 4-tensor field depending on the physical quantum observable ${g}_{\mu \nu}(r,s)$ and realizing a stochastic variable endowed with a stochastic probability density, i.e., dependent on a suitable stochastic field. Such a stochastic field will be identified in the following with the second-order real and observable stochastic displacement field tensor $\Delta g=\left\{\Delta {g}_{\mu \nu}\right\}$ defined by Equation (11) which by assumption depends functionally on ${g}_{\mu \nu}$ (and hence $\delta {g}_{\mu \nu}(r,s)$ too).
- Requirement #3: the PDF $\rho $ is endowed with a Gaussian-like behavior and is non-dispersive in character, namely in the sense of assuming that in the subset of the proper-time axis I in which $\psi $ is defined, its probability density ${\left|\psi \right|}^{2}$ can be identified for all $s\in I\equiv \mathbb{R}$ with a Gaussian-like PDF depending on $\Delta g$ and $\widehat{g}$, and thus by itself realizes a stochastic function. These particular solutions of the CQG-wave equation are generally non-stationary and are required to preserve their Gaussian-like character, and therefore to be non-dispersive, i.e., free of any spreading behavior during the proper-time quantum dynamical evolution.
- Requirement #4: the quantum wave function holds for arbitrary realizations of the deterministic background metric tensor $\widehat{g}\left(r\right)$ and in particular in the case of vacuum solutions of the Einstein field equations.

## 2. Quantum Gravity Theories and Bohmian Formulation in Literature

## 3. Eulerian Representation

## 4. Lagrangian Path (Bohmian) Representation

## 5. Generalized Lagrangian Path Representation

- GLP Requirement #1- The first one is realized by prescribing $\delta {G}_{L\mu \nu}\left(s\right)$ in terms of the displacement tensor $\delta {g}_{L\mu \nu}\left(s\right)$ which is determined according to Equation (5). This yields therefore the identity$${G}_{L\mu \nu}\left(s\right)={\widehat{g}}_{\mu \nu}\left(r\right)+\delta {g}_{L\mu \nu}\left(s\right)-\Delta {g}_{\mu \nu},$$$$\Delta g=g-{G}_{L}\left(s\right)\equiv \delta g-\delta {G}_{L}\left(s\right).$$Notice that, here, ${g}_{\mu \nu}={g}_{L\mu \nu}\left(s\right)$, and hence $\delta {g}_{\mu \nu}\equiv \delta {g}_{L\mu \nu}\left(s\right)$. Consequently, it is understood that $\Delta g$ must be endowed with a suitable stochastic PDF to be suitably prescribed. In this regards, taking $\Delta g$ as an independent stochastic variable, it is natural to assume that the same PDF should be a stationary and spatially uniform probability distribution, i.e., a function independent of $r$, s as well as $\delta {g}_{L}\left(s\right)$, but still allowed to depend in principle on the prescribed metric tensor ${\widehat{g}}_{\mu \nu}\left(r\right)$. More precisely, this means assuming the same PDF to be realized in terms of a smoothly differentiable and strictly positive function of the form$$f=f\left(\Delta g,\widehat{g}\right).$$Hence, the corresponding notion of stochastic average for an arbitrary smooth function $X(\Delta g,r,s)$ is prescribed in terms of the weighted integral$${\u2329X(\Delta g,r,s)\u232a}_{stoch}\equiv \underset{{U}_{g}}{\int}d(\Delta g)X(\Delta g,r,s)f\left(\Delta g,\widehat{g}\right),$$$$\left\{\begin{array}{c}{\u23291\u232a}_{stoch}\equiv \underset{{U}_{g}}{\int}d(\Delta g)f\left(\Delta g,\widehat{g}\right)=1,\\ {\u2329\Delta {g}_{\mu \nu}\u232a}_{stoch}\equiv \underset{{U}_{g}}{\int}d(\Delta g)\Delta {g}_{\mu \nu}f\left(\Delta g,\widehat{g}\right))=\pm {\widehat{g}}_{\mu \nu}\left(r\right),\\ {\sigma}_{\Delta g}^{2}\equiv {\u2329{\left(\Delta g-{\u2329\Delta g\u232a}_{stoch}\right)}^{2}\u232a}_{stoch}\equiv \\ \underset{{U}_{g}}{\int}d(\Delta g){\left(\Delta g-{\u2329\Delta g\u232a}_{stoch}\right)}^{2}f\left(\Delta g,\widehat{g}\right)={r}_{th}^{2},\end{array}\right.$$
- GLP Requirement #2- The second one is obtained requiring that $\Delta g=\left\{\Delta {g}_{\mu \nu}\right\}$ is constant for all $s\in I$ and for an arbitrary Lagrangian Path, i.e., it is prescribed so that identically for all $s,{s}_{o}\in I$ it occurs that$$\Delta {g}_{\mu \nu}\left(s\right)=\Delta {g}_{\mu \nu}\left({s}_{o}\right).$$Notice that here $\frac{D}{Ds}\delta {g}_{L\mu \nu}\left(s\right)=\frac{D}{Ds}\delta {G}_{L\mu \nu}\left(s\right)\equiv {V}_{\mu \nu}({G}_{L}\left(s\right),\Delta g,s)),$ with ${V}_{\mu \nu}({G}_{L}\left(s\right),\Delta g,s)$ being the tensor velocity field in the GLP-representation, namely$${V}_{\mu \nu}({G}_{L}\left(s\right),\Delta g,s)=\frac{1}{\alpha L}\frac{\partial {S}^{\left(q\right)}({G}_{L}\left(s\right),\Delta g,s)}{\partial \delta {g}_{L}^{\mu \nu}\left(s\right)},$$$$\frac{D}{Ds}\Delta {g}_{\mu \nu}\equiv \frac{D}{Ds}\delta {g}_{L\mu \nu}\left(s\right)-\frac{D}{Ds}\delta {G}_{L\mu \nu}\left(s\right)\equiv 0.$$

## 6. GLP Approach: Determination of the Stochastic PDF for $\Delta g$ and of the Quantum PDF

#### 6.1. Prescription of the Stochastic PDF

#### 6.2. The Initial Quantum PDF ${\rho}_{o}$ and Its Invariance Property

**Proposition**

**1.**

**Invariance of the Gaussian PDF**${\rho}_{G}(\Delta g\pm \widehat{g}\left(r\right))$

**Proof.**

#### 6.3. GLP-Quantum and Stochastic Expectation Values

**Proposition**

**2.**

**Equivalent representations of the GLP-quantum expectation value**$\u2329X\left(s\right)\u232a$

**Proof.**

#### 6.4. Generalized Gaussian PDF and Emergent Gravity Interpretation

**Proposition**

**3.**

**Determination of**$\widehat{g}\left(r\right)$

**(Emergent gravity)**

**Proof.**

## 7. GLP Approach: Polynomial Decomposition of the Quantum Phase Function

#### 7.1. Implications of the Polynomial Decomposition for ${S}^{\left(q\right)}({G}_{L}\left(s\right),\Delta g,s)$

**Proposition**

**4.**

**Determination of the Gaussian PDF**$\rho ({G}_{L}\left(s\right),\Delta g,s)$

**Proof.**

#### 7.2. Implications of the Polynomial Decomposition for $V({G}_{L}\left(s\right),\Delta g,s)$

**Proposition**

**5.**

**Harmonic representation of the vacuum effective potential**

**Proof.**

#### 7.3. Construction of the GLP-Equations

#### 7.4. Small-Amplitude Solutions: Conditions of Validity

**Proposition**

**6.**

**Small-amplitude solutions of Equation (104)**

**Proof.**

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Evaluation of p(s) and Differential Iden-Tities

**Proposition**

**A1.**

**Determination of the tensor field**$\frac{\partial \Delta {g}_{\alpha \beta}}{\partial {g}_{L\mu \nu}\left({s}^{\prime}\right)}$

**Proof.**

**Proposition**

**A2.**

**Determination of the 4-scalar function**$p\left(s\right)$

**Proof.**

## Appendix B. Differential Identities for the Tensor Coefficients ${\mathit{a}}_{\mathit{pq}}^{\mathit{\alpha}\mathit{\beta}}\left(\mathit{s}\right)$

## References

- Messiah, A. Quantum Mechanics; Dover Pubs: New York, NY, USA, 1999. [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. In Theoretical Physics; Addison-Wesley: New York, NY, USA, 1957. [Google Scholar]
- Misner, C.W.; Thorne, K.S.; Wheeler, J.A. Gravitation; W.H. Freeman and Company: New York, NY, 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] - 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] - Cremaschini, C.; Tessarotto, M. Synchronous Lagrangian variational principles in general relativity. Eur. Phys. J. Plus
**2015**, 130, 123. [Google Scholar] [CrossRef] - 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] - De Donder, Th. Théorie Invariantive Du Calcul des Variations; Gaultier-Villars & Cia: Paris, France, 1930. (In French) [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] - 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] - Han, M. Einstein equation from covariant loop quantum gravity in semiclassical continuum limit. Phys. Rev. D
**2017**, 96, 024047. [Google Scholar] [CrossRef] - Madelung, E. Quantentheorie in hydrodynamischer form. Z. Phys.
**1927**, 40, 322. (In German) [Google Scholar] [CrossRef] - Tessarotto, M.; Cremaschini, C. Generalized Lagrangian-path representation of non-relativistic quantum mechanics. Found. Phys.
**2016**, 46, 1022. [Google Scholar] [CrossRef] - Tessarotto, M.; Mond, M.; Batic, D. Hamiltonian structure of the Schrödinger classical dynamical system. Found. Phys.
**2016**, 46, 1127. [Google Scholar] [CrossRef] - Nelson, E. Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev.
**1966**, 150, 1079. [Google Scholar] [CrossRef] - Bouda, A. From a mechanical Lagrangian to the Schrödinger equation: A modified version of the quantum Newton law. Int. J. Mod. Phys. A
**2003**, 18, 3347. [Google Scholar] [CrossRef] - Holland, P. Computing the wavefunction from trajectories: Particle and wave pictures in quantum mechanics and their relation. Ann. Phys.
**2005**, 315, 505. [Google Scholar] [CrossRef] - Poirier, B. Bohmian mechanics without pilot waves. Chem. Phys.
**2010**, 370, 4. [Google Scholar] [CrossRef] - Holland, P. Foreword. In Quantum Trajectories; Chattaraj, P., Ed.; Taylor & Francis/CRC: Boca Raton, FL, USA, 2010. [Google Scholar]
- Poirier, B. Trajectory-based derivation of classical and quantum mechanics. In Quantum Trajectories; Hughes, K.H., Parlant, G., Eds.; Daresbury Laboratory: Daresbury, UK, 2011. [Google Scholar]
- Schiff, J.; Poirier, B. Communication: Quantum mechanics without wavefunctions. J. Chem. Phys.
**2012**, 136, 031102. [Google Scholar] [CrossRef] [PubMed] - Parlant, G.; Ou, Y.C.; Park, K.; Poirier, B. Classical-like trajectory simulations for accurate computation of quantum reactive scattering probabilities. Comput. Theoret. Chem.
**2012**, 990, 3. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B.J.; Kaloyerou, P.N. An ontological basis for the quantum theory. Phys. Rep.
**1987**, 144, 321. [Google Scholar] [CrossRef] - Bohm, D. A suggested interpretation of the quantum theory in terms of ”hidden” variables. I. Phys. Rev.
**1952**, 85, 166. [Google Scholar] [CrossRef] - Bohm, D. A suggested interpretation of the quantum theory in terms of ”hidden” variables. II. Phys. Rev.
**1952**, 85, 180. [Google Scholar] [CrossRef] - Bohm, D. Reply to a criticism of a causal re-interpretation of the quantum theory. Phys. Rev.
**1952**, 87, 389. [Google Scholar] [CrossRef] - Weinberg, S. The cosmological constant problem. Rev. Mod. Phys.
**1989**, 61, 1. [Google Scholar] [CrossRef] - Grössing, G. On the thermodynamic origin of the quantum potential. Physica A Stat. Mech. Appl.
**2009**, 388, 811. [Google Scholar] [CrossRef] - Dennis, G.; de Gosson, M.A.; Hiley, B.J. Bohm’s quantum potential as an internal energy. Phys. Lett. A
**2015**, 379, 1224. [Google Scholar] [CrossRef] - Schrödinger, E. Der stetige Übergang von der Mikro- zur Makromechanik. Die Naturwisseschaften
**1926**, 14, 664. (In German) [Google Scholar] [CrossRef] - Ashtekar, A. Gravity and the quantum. New J. Phys.
**2005**, 7, 198. [Google Scholar] [CrossRef] - Etienne, Z.B.; Liu, Y.T.; Shapiro, S.L. Relativistic magnetohydrodynamics in dynamical spacetimes: A new adaptive mesh refinement implementation. Phys. Rev. D
**2010**, 82, 084031. [Google Scholar] [CrossRef] - 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] - Ruchin, V.; Vacaru, O.; Vacaru, S. On relativistic generalization of Perelman’s W-entropy and thermodynamic description of gravitational fields and cosmology. Eur. Phys. J. C
**2017**, 77, 184. [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 Series 169; Springer-Verlag: Berlin, Germany, 1982. [Google Scholar]
- Sudarshan, E.C.G.; Mukunda, N. Classical Dynamics: A Modern Perspective; Wiley-Interscience Publication: New York, NY, USA, 1964. [Google Scholar]
- Mukunda, N. Generators of symmetry transformations for constrained Hamiltonian systems. Phys. Scr.
**1980**, 21, 783. [Google Scholar] [CrossRef] - Castellani, L. Symmetries in constrained hamiltonian systems. Ann. Phys.
**1982**, 143, 357. [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]
- Alcubierre, M. Introduction to 3+1 Numerical Relativity; Oxford University Press: Oxford, UK, 2008. [Google Scholar]
- DeWitt, B.S. Quantum theory of gravity. Phys. Rev.
**1967**, 60, 1113. [Google Scholar] [CrossRef] - Ashtekar, A. New variables for classical and quantum gravity. Phys. Rev. Lett.
**1986**, 57, 2244. [Google Scholar] [CrossRef] [PubMed] - Ashtekar, A. New Hamiltonian formulation of general relativity. Phys. Rev. D
**1987**, 36, 1587. [Google Scholar] [CrossRef] - Jacobson, T.; Smolin, L. Nonperturbative quantum geometries. Nucl. Phys. B
**1988**, 299, 295. [Google Scholar] [CrossRef] - Rovelli, C.; Smolin, L. Knot theory and quantum gravity. Phys. Rev. Lett.
**1988**, 61, 1155. [Google Scholar] [CrossRef] [PubMed] - Rovelli, C.; Smolin, L. Loop space representation of quantum general relativity. Nucl. Phys. B
**1990**, 331, 80. [Google Scholar] [CrossRef] - Rovelli, C. Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: A Report. Class. Quantum Gravity
**1991**, 8, 1613. [Google Scholar] [CrossRef] - Ashtekar, A.; Geroch, R. Quantum theory of gravitation. Rep. Prog. Phys.
**1974**, 37, 1211. [Google Scholar] [CrossRef] - Weinberg, S. Gravitation and Cosmology; John Wiley: New York, NY, USA, 1972. [Google Scholar]
- DeWitt, B.S. Covariant quantum geometrodynamics. In Magic without Magic; Wheeler, J.A., Klauder, J.R., Eds.; W. H. Freeman: San Francisco, CA, USA, 1972. [Google Scholar]
- Batalin, I.A.; Vilkovisky, G.A. Relativistic S-matrix of dynamical systems with boson and fermion constraints. Phys. Lett. B
**1977**, 69, 309. [Google Scholar] [CrossRef] - Batalin, I.A.; Vilkovisky, G.A. Gauge algebra and quantization. Phys. Lett. B
**1981**, 102, 27. [Google Scholar] [CrossRef] - Batalin, I.A.; Vilkovisky, G.A. Feynman rules for reducible gauge theories. Phys. Lett. B
**1983**, 120, 166. [Google Scholar] [CrossRef] - Batalin, I.A.; Vilkovisky, G.A. Quantization of gauge theories with linearly dependent generators. Phys. Rev. D
**1983**, 28, 2567. [Google Scholar] [CrossRef] - Mandal, B.P.; Rai, S.K.; Upadhyay, S. Finite nilpotent symmetry in Batalin-Vilkovisky formalism. Eur. Phys. Lett.
**2010**, 92, 21001. [Google Scholar] [CrossRef] - Upadhyay, S.; Mandal, B.P. BV formulation of higher form gauge theories in a superspace. Eur. Phys. J. C
**2012**, 72, 2059. [Google Scholar] [CrossRef] - Upadhyay, S. Perturbative quantum gravity in Batalin-Vilkovisky formalism. Phys. Lett. B
**2013**, 723, 470. [Google Scholar] [CrossRef] - Fredenhagen, K.; Rejzner, K. Batalin-Vilkovisky formalism in the functional approach to classical field theory. Comm. Math. Phys.
**2012**, 314, 93. [Google Scholar] [CrossRef] - Pinto-Neto, N.; Santos, G.; Struyve, W. Quantum-to-classical transition of primordial cosmological perturbations in de Broglie-Bohm quantum theory. Phys. Rev. D
**2012**, 85, 083506. [Google Scholar] [CrossRef] - Pinto-Neto, N.; Falciano, F.T.; Pereira, R.; Santini, E.S. Wheeler-DeWitt quantization can solve the singularity problem. Phys. Rev. D
**2012**, 86, 063504. [Google Scholar] [CrossRef] - Pinto-Neto, N.; Fabris, J.C. Quantum cosmology from the de Broglie-Bohm perspective. Class. Quantum Gravity
**2013**, 30, 143001. [Google Scholar] [CrossRef] - Falciano, F.T.; Pinto-Neto, N.; Struyve, W. Wheeler-DeWitt quantization and singularities. Phys. Rev. D
**2015**, 91, 043524. [Google Scholar] [CrossRef] - Holland, P.R. The Quantum Theory of Motion; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Wyatt, R. Quantum Dynamics with Trajectories; Springer-Verlag: Berlin, Germany, 2005. [Google Scholar]
- Bhattacharya, S.; Shankaranarayanan, S. How emergent is gravity? Int. J. Mod. Phys. D
**2015**, 24, 1544005. [Google Scholar] [CrossRef] - Padmanabhan, T. Emergent gravity paradigm: Recent progress. Mod. Phys. Lett. A
**2015**, 30, 1540007. [Google Scholar] [CrossRef] - Jacobson, T. Thermodynamics of spacetime: The Einstein equation of state. Phys. Rev. Lett.
**1995**, 75, 1260. [Google Scholar] [CrossRef] [PubMed] - Faizal, M.; Ashour, A.; Alcheikh, M.; Alasfar, L.; Alsaleh, S.; Mahroussah, A. Quantum fluctuations from thermal fluctuations in Jacobson formalism. Eur. Phys. J. C
**2017**, 77, 608. [Google Scholar] [CrossRef]

© 2018 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**

Tessarotto, M.; Cremaschini, C. Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory. *Entropy* **2018**, *20*, 205.
https://doi.org/10.3390/e20030205

**AMA Style**

Tessarotto M, Cremaschini C. Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory. *Entropy*. 2018; 20(3):205.
https://doi.org/10.3390/e20030205

**Chicago/Turabian Style**

Tessarotto, Massimo, and Claudio Cremaschini. 2018. "Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory" *Entropy* 20, no. 3: 205.
https://doi.org/10.3390/e20030205