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

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## Abstract

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## 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 $\rho (g,r,s)\equiv {\left(\right)}^{\psi}2$ 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 $\Delta g=\left(\right)open="\{"\; close="\}">\Delta {g}_{\mu \nu}$ 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$$f=f\left(\right)open="("\; close=")">\Delta g,\widehat{g}$$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$${\left(\right)}_{X}stoch,$$$$\left(\right)$$
- GLP Requirement #2- The second one is obtained requiring that $\Delta g=\left(\right)open="\{"\; close="\}">\Delta {g}_{\mu \nu}$ 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**$\left(\right)$

**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 ${\mathit{a}}_{\mathit{pq}}^{\mathit{\alpha}\mathit{\beta}}\left(\mathit{s}\right)$

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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