# Background Independence and Gauge Invariance in General Relativity Part 2—Covariant Quantum Gravity

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## Abstract

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## 1. Introduction

- Issue #1: The first one is related with the possible direct observation of gravitons, which would of course represent the definitive proof for QG. While many aspects of GR have been tested, and general principles of quantum dynamics demand its quantization, there is yet no direct experimental evidence for the existence of gravitons, i.e., the quanta of the gravitational field. On the basis of elementary physical grounds and according to the theoretical estimate pointed out in Ref. [5], the theoretical graviton rest mass ${m}_{o}$ should scale with respect to the electron rest mass ${m}_{e}$ as $\frac{{m}_{o}}{{m}_{e}}\cong 1.38\times {10}^{-39}$. This suggests that in order to increase the probability of detection of gravitons and enhance measurements of their rest mass one should look for environments characterized by the presence of intense fluxes of gravitons themselves, e.g., those expected to occur at the formation/merger of black holes. Conversely, the realization of graviton detectors sensitive enough to detect individual particles (gravitons) requires detectors massive enough to generate a black hole event [6].
- Issue #2: The second issue that might also lead to an independent firm establishment of quantized gravity is related to the direct observation of a possible cosmological gravitational wave background (CGWB). According to Ref. [7], it is believed that measurement of polarization of the CGWB might unveil the long-wavelength stochastic background of gravitational waves associated with the inflationary stage of the Early Universe.
- Issue #3: A notable issue is certainly the measurement and possible identification of the physical origin of the quantum cosmological constant (CC), as predicted in Ref. [8]. The reason is that, at the classical level, the CC remains undetermined (see also the discussion in the subsequent Section 3). It follows that a meaningful (ideal) target concerns identifying the quantum mechanism responsible for its generation, i.e., the quantum physical interaction responsible for the occurrence of the experimentally observed CC. This, in turn, may lead to the proof of the possible stochastic nature of the same quantum cosmological constant, which might not be deterministic in character as usually considered in the literature.
- Issue #4: The proof of the possible stochastic (i.e., quantum) rather than classical (i.e., deterministic) behavior, which, contrary to conventional wisdom, might characterize event horizons (EH) associated to BHs [9]. This issue is intrinsically related to the search of the possible quantum regularization mechanisms of singular classical space-time solutions associated with the classical Einstein Field Equations (EFE).

#### 1.1. Two Types of Quantum Space-Times: The Multi- and Uni-Verse Space-Time Representations

- Multi-verse quantum space-time: case of the constrained functional setting${\left\{g\left(r\right)\right\}}_{C}$. In this case, the generic quantum tensor $g\left(r\right)\in \left\{g\left(r\right)\right\}$ is itself identified with a metric tensor, which therefore generates its own space-time. For this reason, this representation (of quantum space-time) is denoted intuitively as multi-verse, since each choice of the tensor determines, at least in principle, a new and different space-time. Therefore, the structure of the corresponding quantum space-time is identified with the differential manifold$$\left\{{\mathbf{Q}}^{4},g\left(r\right)\right\}.$$This means that the counter- and covariant components of the symmetric tensor $g\left(r\right)$, namely, $\phantom{\rule{4pt}{0ex}}\left\{{g}_{\mu \nu}\left(r\right)\right\}$ and $\left\{{g}^{\mu \nu}\left(r\right)\right\}$ are necessarily constrained, the constraint conditions being represented by the orthogonality relations$${g}_{\mu \nu}\left(r\right){g}^{\alpha \nu}\left(r\right)={\delta}_{\mu}^{\alpha}.$$These constraints must hold identically at arbitrary $4-$positions $r\equiv \left\{{r}^{\mu}\right\}$ spanning $\left\{{\mathbf{Q}}^{4},g\left(r\right)\right\}$ and arbitrary indices $\mu ,\alpha =0,3.$ As a consequence, the functional setting $\left\{g\left(r\right)\right\}$ is represented by constrained tensor functions subject to the orthogonality constraints (3), i.e., a suitably defined constrained-function space ${\left\{g\left(r\right)\right\}}_{C}$.
- Uni-verse or Background quantum space-time: case of the unconstrained functional setting${\left\{g\left(r\right)\right\}}_{U}$. In this case, the same generic quantum tensor $g\left(r\right)$ defined above is not required to be a metric tensor, which means that it remains unconstrained. However, the same $g\left(r\right)$ is assumed to be a $4-$tensor with respect to a suitably prescribed space-time. For this purpose, the assumption is introduced that there exists a suitable and possibly non-unique quantum background metric field tensor $\widehat{g}\left(r\right)\equiv \left\{{\widehat{g}}_{\mu \nu}\left(r\right)\right\}\equiv \left\{{\widehat{g}}^{\mu \nu}\left(r\right)\right\}$, with $\widehat{g}\left(r\right)$ itself belonging to an appropriate functional set $\left\{\widehat{g}\left(r\right)\right\}$, and an associated “background” quantum space-time$$\left\{{\mathbf{Q}}^{4},\widehat{g}\left(r\right)\right\},$$

#### 1.2. Goals of the Investigation

## 2. The (Minimal) Requirements of QG-Theory

- Feature #1: it satisfies the covariance principle, i.e., it is a $4-$scalar, so its complex value should be frame independent, namely, independent of the choice of the possible coordinate systems that are mutually connected by means of local diffeomorphisms $r\iff {r}^{\prime}={r}^{\prime}\left(r\right)$ (diffeomorphism invariance).
- Feature #2: for the same reason, it should obey a $4-$scalar PDE [12]. This feature automatically preserves for the quantum state $\psi \left(r\right)$ its $4-$scalar character everywhere in the corresponding quantum space-time.
- Feature #3: it admits, as all quantum theories worthy of this name, a Hilbert-space representation and the prescription of appropriate scalar product [11]. For the same reason, it must be prescribed so that its mod square ${\left|\psi \left(r\right)\right|}^{2}\equiv $$\psi \left(r\right){\psi}^{*}\left(r\right)$ represents the quantum probability density for the occurrence of the quantum metric tensor $g\left(r\right)=\left\{{g}_{\mu \rho}\left(r\right)\right\}=\left\{{g}^{\mu \nu}\left(r\right)\right\}$, with respect to a suitably prescribed functional configuration space, i.e., either ${\left\{g\left(r\right)\right\}}_{C}$ or ${\left\{g\left(r\right)\right\}}_{U}$.
- Feature #4: it satisfies, under the assumption of quantum unitarity, the Heisenberg uncertainty principle [11].
- Feature #5: it admits a formulation of quantum logic analogous to that of quantum mechanics [13].
- Feature #6: it permits to unveil the intrinsic stochastic character of Quantum Gravity and the consequent implications on classical counterpart solutions, in particular, as far as it concerns the definition of the quantum nature of the cosmological constant and the quantum regularization of classical singular solutions predicted by EFE (e.g., black-hole singularities and event horizons).

#### Why CQG-Theory?

## 3. The Metric-Ricci and the $\mathbf{g}-$Quantization of GR

#### 3.1. Stochastic Quantization in CQG-Theory

- The first one is the intrinsic stochasticity of space-time quantum trajectories. Their identification with deterministic trajectories, as performed in the original formulation of Bohmian quantum mechanics [21], may be viewed as physically incorrect because this is in potential violation of the Heisenberg indeterminacy principle. The proof of their stochastic character has actually been reached rigorously in the context of QM. Therefore, in close analogy, it appears reasonable to expect that graviton particle trajectories that span the background space-time could similarly acquire a stochastic character and therefore depart from purely deterministic geodetics as typically assumed in QG-theory.
- The second reason that motivates the allowance of stochastic space-time trajectories is the extended feature of quantum particles. Accordingly, all quantum particles (including graviton particles), in a strict sense, should not be treated (any more) as point-like particles. The corresponding physical motivations being either (a) the proper treatment of particle self-interactions, which cannot simply be defined in the case of point-particles (see for example the case of electromagnetic self-interactions), or (b) the issue of possible quantum regularization of space-time singularities.
- Finally, we add here a third and actually crucial motivation: this is realized by the fact that it becomes possible to represent the quantum-modified EFE (which determines the same background field tensor) by performing the quantum-average (or quantum expectation value) of a suitably determined quantum Hamilton equation. However, by construction, the quantum expectation value now also includes suitable stochastic averaging on the stochastic trajectories. This stochastic-averaging effect is believed to be crucial for determining the qualitative behavior—i.e., regularization—of the background metric tensor at microscopic scales, i.e., at the Planck scale length.

**.**Thus, stochastic quantization must necessarily involve the additional synchronous transformation (i.e., in which the proper time s is left unchanged):

#### 3.2. Stochastic Canonical Map and Stochastic CQG-Quantum-Wave Equation

#### 3.3. Quantum Hydrodynamic Equations

#### 3.4. GLP Representation

#### 3.5. Quantum Expectation Value and Stochastic Averaging

## 4. Two Construction Methods for the Quantum-Modified EFE

#### 4.1. First Construction Method: Deterministic Limit

#### 4.2. Second Construction Method: Stochastic Averaging

## 5. Background Independence

#### Background Independence in CQG-Theory

- $\widehat{g}\left(r\right)$ remains still arbitrary as its prescription depends on still undetermined appropriate boundary conditions;
- CQG-theory holds for arbitrary particular solutions $\widehat{g}\left(r\right)$;
- $\widehat{g}\left(r\right)$ can be represented in arbitrary coordinate systems (GR-frames);
- $\widehat{g}\left(r\right)$ has an emergent character, namely, it can be represented in terms of a quantum expectation value in terms of a suitable quantum (i.e., stochastic) field tensor [8];
- finally, as discussed in the previous section, background independence implies also that the same q-modified EFE can be uniquely determined in terms of quantum expectation value (or equivalent stochastic average) of an underlying quantum equation.

## 6. Conclusions

- Besides the introduction of a stochastic quantum cosmological constant, i.e., a physical property of crucial importance for the possible tunneling effect arising at the event horizons (a topic already in part discussed in our earlier paper [9]).

- First, the adoption of stochastic quantum trajectories. As a consequence, the quantum trajectories that are taken into account in the quantum-wave equation of CQG-theory are actually considered stochastic. This effect is believed to be relevant to extend the validity of quantum theory to the Planck scale.
- Second, the inclusion of a stochastic-modified quantum potential appearing through stochastic gauge contributions. This feature is crucial for the establishment of the property of background independence in Quantum Gravity, in the sense that the equation determining the background metric field tensor—i.e., the quantum expectation value of a suitable quantum tensor field—should itself be realized in terms of a suitable quantum expectation value.
- Third, the representation of the quantum-modified EFE (which determines the same background field tensor) in terms of stochastic quantum-average (or quantum expectation value) of a suitably determined quantum Hamiltonian equation that depends on the stochastic trajectories.

- (a)
- The graviton quantum trajectories;
- (b)
- The quantum potential;
- (c)
- The cosmological constant.

- The relevant dynamical equations for the tensor fields are realized by means of evolution-type ODEs, which identify a Hamiltonian system in a proper sense. Such a feature is warranted by the fulfillment of the constraint condition characteristic of the Einstein field equations, namely, the requirement that the components of the Ricci tensor depend on the background metric tensor $\widehat{g}\left(r\right)$ only.
- The corresponding Hamiltonian system $\left\{{x}_{R},{H}_{R}\right\}$ is constraint-free because the tensor components of the Hamiltonian state ${x}_{R}\left(s\right)\equiv (g\left(s\right),\pi \left(s\right))$ are independent.
- Suitable quantum-modified Einstein field equations are obtained that prescribe, in a suitable functional class, the background metric field tensor.
- Quantum Hamilton equations are determined, which are related to the quantum hydrodynamic equations.
- The theory is gauge invariant. As a consequence, both the Hamiltonian and Lagrangian densities are intrinsically non-unique, being determined up to an additive gauge function.And finally, the crucial aspect which characterizes CQG-theory and its classical counterpart described in Part I, namely:
- The property of manifest covariance, i.e., the theory is set in $4-$tensor form, since its quantum state and its quantum Hamiltonian operator have a $4-$tensor and frame-independent character with respect to the background space-time $\left\{{\mathbf{Q}}^{4},\widehat{g}\left(r\right)\right\}$.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Classical Hamiltonian Function of GR and Covariant Derivative

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Tessarotto, M.; Cremaschini, C.
Background Independence and Gauge Invariance in General Relativity Part 2—Covariant Quantum Gravity. *Symmetry* **2022**, *14*, 2229.
https://doi.org/10.3390/sym14112229

**AMA Style**

Tessarotto M, Cremaschini C.
Background Independence and Gauge Invariance in General Relativity Part 2—Covariant Quantum Gravity. *Symmetry*. 2022; 14(11):2229.
https://doi.org/10.3390/sym14112229

**Chicago/Turabian Style**

Tessarotto, Massimo, and Claudio Cremaschini.
2022. "Background Independence and Gauge Invariance in General Relativity Part 2—Covariant Quantum Gravity" *Symmetry* 14, no. 11: 2229.
https://doi.org/10.3390/sym14112229