# Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity

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## Abstract

**:**

## 1. Introduction

- The establishment of the Schroedinger-like CQG-wave equation in manifest covariant form, which is realized by a first-order PDE with respect to the invariant proper-time [4].
- The statistical interpretation of the CQG-wave equation in terms of corresponding quantum hydrodynamic equations [5].
- The fulfillment of generalized Heisenberg inequalities relating the statistical measurement errors of quantum observables, represented in terms of the standard deviations of the quantum gravitational tensor ${g}_{\mu \nu}$ and its quantum conjugate momentum operator [5].
- The construction of generally non-stationary analytical solutions for the CQG-wave equation with non-vanishing cosmological constant and exhibiting Gaussian-like probability densities that are non-dispersive in proper-time [6].
- The proof of the existence of an emergent gravity phenomenon occurring in the context of CQG-theory (previously referred to as “second-type emergent-gravity paradigm” (see Ref. [6])) according to which it can/must be possible to represent the mean-field background space-time metric tensor $\widehat{g}$ in terms of a suitable ensemble average. More precisely, as shown in the same reference, this is identified in terms of a statistical average with respect to stochastic fluctuations of the quantum gravitational field ${g}_{\mu \nu}$, whose quantum-wave dynamics is actually described by means of GLP trajectories [6].

- The proof of existence of a discrete invariant-energy spectrum for stationary solutions of the CQG-wave equation, obtained by implementing the Dirac ladder method (i.e., a second-quantization method) for the stationary wave equation with harmonic Hamiltonian potential [4].
- The analytical estimate for the graviton mass and its quantum discrete invariant energy spectrum, supporting the interpretation of the graviton DeBroglie length as being associated with the quantum ground-state related to the cosmological constant [4]. It must be stressed in this connection that the prediction of massive gravitons represents an intrinsic property of CQG-theory which marks also an important point of distinction with respect to past literature. In fact previous perturbative treatments of quantum gravity based on linearized GR theories typically exhibit—in analogy with the case of the electromagnetic field - massless gravitons. Indeed in the framework of CQG-theory, as discovered in Ref. [4], the existence of massive gravitons and their mass estimate are found to be associated with a non-vanishing cosmological constant.

#### 1.1. Physical Evidence and Open Problems

#### 1.2. Issues about the Cosmological Constant

- ISSUE #1: the possible quantum origin and, more precisely, the quantum self-generation of the cosmological constant $\mathsf{\Lambda}$, i.e., in which the same one is produced merely by the presence of gravitons, as well as its precise estimate in the context of Quantum Gravity.
- ISSUE #2: the possible dynamical behavior of $\mathsf{\Lambda}$ and the search of an admissible dynamical parametrization in terms of physical observables, including the relationship with its constant representation.
- ISSUE #3: the corresponding eventual implications for cosmology, in particular in reference with the large-scale phenomenology of the universe.

#### 1.3. Goals and Structure of the Paper

## 2. Extended Functional Setting for CQG-Theory

## 3. The Role of Proper-Time in Covariant Classical/Quantum Gravity

#### 3.1. Proper-Time as a Local or Global Observable

#### 3.2. Interpretation/Meaning of Proper-Time

## 4. Covariant Classical/Quantum Gravity in the Extended Setting

#### 4.1. The Classical Hamiltonian Structure of GR

#### 4.2. GR—Hamilton-Jacobi Quantization

## 5. Hamiltonian Representation of the CQG-Quantum Hydrodynamics Equations

**Theorem**

**1**

**(Quantum**

**Hamilton**

**equations).**

**Proof.**

**Q.E.D.**□

## 6. Quantum Modified Einstein Field Equations

**Theorem**

**2**

**(Quantum-modified**

**Einstein**

**field**

**equations).**

**Proof.**

**Q.E.D.**□

## 7. GLP-Approach in the Extended Functional Setting

#### 7.1. Formal Solution of GLP in the Extended Functional Setting

**Theorem**

**3**

**(Construction**

**of**

**a**

**formal**

**representation**

**of**

**GLP).**

**Proof.**

**Q.E.D.**□

#### 7.2. Properties of Polynomial GLP-Solutions of the Hamilton-Jacobi Equation

**Theorem**

**4**

**(Polynomial**

**GLP-solutions**

**of**

**the**

**Hamilton-Jacobi**

**equation**

**in**

**the**

**case**

**of**

**vacuum).**

**Proof.**

**Q.E.D.**□

#### 7.3. GLP Gaussian Particular Solutions of the Quantum PDF

#### 7.4. Semiclassical Limit

## 8. Explicit Evaluation of the Bohm Effective Potential and Source Term

- First, as shown in the present paper, the same Equation (66) has been recovered independently also in the GLP-approach, being provided in such a context by the extremal tensor Equation (114) (see THM. 4).
- Second, based on the construction of an analytic solution for the quantum PDF (see Equation (130) above) and of the corresponding quantum phase-function (the function ${\mathcal{S}}^{\left(q\right)}({G}_{L}\left(s\right),\Delta g,s)$ determined via the polynomial representation (108)), the GLP-approach permits one to obtain also an explicit representation of the Bohm effective quantum potential (52) and corresponding source term ${B}_{\mu \nu}\left(s\right)$.
- Third, in the subsequent calculations all integrations are performed with respect to the local extremal geodesic trajectory. As a consequence the initial proper-time ${s}_{o}$ is set equal to ${s}_{o}=0.$

#### Determination of the CQG-Cosmological Constant ${\mathsf{\Lambda}}_{CQG}\left(s\right).$

## 9. Proper-Time Behavior of ${\mathsf{\Lambda}}_{\mathbf{CQG}}\left(\mathbf{s}\right)$ and Physical Implications

**Theorem**

**5 (Asymptotic behavior of the solutions of Equation (152)).**

**Proof.**

**Q.E.D.**□

#### Physical Implications

## 10. Concluding Remarks

- (1)
- The definition of the observer’s proper-time (s), consistent with the treatment adopted in CQG-theory of gravitons as classical point-particles and with the Big Bang hypothesis. This is prescribed as the arc length of a suitable non-null geodesic world-line associated with the background metric tensor $\widehat{g}$, which represents a virtual trajectory, namely one of the infinite possible physically admissible worldlines, associated with a massive graviton. To this end the same curve is identified in a cosmological framework with an observer’s maximal geodetics, i.e., a geodesic curve having the maximal arc length and with origin point ${r}^{\mu}\left({s}_{o}\right),$ the point of creation of the same particle, coinciding with (or suitably close to) the Big Bang event. By construction for the initial 4-position ${r}^{\mu}\left({s}_{o}\right)$ is therefore such that ${r}^{\mu}\left({s}_{o}\right)\equiv {r}^{\mu}({s}_{o}=0)$.
- (2)
- The establishment of the Hamiltonian structure of CQG-theory. This is represented by a set of continuous canonical equations (referred to here as quantum Hamilton equations) whose validity is implied by the quantum-wave equation through its corresponding quantum Hamilton-Jacobi equation. As shown in THM. 1, the same Hamiltonian structure remains preserved also in validity of the said extended setting (i.e., for non-stationary background metric tensor).
- (3)
- The discovery of quantum-modified Einstein field equation. In fact, the quantum Hamilton equations have been shown to admit a particular realization in terms of a set of PDEs which is analogous to the classical Einstein Equation (5) but in which quantum source terms are taken into account. Remarkably also such an equation remains preserved under the same extended functional setting (see THM. 2).
- (4)
- The establishment of the corresponding formulation of the generalized Lagrangian path (GLP) approach. The issues indicated above have been cast in the framework provided by the said, earlier formulated, GLP-approach. The key feature of the GLP-approach unveiled here (THM. 3 and THM. 4) concerns its validity also in the context of the extended functional setting and the determination of explicit vacuum solutions of the quantum hydrodynamic equations associated with the CQG-wave equation, with particular reference to quantum solutions characterized by Gaussian quantum PDFs.
- (5)
- The prescription of the quantum cosmological constant, its estimate achieved in the framework of CQG-theory and its dynamical behavior. In fact it has been shown that the cosmological constant $\mathsf{\Lambda}\equiv {\mathsf{\Lambda}}_{CQG}\left(s\right)$ is non-stationary, i.e., dependent on the observer’s proper-time s. The determination of the proper-time dependence of the quantum cosmological constant has been based on the GLP-approach which permits the construction of dynamically-consistent analytic solutions for the quantum wave-function. As a result the relevant asymptotic properties (for $s\to \infty $) of the s-dependent quantum cosmological constant have been established (THM. 5).
- (6)
- The implications and possible interpretation of the large-scale phenomenology of the universe by means of an extended formulation of CQG-theory in which the background space-time itself is non-stationary. For this purpose the associated background metric field tensor $\widehat{g}\equiv \left(\right)open="\{"\; close="\}">{\widehat{g}}_{\mu \nu}$ has been couched in an extended functional setting in which the same tensor field is considered of the form $\widehat{g}(r,s)\equiv \left(\right)open="\{"\; close="\}">{\widehat{g}}_{\mu \nu}(r,s)$ namely again explicitly dependent on the same proper-time $s.$

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Classical Kinetic and Normalized Effective Potential Densities

## Appendix B. Covariant Partial Derivative

## Appendix C. Determination of the 4-Scalar Factor p(s)

**Proposition A1**

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

**Proposition A2**

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

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Cremaschini, C.; Tessarotto, M.
Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity. *Symmetry* **2018**, *10*, 287.
https://doi.org/10.3390/sym10070287

**AMA Style**

Cremaschini C, Tessarotto M.
Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity. *Symmetry*. 2018; 10(7):287.
https://doi.org/10.3390/sym10070287

**Chicago/Turabian Style**

Cremaschini, Claudio, and Massimo Tessarotto.
2018. "Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity" *Symmetry* 10, no. 7: 287.
https://doi.org/10.3390/sym10070287