# Evidence of Time Evolution in Quantum Gravity

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## Abstract

**:**

## 1. Introduction

## 2. Classical Picture

## 3. Quantum Pictures with Time

#### 3.1. The Schrödinger Equation with a Physical Hamiltonian (Method A)

#### 3.2. Time Evolution from the WDW Equation (Method B)

#### 3.3. An Evolution from the WDW Using the Grassmann Variables (Method C)

#### 3.4. The quasi-Heisenberg Picture (Method D)

#### 3.5. An Evolution Using the Unconstraint Schrödinger Equation in the Extended Space (Method E)

#### 3.5.1. Canonical Gauge

#### 3.5.2. Non-Canonical Gauge

## 4. Discussion and Possible Application of the Above Approaches to the General Case of Gravity Quantization

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Aidala, C.A.; Bass, S.D.; Hasch, D.; Mallot, G.K. The spin structure of the nucleon. Rev. Mod. Phys.
**2013**, 85, 655–691. [Google Scholar] [CrossRef][Green Version] - Kuchar, K.V. Time and interpretations of quantum gravity. Int. J. Mod. Phys.
**2011**, D20, 3–86. [Google Scholar] [CrossRef] - Isham, C.J. Canonical Quantum Gravity and the Problem of Time. In Integrable Systems, Quantum Groups, and Quantum Field Theories; Ibort, L.A., Rodríguez, M.A., Eds.; Springer Netherlands: Dordrecht, The Netherlands, 1993; pp. 157–287. [Google Scholar] [CrossRef]
- Shestakova, T.P.; Simeone, C. The Problem of time and gauge invariance in the quantization of cosmological models. II. Recent developments in the path integral approach. Grav. Cosmol.
**2004**, 10, 257–268. [Google Scholar] - Rovelli, C. “Forget time”. arXiv
**2009**, arXiv:0903.3832. [Google Scholar] - Anderson, E. The Problem of Time in Quantum Gravity. arXiv
**2010**, arXiv:1009.2157. [Google Scholar] [CrossRef][Green Version] - DeWitt, B.S. Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev.
**1967**, 160, 1113–1148. [Google Scholar] [CrossRef][Green Version] - Kiefer, C. Conceptual Problems in Quantum Gravity and Quantum Cosmology. Isrn Math. Phys.
**2013**, 2013, 509316. [Google Scholar] [CrossRef][Green Version] - Shestakova, T.P. Is the Wheeler-DeWitt equation more fundamental than the Schrödinger equation? Int. J. Mod. Phys. D
**2018**, 27, 1841004. [Google Scholar] [CrossRef] - Rovelli, C. Time in quantum Gravity: An hypothesis. Phys. Rev. D
**1991**, 43, 442–456. [Google Scholar] [CrossRef] - Wallace, D. The Emergent Multiverse; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- O’Neill, W. Time and Eternity in Proclus. Phronesis
**1962**, 7, 163. [Google Scholar] [CrossRef] - Bojowald, M. Quantum cosmology: A review. Rep. Prog. Phys.
**2015**, 78, 023901. [Google Scholar] [CrossRef] [PubMed] - Kaku, M. Introduction to Superstrings; Springer: New York, NY, USA, 2012. [Google Scholar]
- Dirac, P.A.M. Generalized Hamiltonian Dynamics. Can. J. Math.
**1950**, 2, 129–148. [Google Scholar] [CrossRef] - Dirac, P.A.M. Generalized Hamiltonian Dynamics. Proc. R. Soc. Lond. A
**1958**, 246, 326–332. [Google Scholar] [CrossRef] - Gitman, D.; Tyutin, I.V. Quantization of Fields with Constraints; Springer: Berlin, Germany, 1990. [Google Scholar]
- Henneaux, M.; Teitelboim, C. Quantization of Gauge Systems; Princeton Univ. Press: Princeton, NJ, USA, 1992. [Google Scholar]
- Bojowald, M. Canonical Gravity and Applications; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Barvinsky, A.O.; Kamenshchik, A.Y. Selection rules for the Wheeler-DeWitt equation in quantum cosmology. Phys. Rev. D
**2014**, 89, 043526. [Google Scholar] [CrossRef][Green Version] - Kamenshchik, A.Y.; Tronconi, A.; Vardanyan, T.; Venturi, G. Time in quantum theory, the Wheeler–DeWitt equation and the Born–Oppenheimer approximation. Int. J. Mod. Phys. D
**2019**, 28, 1950073. [Google Scholar] [CrossRef] - Supplementary Material Represents the Mathematica 12.0 Notebooks Containing the Calculations of $<C|{\varphi}^{2}|C>$ and $<C|{\varphi}^{4}|C>$ by the Methods A,B,C,D, and $<C|{a}^{2}|C>$ by the Method E. Available online: https://info.tuwien.ac.at/kalashnikov/supplemental.zip (accessed on 9 May 2020).
- Wheeler, J. Superspace and Nature of Quantum Geometrodynamics. In Battelle Rencontres; DeWitt, C., Wheeler, J., Eds.; Benjamin: New York, NY, USA, 1968; pp. 615–724. [Google Scholar]
- Wheeler, J.A. Superspace and the Nature of Quantum Geometrodynamics. Adv. Ser. Astrophys. Cosmol.
**1987**, 3, 27–92, [reprint of 1968 Edd.]. [Google Scholar] - Ramírez, C.; Vázquez-Báez, V. Quantum supersymmetric FRW cosmology with a scalar field. Phys. Rev. D
**2016**, 93, 043505. [Google Scholar] [CrossRef][Green Version] - Mostafazadeh, A. Quantum mechanics of Klein-Gordon-type fields and quantum cosmology. Ann. Phys. (N.Y.)
**2004**, 309, 1–48. [Google Scholar] [CrossRef][Green Version] - Halliwell, J.J. Decoherent histories analysis of minisuperspace quantum cosmology. J. Phys. Conf. Ser.
**2011**, 306, 012023. [Google Scholar] [CrossRef] - Ruffini, G. Quantization of simple parametrized systems. arXiv
**2005**, arXiv:gr-qc/0511088. [Google Scholar] - Kimura, T. Explicit description of the Zassenhaus formula. Prog. Theor. Exp. Phys.
**2017**, 2017. [Google Scholar] [CrossRef][Green Version] - Cherkas, S.L.; Kalashnikov, V.L. Quantum evolution of the universe in the constrained quasi-Heisenberg picture: From quanta to classics? Grav. Cosmol.
**2006**, 12, 126–129. [Google Scholar] - Cherkas, S.L.; Kalashnikov, V.L. An inhomogeneous toy model of the quantum gravity with the explicitly evolvable observables. Gen. Relativ. Gravit.
**2012**, 44, 3081–3102. [Google Scholar] [CrossRef][Green Version] - Cherkas, S.L.; Kalashnikov, V.L. Quantization of the inhomogeneous Bianchi I model: quasi-Heisenberg picture. Nonlinear Phenom. Complex Syst.
**2013**, 18, 1–15. [Google Scholar] - Cherkas, S.L.; Kalashnikov, V.L. Quantum Mechanics Allows Setting Initial Conditions at a Cosmological Singularity: Gowdy Model Example. Theor. Phys.
**2017**, 2, 124–135. [Google Scholar] [CrossRef] - Savchenko, V.; Shestakova, T.; Vereshkov, G. Quantum geometrodynamics in extended phase space - I. Physical problems of interpretation and mathematical problems of gauge invariance. Grav. Cosmol.
**2001**, 7, 18–28. [Google Scholar] - Vereshkov, G.; Marochnik, L. Quantum Gravity in Heisenberg Representation and Self-Consistent Theory of Gravitons in Macroscopic Spacetime. J. Mod. Phys.
**2013**, 04, 285–297. [Google Scholar] [CrossRef][Green Version] - Feynman, R.; Hibbs, A. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Faddeev, L.; Slavnov, A. Gauge Fields. Introduction to Quantum Theory; Benjamin: London, UK, 1980. [Google Scholar]
- DeWitt, B.S. Dynamical Theory in Curved Spaces. I. A Review of the Classical and Quantum Action Principles. Rev. Mod. Phys.
**1957**, 29, 377–397. [Google Scholar] [CrossRef] - Cianfrani, F.; Montani, G. Dirac prescription from BRST symmetry in FRW space-time. Phys. Rev. D
**2013**, 87. [Google Scholar] [CrossRef][Green Version] - Kocher, C.D.; McGuigan, M. Simulating 0+1 Dimensional Quantum Gravity on Quantum Computers: Mini-Superspace Quantum Cosmology and the World Line Approach in Quantum Field Theory. In Proceedings of the 2018 New York Scientific Data Summit (NYSDS), New York, NY, USA, 6–8 August 2018; pp. 1–5. [Google Scholar] [CrossRef][Green Version]
- Lloyd, S. A theory of quantum gravity based on quantum computation. arXiv
**2005**, arXiv:quant-ph/0501135. [Google Scholar] - Ganguly, A.; Behera, B.K.; Panigrahi, P.K. Demonstration of Minisuperspace Quantum Cosmology Using Quantum Computational Algorithms on IBM Quantum Computer. arXiv
**2019**, arXiv:1912.00298. [Google Scholar] - Cherkas, S.L.; Kalashnikov, V.L. Eicheons instead of Black holes. arXiv
**2020**, arXiv:2004.03947. [Google Scholar]

1 | The issue of compatibility of gauge invariance and the Schrödinger equation in connection with gravity quantization is discussed in [9]. |

2 | |

3 | One has to note that the methods considered are not the exclusive methods describing the quantum evolution of the universe. For instance, one could take a scale factor or a scalar field [25] as the “time variable.” |

**Figure 1.**The mean value of the square of the scalar field with respect to the wave packet Equation (18).

**Figure 2.**The mean value of ${\widehat{\varphi}}^{4}$ over the wave packet Equation (18) for methods. A, D: solid line and methods B, C: dashed line.

**Figure 3.**The illustration that different methods could have different Hilbert spaces for producing the same set of the mean values for the arbitrarily given operators. Still, there should be correspondence between the state $C\left(k\right)$ of the Hilbert space 1 and the state $\tilde{C}\left(k\right)$ of the Hilbert space 2 producing the same mean values.

**Table 1.**Comparison of the mean values calculated by the different methods. Capital letters denote a method. A plus implies that the values obtained by the different methods coincide. Crosses of two types in a circle mean that the values obtained at least by two different methods coincide.

Method | A | B | C | D | E |
---|---|---|---|---|---|

a | + | + | + | + | |

${a}^{2}$ | + | + | + | + | + |

${\widehat{\varphi}}^{2}$ | + | + | + | + | |

${\widehat{\varphi}}^{4}$ | ⊕ | ⊗ | ⊗ | ⊕ | |

${\widehat{\varphi}}^{6}$ | ⊕ | ⊕ | |||

$a{\widehat{\varphi}}^{2}+{\widehat{\varphi}}^{2}a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$ | ⊕ | ⊕ |

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Cherkas, S.; Kalashnikov, V. Evidence of Time Evolution in Quantum Gravity. *Universe* **2020**, *6*, 67.
https://doi.org/10.3390/universe6050067

**AMA Style**

Cherkas S, Kalashnikov V. Evidence of Time Evolution in Quantum Gravity. *Universe*. 2020; 6(5):67.
https://doi.org/10.3390/universe6050067

**Chicago/Turabian Style**

Cherkas, Sergey, and Vladimir Kalashnikov. 2020. "Evidence of Time Evolution in Quantum Gravity" *Universe* 6, no. 5: 67.
https://doi.org/10.3390/universe6050067