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On A. D. Sakharov’s Hypothesis of Cosmological Transitions with Changes in the Signature of the Metric^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

“…one would expect that quantum gravity would allow all possible topologies of spacetime… It is precisely these other topologies that seem to give the most interesting effects.”

“It is conjectured that there exist states of the physical continuum which include regions with different signatures of the metric…”

“Differences in the signature structure… appear just as natural as differences in the topological structure.”

## 2. The Definition of the Path Integral over Spatial Regions

## 3. Non-Trivial Topology and Gauge Invariance

## 4. Time Coordinates and Evolution in Time

## 5. Discussion

“It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them which give us the key to the understanding of the phenomena of Nature.”

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hawking, S.W. The path-integral approach to quantum gravity. In General Relativity. An Einstein Centenary Survey; Hawking, S.W., Israel, W., Eds.; Cambridge University Press: Cambridge, UK, 1979; pp. 746–789. [Google Scholar]
- Sakharov, A.D. Cosmological transitions with changes in the signature of the metric. Zh. Eksp. Teor. Fiz.
**1984**, 87, 375–383. [Google Scholar] - Einstein, A. Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie. Sitzungsberichte Preußische Akad. Wissenschaften
**1917**, 1, 142–152. [Google Scholar] - Bondi, H.; Gold, T. The steady-state theory of the expanding Universe. Mon. Not. R. Astron. Soc.
**1948**, 108, 252–270. [Google Scholar] [CrossRef] - Hoyle, F. A new model for the expanding Universe. Mon. Not. R. Astron. Soc.
**1948**, 108, 372–382. [Google Scholar] [CrossRef] [Green Version] - Bronstein, M.P. Quantization of gravitational waves. Zh. Eksp. Tear. Fiz.
**1936**, 6, 195–236. [Google Scholar] - Ellis, G.; Sumeruk, A.; Coule, D.; Hellaby, C. Change of signature in classical relativity. Class. Quantum Grav.
**1992**, 9, 1535–1554. [Google Scholar] [CrossRef] - Hayward, S.A. Junction conditions for signature change. arXiv
**1993**, arXiv:gr-qc/9303034. [Google Scholar] - Carlini, A.; Greensite, J. Why is spacetime Lorentzian? Phys. Rev. D
**1994**, 49, 866–878. [Google Scholar] [CrossRef] [Green Version] - KrieIe, M.; Martin, J. Black holes, cosmological singularities and change of signature. Class. Quantum Grav.
**1995**, 12, 503–511. [Google Scholar] [CrossRef] [Green Version] - Dray, T.; Ellis, G.; Hellaby, C.; Manogue, C.A. Gravity and Signature Change. Gen. Rel. Grav.
**1997**, 29, 591–597. [Google Scholar] [CrossRef] [Green Version] - Vilenkin, A. Birth of inflationary universes. Phys. Rev. D
**1983**, 27, 2848–2855. [Google Scholar] [CrossRef] - Hartle, J.B.; Hawking, S.W. Wave function of the Universe. Phys. Rev. D
**1983**, 28, 2960–2975. [Google Scholar] [CrossRef] - Savchenko, V.A.; Shestakova, T.P.; Vereshkov, G.M. Quantum geometrodynamics of the Bianchi IX model in extended phase space. Int. J. Mod. Phys. A
**1999**, 14, 4473–4490. [Google Scholar] [CrossRef] - Savchenko, V.A.; Shestakova, T.P.; Vereshkov, G.M. The exact cosmological solution to the dynamical equations for the Bianchi IX model. Int. J. Mod. Phys. A
**2000**, 15, 3207–3220. [Google Scholar] [CrossRef] - Savchenko, V.A.; Shestakova, T.P.; Vereshkov, G.M. Quantum geometrodynamics in extended phase space—I. Physical problems of interpretation and mathematical problems of gauge invariance. Gravit. Cosmol.
**2001**, 7, 18–28. [Google Scholar] - Savchenko, V.A.; Shestakova, T.P.; Vereshkov, G.M. Quantum geometrodynamics in extended phase space—II. The Bianchi IX model. Gravit. Cosmol.
**2001**, 7, 102–116. [Google Scholar] - Shestakova, T.P. Wave function of the Universe, path integrals and gauge invariance. Gravit. Cosmol.
**2019**, 25, 289–296. [Google Scholar] [CrossRef] [Green Version] - Feynman, R.P. Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys.
**1948**, 20, 367–387. [Google Scholar] [CrossRef] [Green Version] - Cheng, K.S. Quantization of a general dynamical system by Feynman’s path integration formulation. J. Math. Phys.
**1972**, 13, 1723–1726. [Google Scholar] [CrossRef] - Weinberg, S. The cosmological constant problem. Rev. Mod. Phys.
**1989**, 61, 1–23. [Google Scholar] [CrossRef] - Faddeev, L.D.; Popov, V.N. Feynman diagrams for the Yang—Mills field. Phys. Lett. B
**1967**, 25, 29–30. [Google Scholar] [CrossRef] - Fradkin, E.S.; Vilkovisky, G.A. Quantization of relativistic systems with constraints. Phys. Lett. B
**1975**, 55, 224–226. [Google Scholar] [CrossRef] - Batalin, I.A.; Vilkovisky, G.A. Relativistic S-matrix of dynamical systems with boson and fermion constraints. Phys. Lett. B
**1977**, 69, 309–312. [Google Scholar] [CrossRef] - Fradkin, E.S.; Fradkina, T.E. Quantization of relativistic systems with boson and fermion first- and second-class constraints. Phys. Lett. B
**1978**, 72, 343–348. [Google Scholar] [CrossRef] - Halliwell, J.J. Derivation of the Wheeler—DeWitt equation from a path integral for minisuperspace models. Phys. Rev. D
**1988**, 38, 2468–2481. [Google Scholar] [CrossRef] [PubMed] - 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] - Shestakova, T.P. Changing the Hilbert space structure as a consequence of gauge transformations in “extended phase space” version of quantum geometrodynamics. In Physical Interpretations of Relativity Theory, Proceedings of International Scientific Meeting, Moscow, Russia, 4–7 July 2005; Duffy, M.C., Gladyshev, V.O., Morozov, A.N., Rowlands, P., Eds.; Bauman Moscow State Technical University Publishing: Moscow, Russia, 2005; pp. 26–34. [Google Scholar]
- Sakharov, A.D. Cosmological models of the Universe with reversal of time’s arrow. Zh. Eksp. Teor. Fiz.
**1980**, 79, 689–693. [Google Scholar] [CrossRef] [Green Version] - Kiefer, C. Conceptual issues in quantum cosmology. In Towards Quantum Gravity, Proceeding of the XXXV International Winter School on Theoretical Physics, Polanica, Poland, 2–11 February 1999; Kowalski-Glikman, J., Ed.; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2000; Volume 541, pp. 158–187. [Google Scholar]
- Shestakova, T.P. Is the Copenhagen interpretation inapplicable to quantum cosmology? Universe
**2020**, 6, 128. [Google Scholar] [CrossRef] - Sakharov, A.D. Violation of CP invariance, C asymmetry, and baryon asymmetry of the Universe. Pis’ma Zh. Eksp. Teor. Fiz.
**1967**, 5, 32–35. [Google Scholar] - Ehrenfest, P. In that way does it become manifest in the fundamental laws of physics that space has three dimensions? Proc. Amst. Acad. Sci.
**1917**, 20, 200–209. [Google Scholar] - Gott, J.R.; Alpert, M. General relativity in a (2+1)-dimensional space-time. Gen. Rel. Grav.
**1984**, 16, 243–247. [Google Scholar] [CrossRef] - Einstein, A. On the Method of Theoretical Physics; Clarendon Press: Oxford, UK, 1933. [Google Scholar]
- Hawking, S.W. Information loss in black holes. Phys. Rev. D
**2005**, 72, 084013. [Google Scholar] [CrossRef] [Green Version] - Penrose, R. Singularities and time-asymmetry. In General Relativity. An Einstein Centenary Survey; Hawking, S.W., Israel, W., Eds.; Cambridge University Press: Cambridge, UK, 1979; pp. 581–638. [Google Scholar]
- Penrose, R. Shadows of the Mind: An Approach to the Missing Science of Consciousness; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- Prigogine, I. The End of Certainty: Time, Chaos and the New Laws of Nature; The Free Press: New York, NY, USA; London, UK; Toronto, ON, Canada; Sidney, Australia; Singapore, 1997. [Google Scholar]
- Prigogine, I. Time, Structure and Fluctuation (Nobel prize lecture). Science
**1978**, 201, 777–785. [Google Scholar] [CrossRef] [PubMed] [Green Version]

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**MDPI and ACS Style**

Shestakova, T.P.
On A. D. Sakharov’s Hypothesis of Cosmological Transitions with Changes in the Signature of the Metric. *Universe* **2021**, *7*, 151.
https://doi.org/10.3390/universe7050151

**AMA Style**

Shestakova TP.
On A. D. Sakharov’s Hypothesis of Cosmological Transitions with Changes in the Signature of the Metric. *Universe*. 2021; 7(5):151.
https://doi.org/10.3390/universe7050151

**Chicago/Turabian Style**

Shestakova, Tatyana P.
2021. "On A. D. Sakharov’s Hypothesis of Cosmological Transitions with Changes in the Signature of the Metric" *Universe* 7, no. 5: 151.
https://doi.org/10.3390/universe7050151