# Bubble Nucleation from a de Sitter–Planck Background with Quantum Boltzmann Statistics

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## Abstract

**:**

## 1. Introduction

## 2. The Multiverse in the de Sitter–Planck Background with Quantum Boltzmann Statistics

- (1)
- The particle is initially massless (namely, it corresponds to the vacuum state);
- (2)
- Its localization in an interaction event requires an amount of energy equal to the ratio of $\hslash $ and an opportune duration corresponding to the transition hypersurface ${\mathsf{\Sigma}}_{\frac{1}{\sqrt{\mathsf{\Lambda}}}}$ generating the appearance of time; thus, the fluctuations of the quantum vacuum are associated with the appearance of a particle take place;
- (3)
- The particle self-interacts for a duration of $\hslash /{M}_{dim}{c}^{2}$, and therefore on a scale of lengths equal to $\hslash /{M}_{dim}c$. The total mass of the real particle is therefore the sum of the “bare” mass associated with the bubbles of the de Sitter–Planck vacuum and the $\epsilon /{c}^{2}$ mass derived from this self-interaction.

## 3. Bubbles Geometry and the Generalized Uncertainty Relations

## 4. From the Generalized Uncertainty Relations to Sub-Planckian Black Holes

## 5. Thermodynamics of a Quantum Black Hole

## 6. Casimir Energy and Cosmological Wormholes

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Coleman, S.; de Luccia, F. Gravitational effects on and of vacuum decay. Phys. Rev. D
**1980**, 21, 3305–3315. [Google Scholar] [CrossRef] [Green Version] - Lindley, D. The appearance of bubbles in de Sitter space. Nucl. Phys. B
**1984**, 236, 522–546. [Google Scholar] [CrossRef] - Kanno, S.; Soda, J. Exact Coleman—DE Luccia instantons. Int. J. Mod. Phys. D
**2012**, 21, 1250040. [Google Scholar] [CrossRef] - Fialko, O.; Opanchuk, B.; Sidorov, A.I.; Drummond, P.D.; Brand, J. Fate of the false vacuum: Towards realization with ultra-cold atoms. Europhys. Lett.
**2015**, 110, 56001. [Google Scholar] [CrossRef] [Green Version] - Abel, S.; Spannowsky, M. Observing the fate of the false vacuum with a quantum laboratory. PRX Quantum
**2021**, 2, 010349. [Google Scholar] [CrossRef] - Milsted, A.; Liu, J.; Preskill, J.; Vidal, G. Collisions of False-Vacuum Bubble Walls in a Quantum Spin Chain. PRX Quantum
**2022**, 3, 020316. [Google Scholar] [CrossRef] - Brennan, T.D.; Carta, F.; Vafa, C. The String Landscape, the Swampland, and the Missing Corner. arXiv
**2017**, arXiv:1711.00864. [Google Scholar] - Cicoli, M.; De Alwis, S.; Maharana, A.; Muia, F.; Quevedo, F. De Sitter vs. Quintessence in String Theory. Fortschr. Phys.
**2019**, 67, 1800079. [Google Scholar] [CrossRef] [Green Version] - Banerjee, S.; Danielsson, U.; Giri, S. Bubble needs strings. J. High Energy Phys.
**2021**, 2021, 250. [Google Scholar] [CrossRef] - Carr, B.J. The black hole uncertainty principle correspondence. arXiv
**2014**, arXiv:1402.1427. [Google Scholar] - Carr, B.J.; Mureika, J.R.; Nicolini, P. Sub-Planckian black holes and the generalized uncertainty principle. JHEP
**2015**, 2015, 52. [Google Scholar] [CrossRef] [Green Version] - Calmet, X.; Carr, B.J.; Winstanley, E. Quantum Black Holes; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Adler, R.J.; Santiago, D.I. On gravity and the uncertainty principle. Mod. Phys. Lett. A
**1999**, 14, 1371–1381. [Google Scholar] [CrossRef] - Adler, R.J.; Chen, P.; Santiago, D.I. The generalized uncertainty principle and black hole remnants. Gen. Rel. Grav.
**2001**, 33, 2101–2108. [Google Scholar] [CrossRef] - Chen, P.; Adler, R.J. Black hole remnants and dark matter. Nucl. Phys. B—Proc. Suppl.
**2003**, 124, 103–106. [Google Scholar] [CrossRef] [Green Version] - Adler, R.J. Six easy roads to the Planck scale. Am. J. Phys.
**2010**, 78, 925–932. [Google Scholar] [CrossRef] [Green Version] - Spallucci, E.; Smailagic, A. Horizons and the wave function of planckian quantum black holes. arXiv
**2021**, arXiv:2103.03947. [Google Scholar] [CrossRef] - Alharthy, A.; Kassandrov, V.V. On a Crucial Role of Gravity in the Formation of Elementary Particles. Universe
**2020**, 6, 193. [Google Scholar] [CrossRef] - Anderson, P.R.; Mottola, E. On the Instability of Global de Sitter Space to Particle Creation. Phys. Rev. D
**2014**, 89, 104038. [Google Scholar] [CrossRef] [Green Version] - Anderson, P.; Mottola, E. Quantum Vacuum Instability of ‘Eternal’ de Sitter Space. Phys. Rev. D
**2014**, 89, 104039. [Google Scholar] [CrossRef] [Green Version] - Rajaraman, A. de Sitter Space is Unstable in Quantum Gravity. Phys. Rev. D
**2016**, 94, 125025. [Google Scholar] [CrossRef] [Green Version] - Anderson, P.; Mottola, E.; Sanders, D.H. Decay of the de Sitter Vacuum. Phys. Rev. D
**2018**, 97, 065016. [Google Scholar] [CrossRef] [Green Version] - Matsui, H. Instability of De Sitter Spacetime induced by Quantum Conformal Anomaly. JCAP
**2019**, 1, 3. [Google Scholar] [CrossRef] - Licata, I.; Chiatti, L. Event Based quantum mechanics. Symmetry
**2019**, 11, 181. [Google Scholar] [CrossRef] [Green Version] - Feleppa, F.; Licata, I.; Corda, C. Hartle-Hawking boundary conditions as Nucleation by de Sitter Vacuum. Phys. Dark Universe
**2019**, 26, 100381. [Google Scholar] [CrossRef] [Green Version] - Robles-Pérez, S.; Gonzáles-Díaz, P.F. Quantum state of the multiverse. Phys. Rev. D
**2010**, 81, 083529. [Google Scholar] [CrossRef] [Green Version] - Greenberg, O.W. Example of Infinite Statistics. Phys. Rev. Lett.
**1990**, 64, 705–708. [Google Scholar] [CrossRef] - Arzano, M. Quantum Fields, Non-Locality and Quantum Group Symmetries. Phys. Rev. D
**2008**, 77, 025013. [Google Scholar] [CrossRef] [Green Version] - Balachandran, A.P.; Pinzul, A.; Qureshi, B.A.; Vaidya, S. S-Matrix on the Moyal Plane: Locality versus Lorentz Invariance. arXiv
**2007**, arXiv:0708.1379. [Google Scholar] - Arzano, M.; Kephart, T.W.; Ng, Y.J. From spacetime foam to holographic foam cosmology. Phys. Lett. B
**2007**, 649, 243–246. [Google Scholar] [CrossRef] [Green Version] - Ng, Y.J. Spacetime foam: From entropy and holography to infinite statistics and non-locality. Entropy
**2008**, 10, 441–461. [Google Scholar] [CrossRef] [Green Version] - Ng, Y.J. Holographic quantum foam. arXiv
**2010**, arXiv:1001.0411v1. [Google Scholar] - Ng, Y.J. Various facets of spacetime foam. arXiv
**2011**, arXiv:1102.4109. [Google Scholar] - Bouhmadi-Lopez, M.; Kramer, M.; Morais, J.; Robles-Perez, S. What if? Exploring the multiverse through Euclidean whormholes. Eur. Phys. J. C
**2017**, 77, 718. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bouhmadi-Lopez, M.; Kramer, M.; Morais, J.; Robles-Perez, S. The third quantization: To tunnel or not to tunnel? Galaxies
**2018**, 6, 19. [Google Scholar] [CrossRef] [Green Version] - BLee, H.; Lee, W. The vacuum bubbles in de Sitter background and black hole pair creation. Class. Quant. Grav.
**2009**, 26, 225002. [Google Scholar] - Hebecker, A.; Mikhail, T.; Soler, P. Euclidean wormholes, baby universes, and their impact on particle physics and cosmology. Front. Astron. Space Sci.
**2018**, 5, 35. [Google Scholar] [CrossRef] - Petruzziello, L.; Illuminati, F. Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale. Nat. Commun.
**2021**, 12, 4449. [Google Scholar] [CrossRef] - Amelino-Camelia, G. Quantum gravity phenomenology. arXiv
**2008**, arXiv:0806.0339. [Google Scholar] - Rovelli, C. A new look at loop quantum gravity. arXiv
**2010**, arXiv:1004.1780. [Google Scholar] [CrossRef] [Green Version] - Fiscaletti, D. About non-local granular space-time foam as ultimate arena at the Planck scale. In Space-Time Geometry and Quantum Events; Licata, I., Ed.; Nova Science Publishers: New York, NY, USA, 2014. [Google Scholar]
- Hooft’t, G. How does god play dice? (Pre-)determinism at the Planck scale. arXiv
**2001**, arXiv:0104219. [Google Scholar] - Hooft’t, G. Quantum mechanics and determinism. arXiv
**2001**, arXiv:0105105. [Google Scholar] - Hooft’t, G. The fate of the quantum. arXiv
**2013**, arXiv:1308.1007. [Google Scholar] - Hooft’t, G. The Cellular Automaton Interpretation of Quantum Mechanics; Springer: Heidelberg, Germany, 2016. [Google Scholar]
- Licata, I. Quantum mechanics interpretation on Planck scale. Ukr. J. Phys.
**2020**, 65, 17–30. [Google Scholar] [CrossRef] [Green Version] - Hawking, S.W. Spacetime foam. Nucl. Phys. B
**1978**, 144, 349. [Google Scholar] [CrossRef] - Vasileiou, V.; Granot, J.; Piran, T.; Amelino-Camelia, G.A. Planck-scale limit on spacetime fuzziness and stochastic Lorentz invariance violation. Nat. Phys.
**2015**, 11, 344–346. [Google Scholar] [CrossRef] - Licata, I.; Chiatti, L. Timeless approach to quantum jumps. Quanta
**2015**, 4, 10–26. [Google Scholar] [CrossRef] [Green Version] - Chiatti, L.; Licata, I. Particle model from quantum foundations. Quantum Stud. Math Found.
**2017**, 4, 181–204. [Google Scholar] [CrossRef] [Green Version] - Chiatti, L. The transaction as a quantum concept. arXiv
**2012**, arXiv:1204.6636. [Google Scholar] - Rovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Gambini, R.; Pullin, J. Holography in Spherically Symmetric Loop Quantum Gravity. Int. J. Mod. Phys. D
**2008**, 17, 545–549. [Google Scholar] [CrossRef] [Green Version] - Modesto, L. Disappearance of black hole singularity in quantum gravity. Phys. Rev. D
**2004**, 70, 124009. [Google Scholar] [CrossRef] [Green Version] - Modesto, L. Loop quantum black hole. Class. Quant. Grav.
**2006**, 23, 5587–5601. [Google Scholar] [CrossRef] - Modesto, L. Black hole interior from loop quantum gravity. Adv. High Energy Phys.
**2008**, 2008, 459290. [Google Scholar] [CrossRef] [Green Version] - Modesto, L. Semiclassical loop quantum black hole. Int. J. Theor. Phys.
**2010**, 49, 1649–1683. [Google Scholar] [CrossRef] [Green Version] - Modesto, L. Space-time structure of loop quantum black hole. arXiv
**2008**, arXiv:0811.2196. [Google Scholar] - Modesto, L.; Prémont-Schwarz, I. Self-dual black holes in LQG: Theory and phenomenology. Phys. Rev. D
**2009**, 80, 064041. [Google Scholar] [CrossRef] - Dolce, D. Introduction to the Quantum Theory of Elementary Cycles: The emergence of space, time and quantum. arXiv
**2017**, arXiv:1707.00677. [Google Scholar] - Dolce, D. New stringy physics beyond quantum mechanics from the Feynman path integral. arXiv
**2022**, arXiv:2106.05167. [Google Scholar] [CrossRef] - Fontana, R.D.B.; de Oliveira, J.; Pavan, A.B. Dynamical evolution of non-minimally coupled scalar field in spherically symmetric de Sitter spacetimes. Eur. Phys. J. C
**2019**, 79, 1–16. [Google Scholar] [CrossRef] - Hawking, S.W. Black hole explosions? Nature
**1974**, 248, 30–31. [Google Scholar] [CrossRef] - Mann, R.B.; Shiekh, A.; Tarasov, L. Classical and quantum properties of two-dimensional black holes. Nucl. Phys. B
**1990**, 341, 134. [Google Scholar] [CrossRef] - Mureika, J.R.; Nicolini, P. Self-completeness and spontaneous dimensional reduction. Eur. Phys. J. Plus
**2013**, 128, 78. [Google Scholar] [CrossRef] - Strominger, A.; Vafa, C. Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B
**1996**, 379, 99–104. [Google Scholar] [CrossRef] [Green Version] - Rovelli, C. Black Hole Entropy from Loop Quantum Gravity. Phys. Rev. Lett.
**1996**, 77, 3288–3291. [Google Scholar] [CrossRef] - Nicolini, P.; Spallucci, E. Holographic Screens in Ultraviolet Self-Complete Quantum Gravity. Adv. High Energy Phys.
**2014**, 2014, 805684. [Google Scholar] [CrossRef] - Kubizňák, D.; Mann, R.B.; Teo, M. Black hole chemistry: Thermodynamics with Lambda. Class. Quantum Gravity
**2017**, 34, 063001. [Google Scholar] [CrossRef] [Green Version] - Gregory, R.; Kastor, D.; Traschen, J. Black hole thermodynamics with dynamical lambda. J. High Energy Phys.
**2017**, 2017, 118. [Google Scholar] [CrossRef] [Green Version] - Sorge, F. Casimir effect around an Ellis wormhole. Int. J. Mod. Phys. D
**2019**, 29, 2050002. [Google Scholar] [CrossRef] - Ellis, H.G. Ether flow through a drainhole: A particle model in general relativity. J. Math. Phys.
**1973**, 14, 104–118. [Google Scholar] [CrossRef] - Santos, A.C.L.; Muniz, C.R.; Oliveira, L.T. Casimir effect nearby and through a cosmological wormhole. arXiv
**2021**, arXiv:2103.03368. [Google Scholar] [CrossRef] - Sorge, F.; Wilson, J.H. Casimir effect in free-fall towards a Schwarzschild black hole. Phys. Rev. D
**2019**, 100, 105007. [Google Scholar] [CrossRef] [Green Version] - Tamburini, F.; Licata, I. General Relativistic Wormhole Connections from Planck-Scales and the ER = EPR Conjecture. Entropy
**2020**, 22, 3. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Geroch, R.; Hartle, J.B. Computability and physical theories. Found. Phys.
**1986**, 16, 533–550. [Google Scholar] [CrossRef] [Green Version] - Hartle, J.B.; Hawking, S. Wave Function of the Universe. Phys. Rev. D
**1983**, 28, 2960–2975. [Google Scholar] [CrossRef] - Hartle, J.B.; Hawking, S.; Hertog, T. No-Boundary Measure of the Universe. Phys. Rev. Lett.
**2008**, 100, 201301. [Google Scholar] [CrossRef] [Green Version] - Van Raamsdonk, M. Building up spacetime with quantum entanglement. Gen. Relativ. Gravit.
**2011**, 42, 2323–2329. [Google Scholar] [CrossRef] - Kauffmann, S. On Quantum Gravity If Non-Locality Is Fundamental. Entropy
**2022**, 24, 554. [Google Scholar] [CrossRef] [PubMed] - Arias, C.; Diaz, F.; Sundell, P. De Sitter space and entanglement. Class. Quantum Grav.
**2020**, 37, 015009. [Google Scholar] [CrossRef] [Green Version] - Maldacena, J.; Pimentel, G.L. Entanglement entropy in de Sitter space. J. High Energy Phys.
**2013**, 2013, 38. [Google Scholar] [CrossRef] [Green Version] - Coleman, S. Aspects of Symmetry: Selected Erice Lectures; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Preparata, G.; Xue, S.-S. Do we live on a lattice? Fermion masses from the Planck mass. Phys. Lett. B.
**1991**, 264, 35–38. [Google Scholar] [CrossRef] - Tamburini, F.; Thidé, B.; Licata, I.; Bouchard, F.; Karimi, E. Majorana bosonic quasiparticles from twisted photons in free space. Phys. Rev. A
**2021**, 103, 033505. [Google Scholar] [CrossRef] - Jaeger, G. Localizability and elementary particles. J. Phys. Conf. Ser.
**2020**, 1638, 012010. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Fiscaletti, D.; Licata, I.; Tamburini, F.
Bubble Nucleation from a de Sitter–Planck Background with Quantum Boltzmann Statistics. *Symmetry* **2022**, *14*, 2297.
https://doi.org/10.3390/sym14112297

**AMA Style**

Fiscaletti D, Licata I, Tamburini F.
Bubble Nucleation from a de Sitter–Planck Background with Quantum Boltzmann Statistics. *Symmetry*. 2022; 14(11):2297.
https://doi.org/10.3390/sym14112297

**Chicago/Turabian Style**

Fiscaletti, Davide, Ignazio Licata, and Fabrizio Tamburini.
2022. "Bubble Nucleation from a de Sitter–Planck Background with Quantum Boltzmann Statistics" *Symmetry* 14, no. 11: 2297.
https://doi.org/10.3390/sym14112297