# The Relation between Wavefunction and 3D Space Implies Many Worlds with Local Beables and Probabilities

## Abstract

**:**

## 1. Introduction

**Question 1.**

**Question 2.**

**Question 3.**

**Question 4.**

**Question 5.**

**Question 6.**

**Question 7.**

**Question 8.**

## 2. The Wavefunctional and the 3D Space

**Postulate 1**(Unitary evolution)

**.**

## 3. The Wavefunction’S Ontology: A Densitized Set of Classical Worlds

## 4. The World Appears Classical at the Macroscopic Level

**Definition 1.**

**Postulate 2**(Macroclassicality)

**.**

**Postulate 3**(Microstates)

**.**

- is more general, including measurements as particular cases,
- avoids presuming whether the wavefunction collapses or not,
- relates the macrostates to microstates of the form $|\varphi \rangle $, where $\varphi \in \mathcal{C}$ have clear relations with 3D space.

**Rule 1**(Born rule)

**.**

## 5. Naive Counting Gives the Born Rule in the Continuous Limit

**Theorem 1.**

**Proof.**

## 6. Wavefunction Collapse Is Inconsistent with Our Derivation of the Born Rule

**Proposition 1.**

## 7. What Should Be Counted as a World?

## 8. The 3D Geometry as the Preferred Basis

**Observation 1.**

## 9. The 3D Geometry and the Branching Structure

## 10. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Wavefunction as an Object in 3D Space

## Appendix B. The Existence of a Measure on the Configuration Space of Classical Fields

## Appendix C. Possible Worlds Should Form a Basis

**Proof of Proposition 1.**

## References

- Bell, J. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Everett, H. “Relative State” Formulation of Quantum Mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462. [Google Scholar] [CrossRef] - de Witt, B.; Graham, N. (Eds.) The Many-Worlds Interpretation of Quantum Mechanics; Princeton Series in Physics; Princeton University Press: Princeton, NJ, USA, 1973. [Google Scholar]
- Wallace, D. The Emergent Multiverse: Quantum Theory According to the Everett Interpretation; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Hatfield, B. Quantum Field Theory of Point Particles and Strings; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1955. [Google Scholar]
- Dirac, P. The Principles of Quantum Mechanics; Oxford University Press: Oxford, UK, 1958. [Google Scholar]
- McQueen, K.; Vaidman, L. In defence of the self-location uncertainty account of probability in the many-worlds interpretation. Stud. Hist. Philos. Mod. Phys.
**2019**, 66, 14–23. [Google Scholar] - Vaidman, L. Why the Many-Worlds Interpretation? Quantum Rep.
**2022**, 4, 264–271. [Google Scholar] [CrossRef] - Stoica, O.C. Born rule: Quantum probability as classical probability. arXiv
**2022**, arXiv:2209.08621. [Google Scholar] - Schulman, L. Time’s Arrows and Quantum Measurement; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- ’t Hooft, G. The Cellular Automaton Interpretation of Quantum Mechanics; Springer: Cham, Switzerland; Heidelberg, Germany; New York, NY, USA; Dordrecht, The Netherlands; London, UK, 2016; Volume 185. [Google Scholar]
- Stoica, O.C. The post-determined block universe. Quantum Stud. Math. Found.
**2021**, 8, 69–101. [Google Scholar] [CrossRef] - Stoica, O.C. Quantum Measurement and Initial Conditions. Int. J. Theor. Phys.
**2016**, 55, 1897–1911. [Google Scholar] [CrossRef] - Stoica, O.C. Global and local aspects of causality in quantum mechanics. In EPJ Web of Conferences, TM 2012—The Time Machine Factory [unspeakable, speakable] on Time Travel in Turin; EDP Sciences: Les Ulis, France, 2013; Volume 58, p. 01017. [Google Scholar]
- Saunders, S. Branch-counting in the Everett interpretation of quantum mechanics. Proc. R. Soc. Lond. Ser. A
**2021**, 477, 20210600. [Google Scholar] - Deutsch, D. Quantum theory of probability and decisions. Proc. R. Soc. A—Math. Phys. Eng. Sci.
**1999**, 455, 3129–3137. [Google Scholar] [CrossRef] - Wallace, D. Quantum probability and decision theory, revisited. arXiv
**2002**, arXiv:quant-ph/0211104. [Google Scholar] - Vaidman, L. Probability in the many-worlds interpretation of quantum mechanics. In Probability in Physics; Ben-Menahem, Y., Hemmo, M., Eds.; Springer: Berlin, Geramny, 2012; Volume XII, pp. 299–311. [Google Scholar]
- DeWitt, B. Quantum theory of gravity. I. The canonical theory. Phys. Rev.
**1967**, 160, 1113. [Google Scholar] - Arnowitt, R.; Deser, S.; Misner, C.W. The Dynamics of General Relativity. In Gravitation: An Introduction to Current Research; Witten, L., Ed.; Wiley: New York, NY, USA, 1962; pp. 227–264. [Google Scholar]
- Page, D.; Wootters, W. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D
**1983**, 27, 2885. [Google Scholar] [CrossRef] - Sorkin, R. Spacetime and causal sets. In Relativity and Gravitation: Classical and Quantum; World Scienfic: Singapore, 1990; pp. 150–173. [Google Scholar]
- Regge, T. General relativity without coordinates. Il Nuovo Cimento (1955–1965)
**1961**, 19, 558–571. [Google Scholar] - Loll, R. Quantum gravity from causal dynamical triangulations: A review. Class. Quant. Grav.
**2019**, 37, 013002. [Google Scholar] - Rovelli, C.; Smolin, L. Spin networks and quantum gravity. Phy. Rev. D
**1995**, 52, 5743. [Google Scholar] - Ashtekar, A.; Bianchi, E. A short review of loop quantum gravity. Rep. Prog. Phys.
**2021**, 84, 042001. [Google Scholar] - Smolin, L. The Case for Background Independence. In The Structural Foundations of Quantum Gravity; Rickles, D., French, S., Saatsi, J., Eds.; Clarendon Press: Oxford, UK, 2006; pp. 196–239. [Google Scholar]
- Anandan, J. Interference of geometries in quantum gravity. Gen. Relat. Grav.
**1994**, 26, 125–133. [Google Scholar] [CrossRef] - Penrose, R. On gravity’s role in quantum state reduction. Gen. Relat. Grav.
**1996**, 28, 581–600. [Google Scholar] [CrossRef] - Stoica, O.C. Background freedom leads to many-worlds with local beables and probabilities. arXiv
**2022**, arXiv:2209.08623. [Google Scholar] - Stoica, O.C. The Friedmann-Lemaître-Robertson-Walker Big Bang Singularities are Well Behaved. Int. J. Theor. Phys.
**2016**, 55, 71–80. [Google Scholar] [CrossRef] - Stoica, O.C. Singular General Relativity—Ph.D. Thesis; Minkowski Institute Press: Montreal, QC, Canada, 2013. [Google Scholar]
- Stoica, O.C. Why the wavefunction already is an object on space. arXiv
**2021**, arXiv:2111.14604. [Google Scholar] - Klein, F. Vergleichende Betrachtungen über neuere geometrische Forschungen. Math. Ann.
**1893**, 43, 63–100. [Google Scholar] - Wigner, E. Gruppentheorie und ihre Anwendung auf die Quanten Mechanik der Atomspektren; Friedrich Vieweg und Sohn: Braunschweig, Germany, 1931. [Google Scholar]
- Wigner, E. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra; Academic Press: New York, NY, USA, 1959. [Google Scholar]
- Bargmann, V. Note on Wigner’s theorem on symmetry operations. J. Math. Phys.
**1964**, 5, 862–868. [Google Scholar] - Stoica, O.C. Representation of the wave function on the three-dimensional space. Phys. Rev. A
**2019**, 100, 042115. [Google Scholar] [CrossRef] [Green Version] - Hunt, B.; Sauer, T.; Yorke, J. Prevalence: A translation-invariant “almost every” on infinite-dimensional spaces. Bull. Am. Math. Soc.
**1992**, 27, 217–238. [Google Scholar] - Bekenstein, J. Universal upper bound on the entropy-to-energy ratio for bounded systems. Phys. Rev. D
**1981**, 23, 287. [Google Scholar] [CrossRef] - Bekenstein, J. How does the entropy/information bound work? Found. Phys.
**2005**, 35, 1805–1823. [Google Scholar] - Stoica, O.C. Does quantum mechanics requires “conspiracy”? arXiv
**2022**, arXiv:2209.13275. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stoica, O.C.
The Relation between Wavefunction and 3D Space Implies Many Worlds with Local Beables and Probabilities. *Quantum Rep.* **2023**, *5*, 102-115.
https://doi.org/10.3390/quantum5010008

**AMA Style**

Stoica OC.
The Relation between Wavefunction and 3D Space Implies Many Worlds with Local Beables and Probabilities. *Quantum Reports*. 2023; 5(1):102-115.
https://doi.org/10.3390/quantum5010008

**Chicago/Turabian Style**

Stoica, Ovidiu Cristinel.
2023. "The Relation between Wavefunction and 3D Space Implies Many Worlds with Local Beables and Probabilities" *Quantum Reports* 5, no. 1: 102-115.
https://doi.org/10.3390/quantum5010008