# How Everett Solved the Probability Problem in Everettian Quantum Mechanics

## Abstract

**:**

## 1. The Probability Problem

#### Probabilities of What?

[T]here is no sense in which [decoherence] phenomena lead to a naturally discrete branching process: as we have seen in studying quantum chaos, while a branching structure can be discerned in such systems, it has no natural ‘grain’. To be sure, by choosing a certain discretization of (configuration-)space and time, a discrete branching structure will emerge, but a finer or coarser choice would also give branching. And there is no ‘finest’ choice of branching structure: as we fine-grain our decoherent history space, we will eventually reach a point where interference between branches ceases to be negligible, but there is no precise point where this occurs. As such, the question ‘How many branches are there?’ does not, ultimately, make sense.([17], pp. 99–100)

## 2. The Galton Board

The ensuing, most likely state, which we call that of the Maxwellian velocity distribution, since it was Maxwell who first found the mathematical expression in a special case, is not an outstanding singular state, opposite to which there are infinitely many more non-Maxwellian velocity distributions, but it is, on the contrary, distinguished by the fact that by far the largest number of possible states have the characteristic properties of the Maxwellian distribution, and that compared to this number the amount of possible velocity distributions that deviate significantly from Maxwell’s is vanishingly small. The criterion of equal possibility or equal probability of different distributions is thereby always given by Liouville’s theorem.([31], p. 252)

**Definition**

**1.**

- the stationary Liouville measure $\lambda $ as a typicality measure on phase space;
- the empirical distribution ${r}^{N}\left(k\right)\left[X\right]$ that results from the deterministic dynamics and initial conditions $X\in {\Omega}_{0}$;
- the theoretical probability distribution $P\left(k\right)=B(k;M;\frac{1}{2})$ that approximates the empirical distribution for typical initial conditions.

Typicality Principle (TP).Suppose we accept a theory $\mathcal{T}$ and observe a phenomenon A. If A is typical according to $\mathcal{T}$, we should consider it to be conclusively explained. It is irrational to wonder further why our world is, in this particular respect, like nearly all (possible) worlds that the theory and its laws describe. (Conversely, atypical phenomena are, in general, the kind that cry out for further explanation and may ultimately compel us to revise or reject our theory.)

## 3. Everett’s Typicality Argument

We wish to make quantitative statements about the relative frequencies of the different possible results of observation—which are recorded in the memory—for a typical observer state; but to accomplish this we must have a method for selecting a typical element from a superposition of orthogonal states. [...] The situation here is fully analogous to that of classical statistical mechanics, where one puts a measure on trajectories of systems in the phase space by placing a measure on the phase space itself, and then making assertions … which hold for “almost all” trajectories. [...] However, for us a trajectory is constantly branching (transforming from state to superposition) with each successive measurement. To have a requirement analogous to the “conservation of probability” in the classical case, we demand that the measure assigned to a trajectory at one time shall equal the sum of the measures of its separate branches at a later time. This is precisely the additivity requirement which we imposed and which leads uniquely to the choice of square-amplitude measure.([5], pp. 460–461)

- It should be a positive function of the complex-valued coefficients associated with the branches of the universal wave function.
- It should be a function of the amplitudes of the coefficients alone, i.e., not depend on the phases.
- It should satisfy the following additivity requirement: if a branch b is decomposed into a collection $\left\{{b}_{i}\right\}$ of sub-branches, the measure assigned to b should be the sum of the measures assigned to the sub-branches ${b}_{i}$.

- the typicality measure defined in terms of branch amplitudes of the universal wave function $\Psi $ and uniquely determined by the stationarity condition;
- empirical distributions (frequencies) that obtain within world branches, here for a sequence of spin measurements on identically prepared particles;
- the theoretical Born probabilities defined in terms of the quantum state $\phi $ (the wave function or perhaps density matrix) of subsystems, e.g., by $P\left(\mathrm{spin}\phantom{\rule{4.pt}{0ex}}\mathrm{up}\right)=\langle \phi |\uparrow \rangle \langle \uparrow |\phi \rangle ={\alpha}^{2}$. They are shown to approximate relative frequencies in typical branches.

## 4. Living and Dying in the Multiverse

“[I]n Everettian quantum mechanics, the various possible outcomes of any given experiment all obtain. [...] But in probabilistic explanations, that cannot happen. In probabilistic explanations, the event invoked in the explanandum is the only outcome, of the various possible mutually exclusive outcomes, that occurs”.

## 5. Is Everett’s Derivation Circular?

In his original paper (1957) [Everett] proved that if a measurement is repeated arbitrarily often, the combined mod-squared amplitude of all branches on which the relative frequencies are not approximately correct will tend to zero. And of course this is circular: it proves not that mod-squared amplitude equals relative frequency, but only that mod-squared amplitude equals relative frequency with high mod-squared amplitude.

Substitute ‘probability’ for ‘mod-squared amplitude’, though, and the circularity should sound familiar; indeed, Everett’s theorem (as is well known) is just the Law of Large Numbers transcribed into quantum mechanics. So the circularity in Everett’s argument is just the circularity in the simplest form of frequentism, disguised by unfamiliar language. That simplest form of frequentism may indeed be hopeless, but so far Everettian quantum mechanics has neither helped nor hindered it.([17], p. 127)

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Vaidman, L. Many-Worlds Interpretation of Quantum Mechanics. In The Stanford Encyclopedia of Philosophy, Fall 2021 ed.; Zalta, E.N., Ed.; Metaphysics Research Lab, Stanford University: Stanford, CA, USA, 2021. [Google Scholar]
- Bricmont, J. Making Sense of Quantum Mechanics; Springer International Publishing: Cham, Switzerland, 2016. [Google Scholar]
- Maudlin, T. Philosophy of Physics: Quantum Theory; Princeton University Press: Princeton, NJ, USA, 2019. [Google Scholar]
- Everett, H. The Theory of the Universal Wave Function. Ph.D. Thesis, Princeton University, Princeton, NJ, USA, 1956. [Google Scholar]
- Everett, H. “Relative State” Formulation of Quantum Mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462. [Google Scholar] [CrossRef] - Goldstein, S. Boltzmann’s Approach to Statistical Mechanics. In Chance in Physics: Foundations and Perspectives; Bricmont, J., Dürr, D., Galavotti, M.C., Ghirardi, G., Petruccione, F., Zanghì, N., Eds.; Springer: Berlin, Germany, 2001; pp. 39–54. [Google Scholar]
- Lazarovici, D.; Reichert, P. Typicality, Irreversibility and the Status of Macroscopic Laws. Erkenntnis
**2015**, 80, 689–716. [Google Scholar] [CrossRef] - Bricmont, J. Making Sense of Statistical Mechanics; Undergraduate Lecture Notes in Physics; Springer International Publishing: Cham, Switzerland, 2022. [Google Scholar] [CrossRef]
- Goldstein, S. Typicality and Notions of Probability in Physics. In Probability in Physics; Ben-Menahem, Y., Hemmo, M., Eds.; The Frontiers Collection; Springer: Berlin/Heidelberg, Germany, 2012; pp. 59–71. [Google Scholar] [CrossRef]
- Hubert, M. Reviving Frequentism. Synthese
**2021**, 199, 5255–5284. [Google Scholar] [CrossRef] - Lazarovici, D. Typicality versus Humean Probabilities as the Foundation of Statistical Mechanics. In The Probability Map of the Universe: Essays on David Albert’s Time and Chance; Loewer, B., Winsberg, E., Weslake, B., Eds.; Harvard University Press: Cambridge, MA, USA, 2023. [Google Scholar]
- Maudlin, T. The Grammar of Typicality. In Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature; Allori, V., Ed.; World Scientific: Singapore, 2020. [Google Scholar]
- Wilhelm, I. Typical: A Theory of Typicality and Typicality Explanation. Br. J. Philos. Sci.
**2019**, 73, 561–581. [Google Scholar] [CrossRef] - Wilhelm, I. The Typical Principle. Br. J. Philos. Sci. 2022; Online First. [Google Scholar] [CrossRef]
- Dürr, D.; Goldstein, S.; Zanghì, N. Quantum Equilibrium and the Origin of Absolute Uncertainty. J. Stat. Phys.
**1992**, 67, 843–907. [Google Scholar] [CrossRef] - Barrett, J.A. The Quantum Mechanics of Minds and Worlds, 1st ed.; Oxford University Press: Oxford, NY, USA, 2001. [Google Scholar]
- Wallace, D. The Emergent Multiverse: Quantum Theory According to the Everett Interpretation; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Carroll, S.M.; Singh, A. Mad-Dog Everettianism: Quantum Mechanics at Its Most Minimal. arXiv
**2018**, arXiv:1801.08132. [Google Scholar] [CrossRef] - Vaidman, L. Ontology of the Wave Function and the Many-Worlds Interpretation. In Quantum Worlds: Perspectives on the Ontology of Quantum Mechanics; López, C., Holik, F., Lombardi, O., Fortin, S., Eds.; Cambridge University Press: Cambridge, UK, 2019; pp. 93–106. [Google Scholar] [CrossRef]
- Ney, A. The World in the Wave Function: A Metaphysics for Quantum Physics; Oxford University Press: Oxford, NY, USA, 2021. [Google Scholar]
- Maudlin, T. Can the World Be Only Wave-Function? In Many Worlds? Everett, Quantum Theory, and Reality; Saunders, S., Barrett, J., Kent, A., Wallace, D., Eds.; Oxford University Press: Oxford, UK, 2010; pp. 121–143. [Google Scholar]
- Lazarovici, D.; Reichert, P. The Point of Primitive Ontology. Found. Phys.
**2022**, 52, 120. [Google Scholar] [CrossRef] - Vaidman, L. On Schizophrenic Experiences of the Neutron or Why We Should Believe in the Many-worlds Interpretation of Quantum Theory. Int. Stud. Philos. Sci.
**1998**, 12, 245–261. [Google Scholar] [CrossRef] - Sebens, C.T.; Carroll, S.M. Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics. Br. J. Philos. Sci.
**2018**, 69, 25–74. [Google Scholar] [CrossRef] - McQueen, K.J.; Vaidman, L. In defence of the self-location uncertainty account of probability in the many-worlds interpretation. Stud. Hist. Philos. Sci. Part Stud. Hist. Philos. Mod. Phys.
**2019**, 66, 14–23. [Google Scholar] [CrossRef] - Deutsch, D. Quantum Theory of Probability and Decisions. Proc. R. Soc. London Ser. Math. Phys. Eng. Sci.
**1999**, 455, 3129–3137. [Google Scholar] [CrossRef] - Maudlin, T. Critical Study—David Wallace, The Emergent Multiverse: Quantum Theory According to the Everett Interpretation. Noûs
**2014**, 48, 794–808. [Google Scholar] [CrossRef] - Bell, J.S. Speakable and Unspeakable in Quantum Mechanics, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Maudlin, T. What Could Be Objective about Probabilities? Stud. Hist. Philos. Sci. Part Stud. Hist. Philos. Mod. Phys.
**2007**, 38, 275–291. [Google Scholar] [CrossRef] - Dürr, D.; Froemel, A.; Kolb, M. Einführung in Die Wahrscheinlichkeitstheorie Als Theorie Der Typizität; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar] [CrossRef]
- Boltzmann, L. Vorlesungen über Gastheorie; J. A. Barth: Leipzig, Germany, 1896; Volume 1. [Google Scholar]
- Shafer, G.; Vovk, V. The Sources of Kolmogorov’s Grundbegriffe. Stat. Sci.
**2006**, 21, 70–98. [Google Scholar] [CrossRef] - Barrett, J.A. Typicality in Pure Wave Mechanics. Fluct. Noise Lett.
**2016**, 15, 1640009. [Google Scholar] [CrossRef]

**Figure 2.**Branching Many-Worlds histories after three spin measurements. Successive arrows indicate successive outcomes. Adapted from Barrett [33].

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**MDPI and ACS Style**

Lazarovici, D.
How Everett Solved the Probability Problem in Everettian Quantum Mechanics. *Quantum Rep.* **2023**, *5*, 407-417.
https://doi.org/10.3390/quantum5020026

**AMA Style**

Lazarovici D.
How Everett Solved the Probability Problem in Everettian Quantum Mechanics. *Quantum Reports*. 2023; 5(2):407-417.
https://doi.org/10.3390/quantum5020026

**Chicago/Turabian Style**

Lazarovici, Dustin.
2023. "How Everett Solved the Probability Problem in Everettian Quantum Mechanics" *Quantum Reports* 5, no. 2: 407-417.
https://doi.org/10.3390/quantum5020026