Super-determinism via Solomonoff Induction

A super-deterministic theory of Quantum Mechanics is derived from Solomonoff’s theory of Universal Induction, or what is generally known as Occam’s razor.

Aidan Rocke https://github.com/AidanRocke
03-04-2022

Summary:

In this article, I’d like to advance a solution to a problem which dates back to John Wheeler:

How come ‘one world’ out of many observer-participants?-Wheeler

The theory I propose, in the form of a postulate, has a number of important consequences. First, it addresses the Measurement Problem by determining why Wave Function Collapse occurs. Second, it addresses Gerard’t Hooft’s inquiry as to whether Quantum Mechanics is not in fact a deterministic theory.

Finally, this elegant theory answers a question as old as Shannon’s theory of information:

My greatest concern was what to call it. I thought of calling it ‘information,’ but the word was overly used, so I decided to call it ‘uncertainty.’ When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.’-Claude Shannon

In the final analysis, Entropy is an approximation of Kolmogorov Complexity.

The Trilateral Postulate:

In the following, we assume that a Universal Turing Machine, Conscious Observers, and Quantum Observables form a triad so this UTM performs Solomonoff Induction on the sequential observations of Conscious Observers:

Assuming that the Physical Church-Turing thesis is true and that conscious observers perform Universal Induction, the frequency with which a Quantum Measurement is made converges to its Algorithmic Probability P(X|M) where M denotes measurement settings of the experiment.

This explains how one world emerges from many observer-participants.

As Solomonoff Induction is a deterministic algorithm, which may be approximated using Universal Levin Search, this would imply that Quantum Mechanics is super-deterministic. Furthermore, in this setting \(M\) would refer to the Quantum State of the Universe and the probability distribution \(P\) may be derived from the Universal Wave Function.

While the Trilateral Postulate is not provable, it is a necessary and sufficient description of an implicit axiom underlying all of theoretical physics. Occam’s razor.

An important corollary:

Given that the evolution of the Universe is determined by the Universal Wave Function, it would immediately follow that:

The typical probability of an event X(ex. muon decay, earthquake) converges to its Algorithmic Probability.

To make this corollary precise, we may clarify key notions.

Key definitions:

The Physical Church-Turing thesis:

The Physical Church-Turing thesis is equivalent to the assertion that all of theoretical physics, which defines a set of computable predictive models, is consistent with the hypothesis that the Observable Universe may be simulated by a Universal Turing Machine [1].

Typical Probability of an event:

The typical probability of an event \(X\) is defined using the Asymptotic Equipartition Theorem:

\[\begin{equation} H(X) = \lim_{N \to \infty} \frac{-\log_2 P(X_1,...,X_N)}{N} \tag{1} \end{equation}\]

\[\begin{equation} -\log_2 P(X) = H(X) \tag{2} \end{equation}\]

assuming that \(X_i \sim P(X)\) are independently and identically distributed.

Algorithmic Probability of an event:

The Algorithmic Probability of an event \(X\) is defined in terms of the measurement function \(f: M \rightarrow X\) where \(M\) denotes the measurement settings and \(X\) is the observable in question.

Using Levin’s Coding theorem, the Algorithmic Probability of \(X\) is defined as follows [5]:

\[\begin{equation} -\log_2 m(X) = K_U(X) + \mathcal{O}(1) \tag{3} \end{equation}\]

which is independent of the choice of description language \(U\) due to the Invariance Theorem [3]:

\[\begin{equation} |K_U(X)-K_{U'}(X)| \leq \mathcal{O}(1) \tag{4} \end{equation}\]

Discussion:

The Trilateral Postulate is consistent with but not reducible to Markus Mueller’s recent analysis of observer states via Algorithmic Information Theory titled Law without Law [10].

References:

  1. Rocke (2022, Jan. 5). Kepler Lounge: Revisiting the unreasonable effectiveness of mathematics. Retrieved from keplerlounge.com

  2. R. J. Solomonoff A formal theory of inductive inference: Parts 1 and 2. Information and Control, 7:1–22 and 224–254, 1964.

  3. A. N. Kolmogorov Three approaches to the quantitative definition of information. Problems of Information and Transmission, 1(1):1–7, 1965

  4. G. J. Chaitin On the length of programs for computing finite binary sequences: Statistical considerations. Journal of the ACM, 16(1):145–159, 1969.

  5. L.A. Levin. Laws of information conservation (non-growth) and aspects of the foundation of probability theory. Problems Information Transmission, 10(3):206-210, 1974.

  6. John A. Wheeler, 1990, “Information, physics, quantum: The search for links” in W. Zurek (ed.) Complexity, Entropy, and the Physics of Information. Redwood City, CA: Addison-Wesley.

  7. E.T. Jaynes. Information Theory and Statistical Mechanics. The Physical Review. 1957.

  8. von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Mechanics) Princeton University Press., . ISBN 978-0-691-02893-4.

  9. Shannon, Claude E. (1948). A Mathematical Theory of Communication Bell System Technical Journal 27: 379-423. doi:10.1002/j.1538-7305.1948.tb01338.x.

  10. Markus Mueller. Law without law: from observer states to physics via algorithmic information theory. Arxiv. 2020.

  11. Gerard’t Hooft. The Cellular Automaton Interpretation of Quantum Mechanics. Arxiv. 2015.

  12. Wigner, Eugene; Henry Margenau (1967). “Remarks on the Mind Body Question, in Symmetries and Reflections, Scientific Essays”. American Journal of Physics. 35 (12): 1169–1170.

  13. L. A. Levin, Universal sequential search problems. Problems of Information Transmission, 9(3):265–266, 1973.

Citation

For attribution, please cite this work as

Rocke (2022, March 4). Kepler Lounge: Super-determinism via Solomonoff Induction. Retrieved from keplerlounge.com

BibTeX citation

@misc{rocke2022super-determinism,
  author = {Rocke, Aidan},
  title = {Kepler Lounge: Super-determinism via Solomonoff Induction},
  url = {keplerlounge.com},
  year = {2022}
}