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Fundamental theories of physics

The Cellular Automaton Interpretation of Quantum Mechanics

Gerard ’t Hooft

Published Jan 1, 2016• 355 citations
Publisher

Abstract

Quantum Physics; Mathematical Applications in the Physical Sciences; History and Philosophical Foundations of Physics
DOI
10.1007/978-3-319-41285-6

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A deterministic reformulation of quantum mechanics is thought to be able to bypass the usual philosophical interpretations of probability and stochasticity of the standard quantum mechanical scenarios. Recently ’t Hooft proposed a different perspective based on the ontological formulation of quantum mechanics, obtained by writing the Hamiltonian of a quantum system in a way to render it mathematically equivalent to a deterministic system. The ontological deterministic models consist of elementary cells, also called cellular automata, inside which the quantities describing the dynamics oscillate in periodic orbits, extending and replacing the quantum mechanical classical language based on harmonic oscillators. We show that the structure of the cellular automaton sets finds a clear physical interpretation with the Majorana infinite-component equation: the cellular automata are elementary building blocks generated by the Poincaré group of spacetime transformations with positive-definite energy down to the Planck scales, with a close relation to the Riemann Hypothesis.

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Abstract Careful considerations concerning the interpretation of quantum mechanics serves not only for a better philosophical understanding of the physical world, but can also be instrumental for model building. After a resume of the author’s general views and their mathematical support, it is shown what new insights can be gained, in principle, concerning features such as the Standard Model of the elementary particles and the search for new approaches to bring he gravitational force in line with quantum mechanics. Questions to be asked include the cure for the formal non-convergence of renormalised perturbation expansions, the necessary discreteness of physical variables at the Planck scale, and the need to reconcile these with diffeomorphism invariance in General Relativity. Finally, a program is proposed to attempt to derive the propagation laws for cellular automaton models of the universe.

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A deterministic reformulation of quantum mechanics is thought to be able to bypass the usual philosophical interpretations of probability and stochasticity of the standard quantum mechanical scenarios. Recently ’t Hooft proposed a different perspective based on the ontological formulation of quantum mechanics, obtained by writing the Hamiltonian of a quantum system in a way to render it mathematically equivalent to a deterministic system. The ontological deterministic models consist of elementary cells, also called cellular automata, inside which the quantities describing the dynamics oscillate in periodic orbits, extending and replacing the quantum mechanical classical language based on harmonic oscillators. We show that the structure of the cellular automaton sets finds a clear physical interpretation with the Majorana infinite-component equation: the cellular automata are elementary building blocks generated by the Poincaré group of spacetime transformations with positive-definite energy down to the Planck scales, with a close relation to the Riemann Hypothesis.
Quantum mechanics approaches its centenary with an impressive record. It became the backbone of most research in physics, led to applications such as the transistor and laser, and prompted an upheaval in the philosophy of science. Its scope and its precision have been constantly growing, and it is now promising more powerful computers and safer cryptography. This century of conquests has also been a time of ongoing debates about the foundations and interpretation of the theory, which has been referred to as the quantum controversy. The controversy has been unusually long and remains open. Jammer called it “a story without an ending” and Laloë described this coexistence of successes and doubts with the metaphor of the colossus with feet of clay. This Handbook is a survey on this history. Some of the so-called “interpretations” are not interpretations of a fixed formalism, they are different theories, thus it has been a debate about the interpretations and the foundations of quantum mechanics. In this volume, the “history of interpretations” is used for the sake of brevity. The successive parts of the Handbook deal with the scientific and philosophical issues under debate, historical landmarks, places and contexts, historical and philosophical theses, and the proliferation of interpretations. This Handbook may serve those interested in the history of quantum mechanics and those who are engaged with, or attracted by current research on quantum information, as it may give them a flavor of the history that led to or lay the groundwork for the current investigations.
Abstract Careful considerations concerning the interpretation of quantum mechanics serves not only for a better philosophical understanding of the physical world, but can also be instrumental for model building. After a resume of the author’s general views and their mathematical support, it is shown what new insights can be gained, in principle, concerning features such as the Standard Model of the elementary particles and the search for new approaches to bring he gravitational force in line with quantum mechanics. Questions to be asked include the cure for the formal non-convergence of renormalised perturbation expansions, the necessary discreteness of physical variables at the Planck scale, and the need to reconcile these with diffeomorphism invariance in General Relativity. Finally, a program is proposed to attempt to derive the propagation laws for cellular automaton models of the universe.