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Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?

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Author:: Nicolas Gisin

Abstract

It is common procedure to describe the initial conditions of some classical dynamical physical system, with real numbers, which contain an infinite amount of information. It is argued that a finite volume of space can't contain infinite information, hence real numbers are not physically relevant, as their series of bits are truly random. Additionally, these numbers are more accurately termed "random numbers". An alternative Classical Mechanics is proposed, which is empirically equivalent, but uses only finite-information numbers. The alternative mechanics is non-deterministic, despite the use of deterministic equations. Both the alternative classical mechanics, and quantum theories can be supplemented in such a way that the resulting theory is deterministic. Most physicists supplement classical theory with real numbers, assumed to have some physical existence, while rejecting Bohmian mechanics, as supplemented, arguing that Bohmian positions have no physical reality.

I. Introduction

Physics is often described as a way to explain the world deterministically. Furthermore, it is claimed that all good explanations follow that structure.

Typically, classical physics demonstrates this, as the theory has an expressive explanatory power, despite or perhaps because its limits are known.

The domain of validity is limited by relativity and quantum theory, whose predictions are more accurate when speed and size approach the critical values of the universal constants cc and ℏ\hbar.

Classical mechanics is a group of dynamical laws governing the the evolution of systems based on initial conditions, parameters such as position and momentum of point particles, described with real numbers. Except for unique cases, the equations describing these laws, when supplied with the values of the initial conditions, can completely determine the solutions to the equations for future times.

The results of this are widespread, as this is often considered to be the goal of scientific explanation. Many philosophers and physicists seek to submit quantum randomness/uncertainty to classical determinism.

Referenced in

Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?