Old Game, New Rules: Rethinking The Form of Physics

TL;DR

This paper explores how classical harmonic oscillators can model complex physical phenomena, including relativity and quantum mechanics, through algebraic structures like Clifford algebras, offering a novel phase space interpretation of quantum mechanics.

Contribution

It introduces a modeling game linking classical oscillators to Clifford algebras, providing a unified algebraic framework for relativity and quantum phenomena.

Findings

Correlations between coupled oscillators relate to Dirac algebra.

Space-time dimensions emerge from algebraic properties of oscillators.

Wave functions are interpreted as phase space moments.

Abstract

We investigate the modeling capabilities of sets of coupled classical harmonic oscillators (CHO) in the form of a modeling game. The application of simple but restrictive rules of the game lead to conditions for an isomorphism between Lie-algebras and real Clifford algebras. We show that the correlations between two coupled classical oscillators find their natural description in the Dirac algebra and allow to model aspects of special relativity, inertial motion, electromagnetism and quantum phenomena including spin in one go. The algebraic properties of Hamiltonian motion of low-dimensional systems can generally be related to certain types of interactions and hence to the dimensionality of emergent space-times. We describe the intrinsic connection between phase space volumes of a 2-dimensional oscillator and the Dirac algebra. In this version of a phase space interpretation of quantum mechanics the (components of the) spinor wave-function in momentum space are abstract canonical coordinates, and the integrals over the squared wave function represents second moments in phase space. The wave function in ordinary space-time can be obtained via Fourier transformation. Within this modeling game, 3+1-dimensional space-time is interpreted as a structural property of electromagnetic interaction. A generalization selects a series of Clifford algebras of specific dimensions with similar properties, specifically also 10- and 26-dimensional real Clifford algebras.