Abstract
We investigate the modeling capabilities of sets of coupled classical harmonic oscillators (CHO) in the form of a modeling game. The application of the 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 wavefunction 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.
Highlights
Newtonian celestial mechanics, Relativity, and Mendelian genetics are physical theories, because their mathematics was developed by reasoning out the consequences of clearly stated physical principles from which constraint the possibilities”
He continues “To this day we have no constraining principle from which one can deduce the mathematics of QM and quantum electrodynamics (QED); [...] In other words, the mathematical system of the present quantum theory is [...] unconstrained by any physical principle” [9]
Based on three fundamental principles, which describe the form of physics, we have shown that the algebraic structure of coupled classical degrees of freedom is
Summary
We will show that special relativity can be reinterpreted as a transformation theory of the second moments of the abstract canonical variables of coupled oscillator systems We shall show how Maxwell’s equations find their place within the game The motivation for this game is to show that many aspects of modern physics can be understood on the basis of the classical notions of harmonic oscillation if these notions are appropriately reinterpreted.
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