Statistical Dynamics of Global Unitary Invariant Matrix Models as Pre-Quantum Mechanics

TL;DR

This paper demonstrates that quantum mechanics and its probabilistic nature emerge from classical matrix dynamics with unitary invariance, under certain assumptions, including a hierarchy of scales, leading to a pre-quantum framework.

Contribution

It introduces a trace dynamics model where quantum mechanics arises as an emergent phenomenon from classical matrix variables with unitary invariance.

Findings

Quantum mechanics emerges from classical matrix dynamics.

Fluctuations lead to stochastic modifications of Schrödinger's equation.

State vector reduction follows from thermodynamic fluctuations.

Abstract

We consider the classical dynamics of bosonic and fermionic matrix variables in complex Hilbert space, defined by a trace action, assuming cyclic invariance under the trace and the presence of a global unitary invariance. With plausible and explicitly stated assumptions, including the existence of a large hierarchy of scale between the underlying dynamics and observed physics, we show that (1) the equilibrium statistical mechanics of this matrix dynamics, in the canonical ensemble, gives rise to an emergent quantum mechanics for many degrees of freedom, including the standard canonical commutation/anticommutation relations and the usual unitary Heisenberg and Schr\"odinger picture time evolutions of operators and states, and (2) the fluctuation or Brownian motion corrections to this thermodyamics lead to an energy-driven stochastic modification of the Schr\"odinger equation, which is known to imply state vector reduction with Born rule probabilities. Thus, quantum mechanics and its probabilistic interpretation both arise as emergent phenomena in the underlying trace dynamics.