Matrix models as non-local hidden variables theories

TL;DR

This paper proposes that matrix models used in string and M theory can be viewed as non-local hidden variables theories, where eigenvalues behave quantum mechanically due to stochastic dynamics of matrix entries.

Contribution

It demonstrates that matrix models at finite temperature can be interpreted as non-local hidden variables theories, connecting stochastic matrix dynamics with quantum behavior.

Findings

Eigenvalues follow Schrödinger-like evolution at large N.

Quantum uncertainties arise from statistical fluctuations of off-diagonal elements.

The formulation is background independent and based on Nelson's stochastic quantum mechanics.

Abstract

It is shown that the matrix models which give non-perturbative definitions of string and M theory may be interpreted as non-local hidden variables theories in which the quantum observables are the eigenvalues of the matrices while their entries are the non-local hidden variables. This is shown by studying the bosonic matrix model at finite temperature, with T taken to scale as 1/N. For large N the eigenvalues of the matrices undergo Brownian motion due to the interaction of the diagonal elements with the off diagonal elements, giving rise to a diffusion constant that remains finite as N goes to infinity. The resulting probability density and current for the eigenvalues are then found to evolve in agreement with the Schroedinger equation, to leading order in 1/N. The quantum fluctuations and uncertainties in the eigenvalues are then consequences of ordinary statistical fluctuations in the values of the off-diagonal matrix elements. This formulation of the quantum theory is background independent, as the definition of the thermal ensemble makes no use of a particular classical solution. The derivation relies on Nelson's stochastic formulation of quantum theory, which is expressed in terms of a variational principle.