Matrix models as non-local hidden variables theories

Smolin, Lee (2001) Matrix models as non-local hidden variables theories.

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, with N the rank of the matrices.
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, with hbar proportional to the thermal diffusion constant
for the eigenvalues. The quantum uctuations and uncertainties in the eigenvalues are then
consequences of ordinary statistical uctuations in the values of the off-diagonal matrix
elements. Furthermore, 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.