Could quantum mechanics be an approximation to another theory?

Lee Smolin

Published 2006-09-14Version 1

We consider the hypothesis that quantum mechanics is an approximation to another, cosmological theory, accurate only for the description of subsystems of the universe. Quantum theory is then to be derived from the cosmological theory by averaging over variables which are not internal to the subsystem, which may be considered non-local hidden variables. We find conditions for arriving at quantum mechanics through such a procedure. The key lesson is that the effect of the coupling to the external degrees of freedom introduces noise into the evolution of the system degrees of freedom, while preserving a notion of averaged conserved energy and time reversal invariance. These conditions imply that the effective description of the subsystem is Nelson's stochastic formulation of quantum theory. We show that Nelson's formulation is not, by itself, a classical stochastic theory as the conserved averaged energy is not a linear function of the probability density. We also investigate an argument of Wallstrom posed against the equivalence of Nelson's stochastic mechanics and quantum mechanics and show that, at least for a simple case, it is in error.