Comment on ‘‘Why quantum mechanics cannot be formulated as a Markov process’’

Abstract

In the paper with the above title, D. T. Gillespie [Phys. Rev. A 49, 1607, (1994] claims that the theory of Markov stochastic processes cannot provide an adequate mathematical framework for quantum mechanics. In conjunction with the specific quantum dynamics considered there, we give a general analysis of the associated dichotomic jump processes. If we assume that Gillespie's "measurement probabilities" are the transition probabilities of a stochastic process, then the process must have an invariant (time independent) probability measure. Alternatively, if we demand the probability measure of the process to follow the quantally implemented (via the Born statistical postulate) evolution, then we arrive at the jump process which can be interpreted as a Markov process if restricted to a suitable duration time. However, there is no corresponding Markov process consistent with the Z 2 event space assumption, if we require its existence for all times t ∈ R + .Before, [1], we have contested the general statement due to Gillespie [2] about the generic contradiction between the probabilistic concepts appropriate for quantum theory and those proper to the common-sense theory of stochastic (in particular, Markov) processes. Our argument was based on invoking the standard, configuration space, Schrödinger picture quantum dynamics which, if combined with the Born statistical interpretation postulate, allows for a consistent description in terms of Markov processes of diffusion type. In conformity with the rich theory developed so far, [3,4,5,6,7,8,9].

Equilibration and locality

“…It motivated Nelson [14,15] to interpret quantum-mechanical phenomena within stochastic mechanics [15] based on Markov chains. It is disputed [16][17][18] whether transitions of Markov chains may resemble measurements of quantum physics.…”