Quantum processes on phase space

Abstract

Quantum theory predicts probabilities for various events as well as relative phases between different alternatives of the system. A unified description of both probabilities and phases comes through generalisation of the notion of a density matrix for histories; this object is the decoherence functional introduced by the consistent histories approach. If we take phases as well as probabilities as primitive elements of our theory, we abandon Kolmogorov probability and can describe quantum theory in terms of fundamental commutative observables, without being obstructed by Bell’s and related theorems. We develop the description of relative phases and probabilities for paths on the classical phase space. This description provides a theory of quantum processes, which has many formal analogies with the theory of stochastic processes. We show how from standard quantum theory we can construct a quantum process on phase space (using coherent states). Conversely starting from quantum processes on phase space we recover standard quantum theory on Hilbert space from the requirement that the process satisfies (an analogue of) the Markov property together with time reversibility. The statistical predictions of our theory are identical to the ones of standard quantum theory, but the “logic” of events is Boolean; events are not represented by projectors any more. We discuss some implication of this fact for the interpretation of quantum theory, emphasising that it makes plausible the existence of realist theories for individual quantum systems.