Quantum Mechanics with Extended Probabilities

TL;DR

This paper introduces an extended probability framework for quantum mechanics that includes negative probabilities for certain histories, aiming to provide a more comprehensive description of quantum systems and potential testable alternatives.

Contribution

It proposes a new formulation of quantum mechanics using extended probabilities, satisfying all probability sum rules and encompassing all describable histories, including non-decoherent ones.

Findings

Extended probabilities can be assigned to all histories.

All probability sum rules are satisfied exactly.

Comparison with decoherent histories formulation is discussed.

Abstract

The quantum mechanics of closed systems such as the universe is formulated using an extension of familiar probability theory that incorporates negative probabilities. Probabilities must be positive for sets of alternative histories that are the basis of fair settleable bets. However, in quantum mechanics there are sets of alternative histories that can be described but which cannot be the basis for fair settleable bets. Members of such sets can be assigned extended probabilities that are sometimes negative. A prescription for extended probabilities is introduced that assigns extended probabilities to all histories that can be described, fine grained or coarse grained, members of decoherent sets or not. All probability sum rules are satisfied exactly. Sets of histories that are recorded to sufficient precision are the basis of settleable bets. This formulation is compared with the decoherent (consistent) histories formulation of quantum theory. Prospects are discussed for using this formulation to provide testable alternatives to quantum theory or further generalizations of it.