Unitarity and Causality in Generalized Quantum Mechanics for Non-Chronal Spacetimes

TL;DR

This paper develops a generalized quantum mechanics framework for non-chronal spacetimes, ensuring consistent probabilities despite non-unitarity and modified causality, relevant for spacetimes with closed timelike curves.

Contribution

It introduces a quantum mechanics formulation that remains consistent in non-chronal spacetimes, addressing non-unitarity and causality modifications.

Findings

Probabilities obey probability rules even with non-unitarity

No signaling outside the light cone or energy non-conservation

Future non-chronal regions influence present probabilities

Abstract

Spacetime must be foliable by spacelike surfaces for the quantum mechanics of matter fields to be formulated in terms of a unitarily evolving state vector defined on spacelike surfaces. When a spacetime cannot be foliated by spacelike surfaces, as in the case of spacetimes with closed timelike curves, a more general formulation of quantum mechanics is required. In such generalizations the transition matrix between alternatives in regions of spacetime where states {\it can} be defined may be non-unitary. This paper describes a generalized quantum mechanics whose probabilities consistently obey the rules of probability theory even in the presence of such non-unitarity. The usual notion of state on a spacelike surface is lost in this generalization and familiar notions of causality are modified. There is no signaling outside the light cone, no non-conservation of energy, no ``Everett phones'', and probabilities of present events do not depend on particular alternatives of the future. However, the generalization is acausal in the sense that the existence of non-chronal regions of spacetime in the future can affect the probabilities of alternatives today. The detectability of non-unitary evolution and violations of causality in measurement situations are briefly considered. The evolution of information in non-chronal spacetimes is described.