Unitarity of Quantum Theory and Closed Time-Like Curves

TL;DR

This paper examines the unitarity issues in quantum theory with closed time-like curves, critiques existing solutions, and proposes an alternative approach to restore unitarity using an extended inner product space.

Contribution

It provides a rigorous analysis of a recent unitarity-preserving prescription and introduces a new method involving an indefinite inner product space to ensure unitarity.

Findings

Identifies operational problems with current unitarity prescriptions.

Proposes an extension to a larger inner product space to restore unitarity.

Discusses implications of indefinite inner products in quantum theory.

Abstract

Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are subject to a non-unitary evolution $X$. Recently, a prescription has been proposed, which restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more non-unitary evolutions. We propose an alternative method by which unitarity of the evolution may be regained, by extending $X$ to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localised in regions spacelike separated from the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal.