Evolution in Quantum Causal Histories

TL;DR

This paper introduces quantum causal histories as a framework for describing quantum evolution in a discrete, causally ordered spacetime, generalizing traditional approaches and connecting to quantum information theory.

Contribution

It provides a rigorous definition of quantum causal histories and demonstrates how local quantum evolution can be described without assuming global hyperbolicity.

Findings

Unitary evolution can be recovered where expected

The formalism applies to quantum cosmology without global hyperbolicity

Connections to quantum information theory and algebraic quantum field theory

Abstract

We provide a precise definition and analysis of quantum causal histories (QCH). A QCH consists of a discrete, locally finite, causal pre-spacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events. We show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected. This formalism may describe a quantum cosmology without an assumption of global hyperbolicity; it is thus more general than the Wheeler-DeWitt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set.