Bogoliubov Transformation and Schrodinger Representation on Curved Space

Abstract

It is usually accepted that quantum dynamics described by Schrodinger equation that determines the evolution of states from one Cauchy surface to another is unitary. However, it has been known for some time that this expectation is not borne out in the conventional setting in which one envisages the dynamics on a fixed Hilbert space. Indeed it is not even true for linear quantum field theory on Minkowski space if the chosen Cauchy surfaces are not preserved by the flow of a timelike Killing vector. This issue was elegantly addressed and resolved by Agullo and Ashtekar who showed that in a general setting quantum dynamics in the Schrodinger picture does not take place in a fixed Hilbert space. Instead, it takes place on a non-trivial bundle over time, the Hilbert bundle, whose fibre at a given time is a Hilbert space at that time. In this article, we postulate a Schrodinger equation that incorporates the effect of change in vacuum during time evolution by including the Bogoliubov transformation explicitly in the Schrodinger equation. More precisely, for a linear (real) Klein-Gordon field on a globally hyperbolic spacetime we write down a Schrodinger equation that propagates states between arbitrary chosen Cauchy surfaces, thus describing the quantum dynamics on a Hilbert bundle. We show that this dynamics is unitary if a specific tensor on the canonical phase space satisfies the Hilbert-Schmidt condition. Generalized unitarity condition of Agullo-Ashtekar follows quite naturally from our construction.