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It is usually accepted that quantum dynamics described by Schrodingerequation that determines the evolution of states from one Cauchy surface toanother is unitary. However, it has been known for some time that thisexpectation is not borne out in the conventional setting in which one envisagesthe dynamics on a fixed Hilbert space. Indeed it is not even true for linearquantum field theory on Minkowski space if the chosen Cauchy surfaces are notpreserved by the flow of a timelike Killing vector. This issue was elegantlyaddressed and resolved by Agullo and Ashtekar who showed that in a generalsetting quantum dynamics in the Schrodinger picture does not take place in afixed 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 thattime. In this article, we postulate a Schrodinger equation that incorporatesthe effect of change in vacuum during time evolution by including theBogoliubov transformation explicitly in the Schrodinger equation. Moreprecisely, for a linear (real) Klein-Gordon field on a globally hyperbolicspacetime we write down a Schrodinger equation that propagates states betweenarbitrary chosen Cauchy surfaces, thus describing the quantum dynamics on aHilbert bundle. We show that this dynamics is unitary if a specific tensor onthe canonical phase space satisfies the Hilbert-Schmidt condition. Generalizedunitarity condition of Agullo-Ashtekar follows quite naturally from ourconstruction.

Bogoliubov Transformation and Schrodinger Representation on Curved Space