Uniqueness of the propagator in spacetime with cosmological singularity

Abstract

We consider the question of the uniqueness of the Feynman propagator, or equivalently of the initial vacuum state, in a cosmological model with an initial singularity. After discussing the relationship of the propagator to positive-frequency solutions of the field equation and to physical quantities, we turn to the particular example of the linearly expanding universe. The Feynman propagator in this model was obtained by Chitre and Hartle. We show that the boundary conditions they used are not sufficient to determine the propagator uniquely. This is done by displaying a family of propagators obeying the same boundary conditions. We then explore methods of strengthening the boundary conditions by considering the temperature and chemical potential of the created particles, the massless limit of the propagator, and the square integrability of the analytically continued kernel associated with the propagator. We show that the requirement of square integrability is sufficient to determine the Feynman propagator uniquely and that the resulting propagator is that of Chitre and Hartle. We write the square-integrability condition in a way applicable to general open spacetimes. Another approach we consider is the use of conditions such as consistency with the Einstein equations to determine the temperature and chemical potential characterizing the high-momentum part of the spectrum of created particles.