A constraint on physical quantum states in cosmological space-times

Abstract

The problem of constructing the Hilbert space of physical states for a free scalar quantum field propagating on a cosmological background is considered. The concept of energy-momentum for such a field is discussed and it is noted that, according to current renormalization theory, for a state ¦M〉 to have finite energy density its associated anticommutator function\(\left\langle {M\left| {\widehat\phi \left( x \right)\widehat\phi \left( {x'} \right) + \widehat\phi \left( {x'} \right)\widehat\phi \left( x \right)} \right|M} \right\rangle \) must be of a particular form first discussed by Hadamard. This restriction is shown to lead to a constraint on the construction of the Hilbert space of physical states. This constraint is used to reject a recently proposed scheme for the construction of this space which was based on a principle of energy minimization.