On a Quantized Scalar Field in Some Bianchi-Type I Universe

Abstract

To elucidate the relation between the Parker-Fulling momentum-space approach and DeWitt's configuration-space approach to a cosmological gravitational field interacting with quantized matter fields, an attempt is made to study a quantized scalar field in the Bianchi-type I universe whose geodesic bi-scalar and the associated propagators (except for Hadamard's elementary solution) were dealt with in a previous paper. On the basis of the former approach in which we can set up a time-independent set of annihilation and creation operators and the associated vacuum state, an elementary solution is defined by the vacuum expectation value of field operators and a time-dependent vacuum state at any time is also introduced by a suitable Bogoliubov transformation. Some limit of the latter vacuum states at two separate epochs leads to DeWitt's vacuum states |in, vac> and |out, vac> connected causally, in terms of which his elementary solution can be defined. The difference appears in the respective regular parts of both elementary solutions. Contrary to ours, DeWitt's regular part includes an information at the limiting two epochs. Its implication is also touched upon in reference to the requirement that the finite part extractable from the vacuum expectation value of the energy-momentum tensor should be a local tensor.