Quantized Fields and Particle Creation in Expanding Universes. I
Abstract
Spin0 fields of arbitrary mass and massless fields of arbitrary spin are considered. The equations governing the fields are the covariant generalizations of the specialrelativistic freefield equations. The metric, which is not quantized, is that of a universe with an expanding (or contracting) Euclidean 3space. The spin0 field of arbitrary mass is quantized in the expanding universe by the canonical procedure. The quantization is consistent with the time development dictated by the equation of motion only when the boson commutation relations are imposed. This consistency requirement provides a new proof of the connection between spin and statistics. We show that the particle number is an adiabatic invariant, but not a strict constant of the motion. We obtain an expression for the average particle density as a function of the time, and show that particle creation occurs in pairs. The canonical creation and annihilation operators corresponding to physical particles during the expansion are specified. Thus, we do not use an Smatrix approach. We show that in a universe with flat 3space containing only massless particles in equilibrium, there will be precisely no creation of massless particles as a result of the expansion, provided the Einstein field equations without the cosmological term are correct. Furthermore, in a dustfilled universe with flat 3space there will be precisely no creation of massive spin0 particles in the limit of infinite mass, again provided that the Einstein field equations are correct. Conversely, without assuming any particular equations, such as the Einstein equations, as governing the expansion of the universe, we obtain the familiar Friedmann expansions for the radiationfilled and the dustfilled universes with flat 3space. We only make a very general and natural hypothesis connecting the particle creation rate with the macroscopic expansion of the universe. In one derivation, we assume that in an expansion of the universe in which a particular type of particle is predominant, the type of expansion approached after a long time will be such as to minimize the average creation rate of that particle. In another derivation, we use the assumption that the reaction of the particle creation back on the gravitational field will modify the expansion in such a way as to reduce, if possible, the creation rate. This connection between the particle creation and the Einstein equations is surprising because the Einstein equations themselves played no part at all in the derivation of the equations governing the particle creation. Finally, on the basis of a socalled infinitemass approximation, we argue that in the present predominantly dustfilled universe, only massless particles of zero spin might possibly be produced in significant amounts by the present expansion. In this connection, we show that massless particles of arbitrary nonzero spin, such as photons or gravitons, are not created by the expansion, regardless of its form.
 Publication:

Physical Review
 Pub Date:
 July 1969
 DOI:
 10.1103/PhysRev.183.1057
 Bibcode:
 1969PhRv..183.1057P