Quantized Fields and Particle Creation in Expanding Universes. II

Abstract

We consider the spin-\textonehalf{} field which satisfies the fully covariant generalization of the Dirac equation. The metric, which is not quantized, is that of an expanding universe with Euclidean 3-space. The field is quantized in a manner consistent with the time development dictated by the equation of motion. Consideration of the special-relativistic limit then provides a new proof of the connection between spin and statistics. In general, there will be production of spin-\textonehalf{} particles as a result of the expansion of the universe. However, we show that in the limits of zero and infinite mass there is no spin-\textonehalf{} particle production. For arbitrary mass, we obtain an upper bound on the creation of particles of given momentum. We treat the case of an instantaneous expansion exactly (but not taking into account the reaction of the particle creation back on the gravitational field). For such an expansion, the created particle density, when integrated over all momenta, diverges as a result of the high-momentum behavior. We also consider the Friedmann expansion of a radiation-filled universe, emphasizing the effect of the initial stage of the expansion. We obtain the asymptotic form of the created particle density for large momenta, and thus show that the particle density, integrated over all momenta, is finite, in contrast to the previous case.