Scalar quantum field theory in a closed universe of constant curvature
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Quantization of a massive neutral scalar field without self-interaction defined on a space-time manifold with given metric is studied, with emphasis on the two-dimensional de Sitter space. Applications in both general relativity and constructive quantum field theory are envisaged. The canonical formalism is developed for an arbitrary metric, and for special classes of metrics a Fock space can be constructed in analogy to the case of flat space. However, in this way one is led to different theories for the same manifold, with different definitions of particle observables and energy density. In particular, two nonstandard quantizations of the free field in flat space are exhibited, and three approaches to the two-dimensional de Sitter space are compared: a covariant theory in which the states of a particle transform according to a representation of the symmetry group of the space, a quantization exploiting the static nature of a portion of the universe bounded by horizons, and an "expanding universe" theory in which the particle observables diagonalize the field Hamiltonian at each time and the particle number is not constant. The representations of the canonical commutation relations in the first and third cases are unitarily equivalent. It is concluded that in this context choice of a unique physical representation of the fields is impossible. One must deal with an abstract algebra of observables associated with the field. Nevertheless, some representations are more likely to be useful than others. In this spirit a proposal is made for a definition of particle observables based on diaqonalization of the Hamiltonian on geodesic hypersurfaces. In Minkowski space this condition distinguishes the standard theory from the others. In two-dimensional de Sitter space such a theory predicts finite and reasonably small creation of particles. The relation of "contraction" between the irreducible unitary representations of the de Sitter qroup and those of the Poincar group of the same dimension is discussed in some detail. It is indicated that some arbitrariness can be removed from the treatment by considering concrete realizations of the representations by functions on the respective homogeneous spaces. The analogous case of the three-dimensional rotation group and the Euclidean group of the plane is treated in an appendix.