Elementary particles in a curved space. II

Abstract

This is an attempt to develop conventional, contemporary, elementary-particle physics in a Riemannian space of constant curvature. We study the global structure of the 3 + 2 de Sitter space, which we take to mean the covering space of the hyperboloid $y_0^{}{}_{}{}^2-{\overrightarrow{\mathrm{y}}}^2+y_5^{}{}_{}{}^2={\rho }^{-1\mathrm{}}$ in a five-dimensional Minkowski space. This space is not periodic in time. A causal structure is shown to exist and the commutation relations between free fields are shown to be causal. Elementary massive particles are associated with a class of irreducible representations of the universal covering group of SO(3, 2) for which the Hamiltonian has a discrete spectrum with a lower (positive) bound. A detailed study is made of the wave functions in "momentum space" and in configuration space. Free quantum fields are introduced with the help of a discrete set of creation and destruction operators and the commutator [${\phi }_0\mathrm{}\left(x\right), {\phi }_0\left(x^{\prime }\right)$] is calculated. An appendix describes what we think is an interesting way to realize irreducible representations of the "discrete series."