Group Theoretical Examination of the Relativistic Wave Equations on Curved Spaces. III. Real reducible spaces

Abstract

The group theoretical approach to the relativistic wave equations on the real reducible spaces for spin~0, 1/2 and~1 massless particles is considered. The invariant wave equations which determine the appropriate irreducible representations are constructed. The coincidence of these equations with the general-covariant Klein-Gordon, Weyl and Maxwell equations on the corresponding spaces is shown. The explicit solutions of these equations possessing a simplicity and physical transparency are obtained in the form of so-called "plane waves" without using the method of separation of variables. The invariance properties of these "plane waves" for the spinless particles under the group $SO(3,1)$ were used for the construction of the invariant spin~0,1/2 and~1 two-point functions on the $H^3$. Secondly quantized spin~0,1/2 and~1 fields on the ${\Bbb R}^{1}\otimes H^{3}$ are constructed; their propagators which are their (anti)commutators in different points, are expressed in terms of the mentioned two-point functions. From here the ${\Bbb R}^{1}\otimes SO(3,1)$-invariance follows.