Group Theoretical Examination of the Relativistic Wave Equations on Curved Spaces. I.Basic Principles

Abstract

The basic principles of generalization of the group theoretical approach to the relativistic wave equations on curved spaces are examined. The general method of the determination of wave equations from the known symmetry group of a symmetrical curved space is described. The method of obtaining the symmetrical spaces in which invariant wave equations admit the limiting passage to the relativistic wave equations on the flat (not necessarily real) spaces is explained. Starting from the equations for massless particles and from the Dirac equation in the Minkowski space, admissible real symmetrical spaces are founded. The mentioned procedure is carried out also for the complex spaces. As the basic point, the wave equations on the flat complex space are considered. It is shown that the usual Dirac equation written down in the complex variables does not lead to any curved space. By "decomposition" of the mentioned form of the Dirac equation the equation leading to the space ${\Bbb C}{\Bbb P}^{n}$ and being alternative to the usual Dirac equation on ${\Bbb C}^{n}$ is constructed.