Some Points in the Quantization of Relativistic Grassmann Dependent Interaction Systems

Abstract

The Grassmann variables in the context of a physical theory are now widely accepted. We can think, for instance, in supergravity or in superstring theories as examples of the importance they have now in contemporary Physics. These already ubiquitous examples are not certainly the only ones. Since the Grassmann variables are used as the classical equivalent of quantum spin, they also appear in the description of systems where the spin plays an important role1. In all these cases, the central idea is based in Dirac’s point of view that we should first try to understand a physical theory, and only then try to quantize it2. To illustrate this point, we quote the work by Crater and Van Alstine3, where they construct two body relativistic wave equations for particles with spin. It is not easy to work at the quantum level with this problem, because one must first try to impose certain conditions of compatibilty on the wave function and in the equations themselves, which restrict the class of available potentials for the problem. Crater and Van Alstine translate the problem to the classical level and then they use the general theory of constraints to analyze it. They find that a supersymmetry condition is needed to impose, in order to obtain consistent equations for the problem. Of course, the final theory is obtained after quantizing their results.