Fermionic anomalies in quantum-mechanical relativistic problems

Abstract

We discuss the existence of ambiguities in the quantization of systems with odd Grassmann degrees of freedom, or anomalies of fermionic nature as we call them, which could lead us to an incorrect quantum counterpart. We propose in this work a way of avoiding such ambiguities in the case of relativistic mechanics, which consists in including the odd degrees of freedom into a generalization of the momentum, that fully contains the anomalies. We consider this generalized momentum as the relevant variable to quantize, mentioning their gauge transformation properties. We illustrate our results with some examples. In particular, we discuss the Dirac oscillator as a typical case of the problems we are dealing with.