Nilpotent elements in physics

Abstract

Institute of Theoretical Physics, University of Wroclaw,pl. M. Borna 9, 50–204 Wroclaw, Poland(Received January 4, 2007)We briefly review some issues of the nilpotent objects in theoretical physics using simple modelsas an illustration. Nilpotent elements appear at quantum and classical level in several ways. On theone hand there are celebrated BRST operators, external derivative and related with them cohomolo-gy theory, on the other hand there are dual numbers which are a less known structure than complexnumbers but important in many approaches, then there are commuting nilpotent variables, some-how generalizing the nilpotent but anticommuting Grassmann variables used in super-mathematicsand SUSY models. As an interesting fact we note that nilpotent variables demand the use of thegeneralized light-cone geometry with the metric of the null signature.Key words: infrared divergence, hydrodynamical approach, renormalization.PACS number(s): 11.30.Pb, 45.20.JjI. INTRODUCTION