CLASSICAL YANG-MILLS THEORY WITH FERMIONS I: GENERAL PROPERTIES OF A SYSTEM WITH CONSTRAINTS

Abstract

A review is made of the basic properties of the Hamiltonian description of classical Yang-Mills theory with fermions described by Grassmann-valued Dirac spinors, in the case of a trivial principal bundle with a compact, semisimple, connected, simply connected Lie structure group over Minkowski space-time. The Poincaré group is assumed to be globally implementable and only the field configurations producing finite Poincaré generators are considered. A detailed study of the Hamiltonian group of gauge transformations is made, trying to elucidate the meaning of the global gauge transformations (connected with the non-Abelian charges and with the center of the gauge group), of the winding number (connected with the large gauge transformations and with the topological charge) and of the small gauge transformations generated by the first class constraints. This leads to the identification of boundary conditions on the gauge potentials and their conjugate momenta suitable for the Hamiltonian description and allowing covariance of the non-Abelian charges. Finally, a review is made of the problem of the Gribov ambiguity, whose basis is connected with the existence of stability subgroups of gauge transformations for certain gauge potentials (gauge symmetries) and/or certain field strengths (gauge copies) in generic Sobolev spaces.