Anticommuting variables, internal degrees of freedom, and the Wilson loop

Abstract

Abstract In this paper we show that it is possible to give a real physical meaning to theories in which internal degrees of freedom are described by Grassman variables. The physical theory is defined by means of an averaging procedure in terms of a distribution function in the Grassmann restricted space satisfying all the physical requirements. If we use this result for a scalar particle with inner degrees of freedom (electric charge, colour, …) interacting with Yang-Mills gauge fields, it turns out that we can define two different classical theories. Taking the average of the coupled particle-field equations of motion, we recover the usual classical theory. Taking instead the average of the solution of such equations we get a theory which is free from all the classical infinities (and so of the causal defects, like runaway) solution or pre-acceleration) but also of all the effects of the same order in the charges (like radiation). The main point is that the processes of averaging and integrating the equations of motion do not commute. Then for the case of colour degrees of freedom we study the quantization of the theory by the path-integral method and we show that the functional integration can be done for an arbitrary gluon field simply by using the classical solution. As a result we obtain an expression for the Wilson loop as a functional integral for the internal fermionic degrees of freedom.