ar X iv : h ep - l at / 0 30 90 23 v 1 7 S ep 2 00 3 On the non - Abelian Stokes theorem for SU ( 2 ) gauge fields ITEP - LAT

We derive a version of non-Abelian Stokes theorem for SU(2) gauge fields in which neither additional integration nor surface ordering are required. The path ordering is eliminated by introducing the instantaneous color orientation of the flux. We also derive the non-Abelian Stokes theorem on the lattice and discuss various terms contributing to the trace of the Wilson loop. Introduction The usual Abelian Stokes theorem relates the integral along closed curve C bounding some surface SC and an integral defined on this surface. This version of the Stokes theorem is most relevant in physical applications since it allows to express the holonomy of gauge noninvariant electromagnetic potential via physically observable magnetic flux. The invention of non-Abelian gauge theories necessitated a non-Abelian generalization of the Stokes theorem. Nowadays, there are quite a few formulations of non-Abelian Stokes theorem (NAST) available (for review see, e.g., Ref. [1] and references therein). Generically there exist two principal approaches, the operator [2]—[7] and the path-integral [8]—[13] one. It is worth to mention that the central issue of any formulation of NAST is how to make sense of the path ordering prescription inherent to the non-Abelian holonomy (Wilson loop). In this respect neither operator nor path-integral approaches are helpful for concrete calculation of the Wilson loop. Indeed, while in the former case the path ordering is traded for much more complicated surface ordering prescription, in the latter case the additional path integral over auxiliary variables is introduced which cannot be calculated even approximately. In this paper we derive a new version of non-Abelian Stokes theorem focusing exclusively on SU(2) valued Wilson loops in the fundamental representation. Moreover, the central object of our discussion is the phase of the Wilson loop φw. It determines the Wilson loop trace, 1/2TrW = cosφw, which is the only gauge invariant quantity associated with gauge holonomy. The basic idea is to introduce the instantaneous color orientation of the chromomagnetic flux piercing the loop. Evidently this color orientation remains unknown until the Wilson loop is calculated by some other means. Nevertheless, it allows us to avoid the path ordering and represent the Wilson loop phase as an ordinary integral to which Abelian Stokes theorem applies. Furthermore, we relate the resulting surface integral with properties of gauge fields on this surface. As we noted above, in order to get explicitly the color orientation of the flux one has to calculate the Wilson loop first. In this respect our formulation is well suited for the lattice where the gauge holonomy is to be calculated numerically. We derive the lattice version of the non-Abelian Stokes theorem. Finally, we discuss the physical meaning and origin of various terms contributing to the trace of the Wilson loop and present the results of our qualitative numerical simulations. Non-Abelian Stokes theorem in the continuum limit Consider Wilson loop operator in the fundamental representation, W (T ), evaluated on a closed contour C = {xμ(t), t ∈ [0;T ], xμ(0) = xμ(T )}, which is parameterized by differentiable functions xμ(t). By definition the operator W (T ) provides a solution to the first-order differential equation 〈ψ(t) | (i∂t + A) = 0 , (1) 〈ψ(t) | = 〈ψ(0) |W (t) , (2)