Abelian Stokes theorem

In order to understand interferometry at a fundamental level in gauge field theory, the starting point must be the non-Abelian Stokes theorem [4]. The theorem is generated by a round trip or closed loop in Minkowski spacetime using covariant derivatives, and in its most general form is given [17] by... 

The space part of this expression is the ordinary, or Abelian, Stokes theorem... 

From the foregoing, it becomes clear that fields and potentials are freely intermingled in the symmetry-broken Lagrangians of the Higgs mechanism. To close this section, we address the question of whether potentials are physical (Faraday and Maxwell) or mathematical (Heaviside) using the non-Abelian Stokes theorem for any gauge symmetry ... 

The non-Abelian Stokes theorem is a relation between covariant derivatives for any gauge group symmetry ... 

The aim of this chapter is to present a short review of the non-Abelian Stokes theorem. At first, we will give an account of different formulations of the non-Abelian Stokes theorem and next of various applications of thereof. 

Before we engage in the non-Abelian Stokes theorem it seems reasonable to recall its Abelian version. The (Abelian) Stokes theorem says (see, e.g., Ref. 1 for an excellent introduction to the subject) that we can convert an integral around a closed curve C bounding some surface S into an integral defined on this surface. Specifically, in three dimensions... 

Figure 1. Integration areas for the lowest-dimensional (nontrivial) version of the Abelian Stokes theorem.

In turn, in geometry A plays the role of connection (it defines the parallel transport around C) and F is the curvature of this connection. A global version of the Abelian Stokes theorem... 

The birth of the ideas related to the (non-)Abelian Stokes theorem dates back to the ninetenth century, with the emergence of the Abelian Stokes theorem. The Abelian Stokes theorem can be treated as a prototype of the non-Abelian Stokes theorem or a version of thereof when we confine our discussion to an Abelian group. 

What is the non-Abelian Stokes theorem To answer this question, we should first recall the form of the well-known Abelian Stokes theorem [see Eq. (2)]... 

A physicist would view the expression (10) as typical in quantum mechanics and as corresponding to the evolution operator. Equations (8) and (9) are, incidentally, very typical in gauge theory, such as in QCD. Thus, guided by our intuition, we can reformulate our chief problem as a quantum-mechanical one. In other words, the approaches to the l.h.s. of the non-Abelian Stokes theorem are analogous to the approaches to the evolution operator in quantum mechanics. There are the two main approaches to quantum mechanics, especially to the construction of the evolution operator opearator approach and path-integral approach. Both can be applied to the non-Abelian Stokes theorem successfully, and both provide two different formulations of the non-Abelian Stokes theorem. 

Unfortunately, it is not possible to automatically generalize the Abelian Stokes theorem [e.g., Eq. (4)] to the non-Abelian one. In the non-Abelian case one faces a qualitatively different situation because the integrand on the l.h.s. assumes values in a Lie algebra g rather than in the field of real or complex numbers. The picture simplifies significantly if one switches from the local language to a global one [see Eq. (5)]. Therefore we should consider the holonomy (7) around a closed curve C ... 

The non-Abelian Stokes theorem is as follows. The non-Abelian generalization of Eq. (5) should read as... 

The non-Abelian Stokes theorem in its original operator form roughly claims that the holonomy around a closed curve C = 05 equals a surface-ordered... 

A more precise form of the non-Abelian Stokes theorem as well as an exact meaning of the notions appearing in the theorem will be given in the course of the proof. 

Proof. Following Aref eva [3] and Menski [8], we will present a short, direct proof of the non-Abelian Stokes theorem. 

There are may other approaches to the (operator) non-Abelian Stokes theorem, which are more or less interrelated, including an analytical approach advocated by Bralic [4] and Hirayama and Ueno [9]. An approach using product integration [10], and last, but not least, a (very interesting) coordinate gauge approach [11,12]. 

In order to formulate the non-Abelian Stokes theorem in the path integral language, we will perform the following three steps ... 

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