Abelian integral

Vol. 1480 F. Dumortier, R. Roussarie, J. Sotomayor, H. Zoladek, Bifurcations of Planar Vector Fields Nilpotent Singularities and Abelian Integrals. VIII, 226 pages. 1991. 

Equation (33) has the form of an Abelian integral equation which is solved by... 

A. Andreotti, A. Mayer, On period relations for Abelian integrals on algebraic curves (Ann. Scu. Norm. Sup. Pisa (1967)). 

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.

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. 

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 ... 

We will apply the Abelian Stokes theorem to the exponent of the integrand of the path integral yielding the r.h.s. of the non-Abelian Stokes theorem [a counterpart of the r.h.s. in Eq (8)]. 

First, let us derive the path integral expression for the parallel-transport operator U along L. To this end, we should consider the non-Abelian formula (differential equation) analogous to Eq. (23)... 

At present, we are prepared to formulate a holomorphic path-integral version of the non-Abelian Stokes theorem... 

The non-Abelian Stokes theorem gives the homogeneous field equation of 0(3) electrodynamics, a Jacobi identity in the following integral form ... 

Therefore, the distinction between the topological and dynamical phase has vanished, and the realization has been reached that the phase in optics and electrodynamics is a line integral, related to an area integral over Bt3> by a non-Abelian Stokes theorem, Eq. (553), applied with 0(3) symmetry-covariant derivatives. It is essential to understand that a non-Abelian Stokes theorem must be applied, as in Eq. (553), and not the ordinary Stokes theorem. We have also argued, earlier, how the non-Abelian Stokes explains the Aharonov-Bohm effect without difficulty. 

Barrett [50] has interestingly reviewed and compared the properties of the Abelian and non-Abelian Stokes theorems, a review and comparison that makes it clear that the Abelian and non-Abelian Stokes theorems must not be confused [83,95]. The Abelian, or original, Stokes theorem states that if A(x) is a vector field, S is an open, orientable surface, C is the closed curve bounding S, dl is a line element of C, n is the normal to S, and C is traversed in a right-handed (positive direction) relative to n, then the line integral of A is equal to the surface integral over 5 of V x A-n ... 

In the non-Abelian Stokes theorem (482), on the other hand, the boundary conditions are defined because the phase factor is path-dependent, that is, depends on the covariant derivative [50]. On the U(l) level [50], the original Stokes theorem is a mathematical relation between a vector field and its curl. In 0(3) or SU(2) invariant electromagnetism, the non-Abelian Stokes theorem gives the phase change due to a rotation in the internal space. This phase change appears as the integrals... 

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