Gauge Abelian

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

In conclusion, we have shown that the non-Abelian gauge-field intensity tensor fi sc(X) shown in Eq. (113) vanishes when... 

Neumann boundary conditions, electronic states, adiabatic-to-diabatic transformation, two-state system, 304-309 Newton-Raphson equation, conical intersection location locations, 565 orthogonal coordinates, 567 Non-Abelian theory, molecular systems, Yang-Mills fields nuclear Lagrangean, 250 pure vs. tensorial gauge fields, 250-253 Non-adiabatic coupling ... 

Classical vacuum in non-abelian gauge theory is infinitely degenerate and numbered by Chern-Simons number Ncs of vacuum gauge fields Af ... 

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 two parts of the twisted bundle are copies of SU(2) with a doublet fermion structure. One of the fermions has a very large mass, m = Yn (y)1 ), which is assumed to be unstable and not observed at low energies. So one sector of the twisted bundle is left with the same Abelian structure, but with a singlet fermion, meaning that the SU(2) gauge theory becomes defined by the algebra over the basis elements... 

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 ... 

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]. 

The non-Abelian field naturally appears in the context of (topological) gauge theory [see Eq. (45)]. Now, the Abelian Stokes theorem should suffice. 

In the mid-fifties the violation of parity was discovered, and a universal theory of weak interactions—the (F-A)-theory—was created. Construction of composite hadron models was begun. The first non-abelian gauge theory was developed. 

However, there remains the problem of how to obtain a locally gauge-invariant Proca equation. To address this problem rigorously, it is necessary to use a non-Abelian Higgs mechanism applied within gauge theory. 

In order to form a self-consistent description [44] of interferometry and the Aharonov-Bohm effect, the non-Abelian Stokes theorem is required. It is necessary, therefore, to provide a brief description of the non-Abelian Stokes theorem because it generalizes the ordinary Stokes theorem, and is based on the following relation between covariant derivatives for any internal gauge group symmetry ... 

Equation (482) is a simple form of the non-Abelian Stokes theorem, a form that is derived by a round trip in Minkowski spacetime [46]. It has been adapted directly for the 0(3) invariant phase factor as in Eq. (547), which gives a simple and accurate description of the Sagnac effect [44], A U(l) invariant electrodynamics has failed to describe the Sagnac effect for nearly 90 years, and kinematic explanations are also unsatisfactory [50], In an 0(3) or SU(2) invariant electrodynamics, the Sagnac effect is simply a round trip in Minkowski space-time and an effect of special relativity and gauge theory, the most successful theory of the late twentieth century. There are open questions in special relativity [108], but no theory has yet evolved to replace it. 

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