Non-Abelian gauge theories

Gauge theories are not restricted to local f7(l) symmetry. If one allows for local transformations which lead to mixtures of two fermionic fields 4 a(x) and /3(x) according to 

The first term in the second line is the Lagrangian density of the total free fermion field i x), which represents here a doublet of two fermion fields. The third term corresponds to the Lagrangian density of the free boson 

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

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. 

This non-Abelian gauge theory satisfies the usual transformation properties. If Jl is the base manifold in four dimensions, then the gauge theory is determined by an internal set of symmetries described by a principal bundle. Let Ua, where a = 1,2,be an atlas of charts on the Ji. The transitions from one chart to another is given by gap f/p —> Ua, where these determine the transition functions between sections on the principal bundle. The transform between one section to another is given by... 

T. Kibble, Symmetry breaking in non-abelian gauge theories, Phys. Rev. 155 (1966) 1554-1561. 

The whole of the above analysis can be adapted to any non-Abelian gauge theory. 

The developments of the previous chapters provide the ingredients that one can use to construct models of spontaneously broken non-Abelian gauge theories. In Chapters 21-23 we shall discuss quantum chromodynamics (QCD) which is proposed as a model for strong interactions. Here we shall concentrate on models which attempt to unify the weak and electromagnetic interactions into a unique renormalisable theory. 

Equation (B.26) has the structure of a quaternion-valued non-Abelian gauge field theory. If we denote... 

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 ... 

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... 

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. 

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

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. 

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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