Gauge theories

O RalfeartaIgh L 1997 The Dawning of Gauge Theory (Prinoeton, NJ Prinoeton University Press)... 

Altohlson I J R and Fley A J G 1996 Gauge Theories in Particle Physics A Practical Introduction (Bristol Institute of Physios Publishing)... 

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

L. O RaifearCaigh, The Dawning of Gauge Theory, Princeton University Press, Princeton, N. J., 1997. 

The Coulomb gauge theory and the Lorentz gauge theory thus both describe the same physical phenomena, but they handle one aspect of the physical situation, namely, the Coulomb interaction, in fundamentally different ways. In the Coulomb gauge the interaction is... 

It is possible to formulate the Coulomb gauge theory in terms of radiation operators which satisfy the subsidiary condition... 

Gauge theories, Yang-Mills field, 204-205 Gauge transformation ... 

The electromagnetic field is a well-known example of a gauge theory. A gauge transformation of the form... 

Gauge theories are commonly formulated in terms of group theory. Under a phase rotation... 

C. Hong-Mo and T.S. Tsun, Some Elementary Gauge Theory Concepts, 1993, World Scientific, Singapore. 

Figure 5. Nearest-neighbor spacing distribution P(s) for U(l) gauge theory on an 83 x 6 lattice in the confined phase (left) and in the Coulomb phase (right). The theoretical curves are the chUE result, Eq. (14), and the Poisson distribution, -Pp(s) = exp(-s).

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

Gauge theory on resolution of simple singularities and simple Lie algebras. Inter. Math. Res. 

It should finally be mentioned that the basic equations (l)-(8) have been derived from gauge theory in the vacuum, using the concept of covariant derivative and Feynman s universal influence [38]. These equations and the Proca field equations are shown to be interrelated to the well-known de Broglie theorem, in which the photon rest mass m can be interpreted as nonzero and be related to a frequency v = moc2/h. A gauge-invariant Proca equation is suggested by this analysis and relations (l)-(8). It is also consistent with the earlier conclusion that gauge invariance does not require the photon rest mass to be zero [20,38]. 

A simple example in classical electrodynamics of what is now known as gauge invariance was introduced by Heaviside [3,4], who reduced the original electrodynamical equations of Maxwell to their present form. Therefore, these equations are more properly known as the Maxwell-Heaviside equations and, in the terminology of contemporary gauge field theory, are identifiable as U(l) Yang-Mills equations [15]. The subj ect of this chapter is 0(3) Yang-Mills gauge theory applied to electrodynamics and electroweak theory. 

The systematic development of gauge theory relies on a rotation of dimensional function / of the spacetime coordinate in special relativity. The... 

From the foregoing, U(l) electrodynamics was never a complete theory, although it is rigidly adhered to in the received view. It has been argued already that the Maxwell-Heaviside theory is a U(l) Yang-Mills gauge theory that discards the basic commutator A(1) x A(2). However, this commutator appears in the fundamental definition of circular polarity in the Maxwell-Heaviside theory through the third Stokes parameter... 

The inverse Faraday effect depends on the third Stokes parameter empirically in the received view [36], and is the archetypical magneto-optical effect in conventional Maxwell-Heaviside theory. This type of phenomenology directly contradicts U(l) gauge theory in the same way as argued already for the third Stokes parameter. In 0(3) electrodynamics, the paradox is circumvented by using the field equations (31) and (32). A self-consistent description [11-20] of the inverse Faraday effect is achieved by expanding Eq. (32) ... 

U(l), whose group space is a circle. This result is another internal inconsistency, because the group space of a gauge theory is a circle, there can be no physical quantity in free space perpendicular to that plane. It is necessary but not sufficient, in this view, that the Lagrangian in U(l) field theory be invariant [6] under U(l) gauge transformation. 

All gauge theory depends on the rotation of an -component vector whose 4-derivative does not transform covariantly as shown in Eq. (18). The reason is that i(x) and i(x + dx) are measured in different coordinate systems the field t has different values at different points, but /(x) and /(x) + d f are measured with respect to different coordinate axes. The quantity d i carries information about the nature of the field / itself, but also about the rotation of the axes in the internal gauge space on moving from x + dx. This leads to the concept of parallel transport in the internal gauge space and the resulting vector [6] is denoted i(x) + d i. The notion of parallel transport is at the root of all gauge theory and implies the introduction of g, defined by... 

The P on the left-hand side of Eq. (162) denotes path ordering and the P denotes area ordering [4]. Equation (162) is the result of a round trip or closed loop in Minkowski spacetime with 0(3) covariant derivatives. Equation (161) is a direct result of our basic assumption that the configuration of the vacuum can be described by gauge theory with an internal 0(3) symmetry (Section I). Henceforth, we shall omit the P and P from the left- and right-hand sides, respectively, and give a few illustrative examples of the use of Eq. (162) in interferometry and physical optics. 

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