Gauge transformation theory

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... 

This method has the advantage of formulating the theory in terms of the (observable) field intensities. The more usual procedure which starts from a lagrangian formulation22 expresses the theory in terms of the potentials Alt(x). A gauge transformation... 

The theory is, however, invariant under a gauge transformation whereby... 

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... 

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. 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

Therefore the Lehnert equation (253) correctly conserves action under a local U(l) gauge transformation in the vacuum. Such a transformation leads to a vacuum charge current density as the result of gauge theory itself, because U(l) gauge theory has a scalar internal space that supports A and A. These must be complex in order to define the globally conserved charge ... 

So it becomes clear that the description of the vacuum in gauge theory can be developed systematically by recognizing that, in general, A is an -dimensional vector. On the U(l) level, it is one-dimensional on the 0(3) level, it is three-dimensional and so on. The internal gauge space in this development is a physical space that can be subjected to a local gauge transform to produce physical vacuum charge current densities. 

In the 0(3) invariant theory of the Aharonov-Bohm effect, the holonomy consists of parallel transport using 0(3) covariant derivatives and the internal gauge space is a physical space of three dimensions represented in the basis ((1),(2),(3)). Therefore, a rotation in the internal gauge space is a physical rotation, and causes a gauge transformation. The core of the 0(3) invariant explanation of the Aharonov-Bohm effect is that the Jacobi identity of covariant derivatives [46]... 

In the case of quantum field theory the section determines the Hilbert space of states under a certain gauge. This choice of gauge then determines the unitary representation of the Hilbert space. We may then replace the section with the fermion field /, which acts on the Fock space of states. It is then apparent that a gauge transformation A t > A t + 84 is associated with a unitary transform of the fermion field v / > v / I 8 /. The unitary transformation of the fermion... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

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