Transformation, gauge

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... 

One can define a phase that is given as an integral over the log of the amplitude modulus and is therefore an observable and is gauge invariant. This phase [which is unique, at least in the cases for which Eq. (9) holds] differs from other phases, those that are, for example, a constant, the dynamic phase or a gauge-transformation induced phase, by its satisfying the analyticity requirements laid out in Section I.C.3. 

The vanishing of the YM field intensity tensor can be shown to follow from the gauge transformation properties of the potential and the field. It is well known (e.g., Section II in [67]) that under a unitary transfoiination described by the matrix... 

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

We now describe the relation between a purely formal calculational device, like a gauge transformation that merely admixes the basis states, and observable effects. 

Now, we discuss how the geometric phase is related to the mixing angle in this two-state model. We begin by writing Eq. (A.ll) as the gauge transformation... 

Equation (9-502) is automatically satisfied.19 The potentials Cv(x) are, however, not fixed by Eq. (9-506) since under the gauge transformation ... 

Actually transversality in all the k variables already follows from transversality in any one of the k variables because of the symmetric character of the tensor alll...Un(k1, , kn). Again due to the freedom of gauge transformations an n photon configuration is not described by a unique amplitude but rather by an equivalence class of tensors. We define the notion of equivalence for these tensors, as follows a tensor rfUl. ..Bn( i, , kn) will be said to be equivalent to zero ... 

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 related requirement that when A (x) = 8uO(x) (i.e., zero external field) there be no induced current, i.e., m<0 i (a ) 0)in = 0 (since by a gauge transformation we can revert to the free field case) implies that... 

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

Again, the definition of Uc lacks uniqueness because of the freedom of gauge transformations... 

Hence, the method of Mead and Truhlar [6] yields a single-valued nuclear wave function by adding a vector potential A to the kinetic energy operator. Different values of odd (or even) I yield physically equivalent results, since they yield (< )) that are identical to within an integer number of factors of exp(/< )). By analogy with electromagnetic vector potentials, one can say that different odd (or even) I are related by a gauge transformation [6, 7]. 

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... 

Together, these two equations constitute a compensating gauge transformation. 

For a time-independent gauge vector 0, like an electrostatic field, the gauge transformation after time x4 would yield... 

Now consider two neighbouring points with local phases differing by the amount qApdx 1. Perform a gauge transformation by rotating the phase of the xp function at x by an amount qa(x), i.e. 

The electromagnetic field may now formally be interpreted as the gauge field which must be introduced to ensure invariance under local U( 1) gauge transformation. In the most general case the field variables are introduced in terms of the Lagrangian density of the field, which itself is gauge invariant. In the case of the electromagnetic field, as before,... 

The particle spectrum consists of a massless Goldstone boson 2, a massive scalar i, and more crucially a massive vector A. The Goldstone boson can be eliminated by gauge transformation. For infinitesimal gauge factor a(x),... 

---