Electromagnetic potentials gauge invariance

In the next fourth order, there appear diagrams, whose contribution into the ImA accounts for the core polarization effects. This contribution describes collective effects, and it is dependent upon the electromagnetic potential gauge (the gauge non-invariant contribution). Let us examine the multielectron atom with... 

The identical transformation, equation (6), of the electromagnetic vector potentials was found before to leave the fields unaffected or gauge invariant. The fields Atl are not gauge invariant, but the fields described by the tensor, equation (33)... 

As is well known in classical electromagnetics, the fields described by the Maxwell equations can be derived from a vector potential and a scalar potential. However, there are various forms that are possible, all giving the same fields. This is referred to as gauge invariance. In making measurements at some point... 

So now we have the question poased in an interesting form. There are two quite different kinds of antennas, both of which produce electric dipole fields, but different Lorenz potentials, one emphasizing the vector potential and the other, the scalar potential. In a classical electromagnetic sense, one cannot distinguish these two cases by measurements of the fields (the measurable quantities) at distances away from the source region. The gauge invariance of QED implies the same in quantum sense. 

So our choices of the two antennas is not unique for separately emphasizing the Lorenz vector and scalar potentials. All that is required is for the two to have the same exterior fields (say, electric dipole fields, or more general multipole fields) with different potentials (related by the gauge condition). In a classical electromagnetic sense, these antennas cannot be distinguished by exterior measurements. This is a classical nonuniqueness of sources. In a QED sense, the same is the case due to gauge invariance in its currently accepted form. 

Therefore, it has been shown convincingly that electrodynamics is an 0(3) invariant theory, and so the 0(3) gauge invariance must also be found in experiments with matter waves, such as matter waves from electrons, in which there is no electromagnetic potential. One such experiment is the Sagnac effect with electrons, which was reviewed in Ref. 44, and another is Young interferometry with electron waves. For both experiments, Eq. (584) becomes... 

The total Lagrangian if I X I if2 is now invariant under the local gauge transformation because of the introduction of the 4-potential A, which couples to the current of the complex A of the pure gauge vacuum. The field A also contributes to the Lagrangian, and since if + ifj + if2 is invariant, an extra term if3 appears, which must also be gauge-invariant. This can be so only if the electromagnetic field is introduced... 

We depart from former treatments in other ways. Commencing with a correct observation that the Aharonov-Bohm effect depends on the topology of the experimental situation and that the situation is not simply connected, a former treatment then erroneously seeks an explanation of the effect in the connectedness of the U(l) gauge symmetry of conventional electromagnetism, but for which (1) the potentials are ambiguously defined, (the U(l) A field is gauge invariant) and (2) in U(l) symmetry V x A = 0 outside the solenoid. 

The action integral A is not changed if the trial function f is multiplied by a phase factor exp(/f y t )dt /ifi), while Ti is increased by a time-dependent but spatially uniform potential y(t). This is an example of gauge invariance, taken out of the usual context of electromagnetic theory. Indicating the modified wave function by fy, the modified action integral is... 

Since ip depends on space-time coordinates, the relative phase factor of ip at two different points would be completely arbitrary and accordingly, a must also be a function of space-time. To preserve invariance it is necessary to compensate the variation of the phase a (a ) by introducing the electromagnetic potentials (T4.5). In similar vein the gravitational field appears as the compensating gauge field under Lorentz invariant local isotopic gauge transformation [150]. 

Section II includes an outline of the generalization of the vector potential of electromagnetic theory so as to include a gauge-invariant pseudovector part. 

That is, this pair of A and also leads to the same B and E. This kind of transformation of potential functions is called a gauge transformation. Thus, we can say that the electromagnetic field is invariable against the gauge transformation. 

D.H. Kobe, Conventional and gauge-invariant probability amplitudes when electromagnetic potentials are turned on and off, Eur. J. Phys. 5 (1984) 172. 

Applied to the potentials of the electromagnetic field the coordinate system is determined only to within an additive gradient, which is the well-known property of the vector potential of the Maxwell field. In common practice it is necessary to assume the gauge invariance, which appears naturally in projective relativity. 

This unwelcome discovery is potentially catastrophic for our unified weak and electromagnetic gauge theory. There we have lots of gauge invariance, many conserved currents, both vector and axial-vector, and hence many Ward identities. Moreover the Ward identities play a vital role in proving that the theory is renormalizable. It is the subtle interrelation of matrix elements that allows certain infinities to cancel out and render the theory finite. Thus we cannot tolerate a breakdown of the Ward identities, and we have to ensure that in our theory these triangle anomalies do not appear. 

The exact wave function changes in a characteristic way under gauge transformations of the potentials associated with electromagnetic fields, thereby ensuring that all molecular properties that may be calculated or extracted from the wave function are unaffected by such transformations [5]. It is desirable to incorporate the same gauge invariance in the calculation of properties from approximate wave functions so as to make the calculations unambiguous and well defined. 

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... 

Here,, 4 ( is the vector 4-potential introduced in the vacuum as part of the covariant derivative, and therefore introduced by spacetime curvature. The electromagnetic field and the topological charge g are the results of the invariance of the Lagrangian (868) under local U(l) gauge transformation, in other words, the results of spacetime curvature. 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

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