The Electromagnetic Potential

In this section, we are concerned with the canonical equations of the radiation field. We consider the fact that the electromagnetic wave is a transverse wave, and convert it into the form of Hamilton kinetic equations which are independent of the transformation parameter. In this process we will reach the conclusion that the radiation field is an ensemble of harmonic oscillators. During this process we will stress the concepts of vector potential and scalar potential. The equations of an electromagnetic wave in the vacuum are summarized as follows  

The general solution of these equations of the electromagnetic wave yields a rather complicated mathematical expression. So, by using the second equation of Eq. (1.60) and remembering the formula, V(Vx ) = 0 given in Table 1.3, we introduce the vector potential, A(x,y,z,t), which satisfies the following equations  

This scalar potential, 0, automatically satisfies Eq. (1.65). Thus, the electric field, E, can be expressed with no more than two potential functions, A and j . 

The properties of Eqs. (1.62) and (1.67) can be examined from a different viewpoint. We already understand that the electromagnetic potentials introduced, A and j , lead to the physical quantities, E and B, but in practice we measure the electric field, E, or the magnetic flux density, B. That is to say, there exist innumerable pairs of functions, A and p, which lead to E and B by the above-mentioned differential operations. Hence, it is very difficult to choose a physically-reasonable pair of A and 0 among all the possible pairs. For example, A and j can be transformed to A and j , respectively, as in Eq. (1.68) by using an arbitrary scalar function, ip(x,y, t). 

When we introduce these transformed pairs to the definitions (1.62) and (1.67), we get 

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Essentially what has enabled us to rewrite the matrix element in this form is the fact that the sources of the electromagnetic potential, that is, the currents of the charged particles obey the continuity equation d j x) — 0. 

If we take the transformation properties of the electromagnetic potential operators to be... 

The electromagnetic potentials A and are not determined uniquely by the expressions for B and E. Thus the magnetic field is not affected by the addition of a scalar gradient to A... 

We note that the expression (58) for the electromagnetic potential in a quark core coincides with the analogous one in the 2SC phase [15], but the expression (59) for the gluonic potential differs from analogous one in Ref [15] by its sign. 

To conclude this section we mention that the electromagnetic potential in the hadronic phase of a neutron star can be found from the solution (51) by replacing the penetration depth for quark matter q with that for hadronic matter Xp. 

In standard classical electrodynamics, the Maxwell equation d = 0 becomes a Bianchi identity by using the electromagnetic potential s J, defined as = ds J. The dynamical equation for this field in empty space is d 0. 

What has been presented here is a semiclassical theory of TJ 1) quantum electrodynamics. Here the electromagnetic field is treated in a purely classical manner, but where the electromagnetic potential has been normalized to include one photon per some unit volume. Here the absorption and emission of a photon is treated in a purely perturbative manner. Further, the field normalization is done so that each unit volume contains the equivalent of n photons and that the energy is computed accordingly. However, this is not a complete theory, for it is known that the transition probability is proportional to n + 1. So the semiclassical theory is only appropriate when the number of photons is comparatively large. 

We could now proceed to substitute the second term of (3.133) for A in (3.142) but, as mentioned earlier, the second term of (3.133) is incorrect. It has been found that the correct form of the interaction between the momentum P and this part of the vector potential is obtained only if the retardation of the electromagnetic potentials is included. We do not go into the details here but simply quote the resulting contribution to (3.142),... 

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]. 

Electromagnetic potential equations and boundary conditions Another approach to the formulation of electromagnetic boundary-value problem is to use the electromagnetic potentials introduced in Chapter 8. This approach has been used in a number of publications on numerical electromagnetic methods (Biro and Preis, 1990 Everett and Schultz, 1996 Everett, 1999 Haber et aJ., 1999). 

Similar to the field separation into the background and anomalous parts, one can represent the electromagnetic potentials as the sums of the corresponding potentials for the background and anomalous fields ... 

Figure 12-5 The staggered grid for electromagnetic field discretization in the electromagnetic potential method. The normal components of electric field E are defined at the centers of the cell faces, while the tangential components of magnetic field H are defined on the cell edges.

In other respects, one introduces in the expression of the electromagnetic potentials the factor l/(47reo) (the presence of 4n is due to the writing 47rjfl instead of jfl in the current term of the Maxwell equations), where eo is the permittivity of free space, and e is expressed in e.m.u ... 

If we perform a Poincare transformation of the electromagnetic potentials according to (84), then the corresponding new field strengths E and B will again satisfy Maxwell s equations. 

It was slowly realized [39] that whereas the electromagnetic potentials uniquely defines the electromagnetic fields, they are themselves unique only to within gauge transformations... 

The introduction of the electromagnetic potentials, A and 0, into Maxwell equations, the derivation of the wave function of A, and the canonicalization of the space consisting of this... 

Here r is a position of the electron moving at a constant velocity relative to the observer. The electromagnetic potential evaluated at the time of observation has the components... 

With no loss of generality, the electromagnetic potentials become... 

In the presence of the electromagnetic potential a new term appears in the quadratic form of the Dirac equation. This allows an introduction of the intrinsic magnetic moment of an electron, generated by its spin, which interacts with the external magnetic field. 

This is analogous to a gauge transformation, suggesting identihcation of 7 4 with the electromagnetic potential tpi, apart from a proportionality constant. The antisymmetric tensor... 

The polar plane contains the contact point of the tangent cone with the surface. The polar hyperplane should represent the electromagnetic potentials and the cone, alternatively, the gravitational potentials (Fig. 3). 

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... 

Measurements of the London penetration depth A(T) point to the existence of aniostropic superconducting phases in UBejj (Einzel et al. 1986) and CeCu2Si2 (Gross et al. 1988), as well. Figure 76 demonstrates for the former system clear deviations from the temperature dependence expected for an ordinary, isotropic superconductor. As an explanation, it has been proposed (Gross et al. 1986, Millis 1987) to use the standard connection between the condensate velocity and the electromagnetic potentials,... 

The differential operator Vx consisting of the cross product of the differential vector V with another vector is called the rotational or curl operator, and the notation in terms of a function rot or curl is sometimes used. An example of such a link is the relationship between the electromagnetic induction B and the electromagnetic potential vector A, written as... 

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