Current density, 0 electrodynamics

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

Though the ESR Hamiltonian is typically expressed in terms of effective electronic and nuclear spins, it can, of course, also be derived from the more fundamental Breit-Pauli Hamiltonian, when the magnetic fields produced by the moving nuclei are explicitly taken into account. In order to see this, we shall recall that in classical electrodynamics the magnetic dipole equation can be derived in a multipole expansion of the current density. For the lowest order term the expansion yields (59)... 

This result, as well as the form of expressions (23) and (24), shows that the charge and current density relations (3), (4), and (8) of the present extended theory become consistent with and related to the Dirac theory. It also implies that this extended theory can be developed in harmony with the basis of quantum electrodynamics. 

Therefore, the vacuum charge and current densities of Panofsky and Phillips [86], or of Lehnert and Roy [10], are given a topological meaning in 0(3) electrodynamics. In this condensed notation, the vacuum 0(3) field tensor is given by... 

The Lagrangian (850) shows that 0(3) electrodynamics is consistent with the Proca equation. The inhomogeneous field equation (32) of 0(3) electrodynamics is a form of the Proca equation where the photon mass is identified with a vacuum charge-current density. To see this, rewrite the Lagrangian (850) in vector form as follows ... 

Equation (C.5) means that there are no magnetic charge or current densities in 0(3) electrodynamics. 

It is emphasized, however, that there is no reason to assume plane waves. These are used as an illustration only, and in general the vacuum charge current densities of 0(3) electrodynamics are richly structured, far more so than in U(l) electrodynamics, where vacuum charge current densities also exist from the first principles of gauge theory as discussed already. 

Therefore, a check for self-consistency has been carried out for indices p 2 and v = 1. It has been shown, therefore, that in pure gauge theory applied to electrodynamics without a Higgs mechanism, a richly structured vacuum charge current density emerges that serves as the source of energy latent in the vacuum through the following equation ... 

It thus becomes clear that the vacuum charge current density introduced by Lehnert is an excitation above the true vacuum in classical electrodynamics. The true vacuum is defined by Eq. (337). It follows that in the true classical vacuum, the electromagnetic field also disappears. 

We have established that, in 0(3) electrodynamics, the vacuum charge current densities first proposed by Lehnert [42,45,49] take the form... 

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... 

The current density has a clear physical meaning and this facilitates the construction of phenomenological approaches. In addition, expression (1) itself is, in fact, familiar. It reproduces the Joule-Lentz rule of classical electrodynamics for energy dissipation in a medium when an electric field is applied. 

In order to determine fluxes and current density i, it is necessary to know Vp or E For their deflnition it is necessary to use the Maxwell equations. In general case the external electric held induces secondary electric and magnetic fields (the medium s response), which in turn influence the external field. However, if the external magnetic field is absent, and the external electric field is quasi-stationary, then the electrodynamical problem reduces to electrostatic one, namely, to determining of the electric potential distribution in liquid, described by Poisson equation... 

We begin by introducing the basic ingredients of electrodynamics in covariant notation. The charge-current density... 

We now have to construct the interaction term between the dynamical variables, i.e., the gauge field and the sources of the electrodynamical field, i.e., charged particles giving rise to a charge-current density cf. Eq. (3.161). Again, the contribution of this interaction term to the action S has to be Lorentz and gauge invariant, and the simplest choice is therefore given by... 

This interaction Lagrangian density may depend explicitly on the space-time coordinates x and the 4-velocity u via the charge-current density T. However, as far as only the equation of motion for the electrodynamic field is concerned they do not represent dynamical variables. Lorentz invariance of this interaction term is obvious, and gauge invariance of the corresponding action is a direct consequence of the continuity equation for the charge-current density f, cf. Eq. (3.162),... 

In hie most recent papers (20) Fulton has put his case in the statement e take the view that if nature does not endow herself with cavities we should not have to introduce them. . He avoids such introduction by taking an external charge and current density as sources of electromagnetic field in the medium and by using methods of quantum electrodynamics obtains solutions of microscopic and macroscopic field equations for the polarizations and fields with susceptibility and permittivity obtained as functional derivatives of polarization with respect to source field and macroscopic . 

The response properties introduced via Eqs. (17)-(19) are conveniently re-expressed via the induced current densities allowing for the relationships of classical electrodynamics [6]... 

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