Dirac electromagnetic interaction energy

If there is no explicit external electromagnetic field, the covariant field equations determine a self-interaction energy that can be interpreted as a dynamical electron mass Sm. Since this turns out to be infinite, renormalization is necessary in order to have a viable physical theory. Field quantization is required for quantitative QED. The classical field equation for the electromagnetic field can be solved explicitly using the Green function or Feynman propagator GPV, whose Fourier transform is —gllv/K2, where k = kp — kq is the 4-momentum transfer. The product of y0 and the field-dependent term in the Dirac Hamiltonian, Eq. (10.3), is... 

We have already seen in section 8.1 that (i) a Dirac electron with electromagnetic potentials created by all other electrons [cf. Eq. (8.2)] cannot be solved analytically, which is the reason why the total wave function as given in Eq. (8.4) cannot be calculated, and also that (ii) the electromagnetic interactions may be conveniently expressed through the 4-currents of the electrons as given in Eq. (8.31) for the two-electron case. Now, we seek a one-electron Dirac equation, which can be solved exactly so that a Hartree-type product becomes the exact wave function of this system. Such a separation, in order to be exact (after what has been said in section 8.5), requires a Hamiltonian, which is a sum of strictly local operators. The local interaction terms may be extracted from a 4-current based interaction energy such as that in Eq. (8.31). Of course, we need to take into account Pauli exchange effects that were omitted in section 8.1.4, and we also need to take account of electron correlation effects. This leads us to the Kohn-Sham (KS) model of DFT. 

The total Hamiltonian describing the total interacting many-body problem—Dirac particles + radiation field + nucleus—may be obtained from the T 00-component of the energy-momentum tensor. The part of the Hamiltonian relevant for the relativistic description of the atomic many-body problem in the presence of the external electromagnetic held of the nucleus including radiative corrections and possible interactions... 

The confirmation by Lamb and Retherford of the inadequacy of the Dirac theory stimulated a re-examination of a theoretical problem to which only a very incomplete solution had so far been found the problem of the interaction between charged particles and the electromagnetic field. We shall briefly refer to the problem as it presented itself in classical physics, and then (following Weisskopf [135]) notice the further difficulties which the quantum theory introduces. Finally we shall see how these difficulties have been circumvented by the new quantum electrodynamics, and how a small correction is thereby introduced to the energy levels predicted by Dirac s theory. The new theory, however, is not a complete and logically satisfactory solution to the problems we shall state a difficulty of principle remains now, as formerly. 

In order to derive an expression for the energy of interaction between an intrinsic magnetic moment (as for a particle with nonzero spin angular momentum) and the electromagnetic field, one cannot proceed via classical mechanics as for the charged particle. Rather one proceeds via the relativistic Dirac equation and seeks the nonrelativistic limit... 

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