The non-relativistic limit of electrodynamics

The action integral (102) - on which the Lagrangean equations of motion are based - consists of two parts, one of which (involving T) is Galilei-invariant, the other (involving U) is Lorentz invariant. There are two ways to remedy this inconsistency. One is to replace (101) by (104), i.e. to use a fully Lorentz invariant theory. The other possibility is to choose the nonrelativistic (Galilei-invariant) limit of electrodynamics. 

To construct this limit we can proceed as we did for the nrl of the Dirac equation. We choose a system of units, for which in the Maxwell equations only appears, but not c. There are two allowed choices, either b — c or 6 = 1, but not b = c like in the Gaussian system. 

The scalar potential satisfies in the nonrelativistic limit the Poisson equation (95) with no difference between the Coulomb and the Lorentz gauge. 

There is no wave equation for or A, i.e. in the nrl there are no electromagnetic waves. The nonrelativistic Hamiltonian is 

The difference between the nrl of electrodynamics and full electrodynamics does not show up explicitly in (115) and (116), but the fields and A transform differently between two inertial systems. 

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T. Sane. Spin-Interactions and the Non-relativistic Limit of Electrodynamics. Adv. Quantum Chem., 48 (2005) 383 05. 

The above considerations leads to the somewhat troubling question of whether (128) represents the true non-relativistic limit of the Dirac equation in the presence of external fields. Referring back to (110) we have certainly obtained the non-relativistic limit of the free-particle part Lm, but we have in fact retained the interaction term as well as the Lagrangian of the free field. In order to obtain the proper non-relativistic limit, we must consider what is the non-relativistic limit of classical electrodynamics. This task is not facilitated by the fact that, contrary to purely mechanical systems, the laws of electrodynamics appear in different unit systems in which the speed of light appears differently. In the Gaussian system Maxwell s laws are given as... 

If we approach the non-relativistic limit in the standard manner, that is by letting the speed of light c go to infinity, we find that classical electrodynamics reduces to electrostatics, that is, all magnetic fields disappear. There is thus no need to introduce a vector potential A. Furthermore, we can see that the concept of... 

We shall stress here another aspect in favor of PT. Actually in both non-relativistic and relativistic quantum mechanics one studies the motion (mechanics) of charged particles, that interact according to the laws of electrodynamics. The marriage of non-relativistic mechanics with electrodynamics is problematic, since mechanics is Galilei-invariant, but electrodynamics is Lorentz-invariant. Relativistic theory is consistent insofar as both mechanics and electrodynamics are treated as Lorentz-invariant. A consistent non-relativistic theory should be based on a combination of classical mechanics and the Galilei-invariant limit of electrodynamics as studied in subsection 2.9. 

We start this chapter with a discussion of the non-relativistic limit (nrl) of relativistic quantum theory (section 2). The Levy-Leblond equation will play a central role. We also discuss the nrl of electrodynamics and study how properties differ at their nrl from the respective results of standard non-relativistic quantum theory. We then present (section 3) the Foldy-Wouthuysen (FW) transformation, which still deserves some interest, although it is obsolete as a starting point for a perturbation theory of relativistic corrections. In this context we discuss the operator X, which relates the lower to the upper component of a Dirac bispinor, and give its perturbation expansion. The presentation of direct perturbation theory (DPT) is the central part of this chapter (section 4). We discuss the... 

In order to complete our derivation of the molecular Hamiltonian we must consider the nuclear Hamiltonian in more detail. A thorough relativistic treatment analogous to that for the electron is not possible within the limitations of quantum mechanics, since nuclei are not Dirac particles and they can have large anomalous magnetic moments. However, the use of quantum electrodynamics [18] shows that we can derive the correct Hamiltonian to order 1 /c2 by taking the non-relativistic Hamiltonian ... 

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