Relativistic Electromagnetic Interactions

In dealing with fields that vary over time and space, we will need various differential operators. In the nonrelativistic theory of electrodynamics the gradient operator, V, and the time derivative, d/dr, are used. From our experience in the previous chapter with mixing of space and time coordinates under Lorentz transformations, we might expect these to combine in a four-space differential operator also. Indeed, in our notation. 

The four-vector analogue of the Laplacian is the d Alembertian and is correspondingly defined as 

This operator is the square of a Lorentz-invariant four-vector, and is therefore itself Lorentz invariant. 

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J. H. Detrich, C. C. J. Roothaan. Relativistic Electromagnetic Interaction Without Quantum Electrodynamics. In W. C. Price, S. S. Chissick, Ed., The Uncertainty Principle and Foundations Of Quantum Mlechanics, p. 395-437, London, 1977. John Wiley Sons. 

From the discussion of classical relativistic electromagnetic interactions in chapter 3, we know that the Coulomb interaction between charged particles is not Lorentz invariant, and that we need to take into account the finite transmission speed of the signal between particles (the retardation). If we make the replacement u = ca in (3.84), we get an interaction for low velocities that might well be suitable for relativistic quantum chemistry ... 

Unlike a-decay (and the related nuclear processes involving strong and electromagnetic interactions), -decay involves creation of new particles and relativistic... 

Relativistic electromagnetic solitons (RES) play an important role in the strongly nonlinear interaction between EM radiation and plasma [2]. They... 

This tensor, in a lowest order calculation, is the familiar relativistic Lindhard-type function. The other self-energies appearing in the above equations expressed in terms of the functional Bj are in the functional scheme, whereas those expressed in terms of F have their origins in the -derivable scheme. In both formulations, the contributions due to electronic, ionic, and electromagnetic interactions are incorporated into the respective functionals mentioned above. A direct comparison of the two procedures should be illuminating to see how... 

Transition probabilities in fact provide a particularly severe test of atomic calculations, because they are rather sensitive to the wavefunctions of both levels involved in the transition and to the approximations used, especially when electron correlations and relativistic effects are considered. On a more fundamental level, measurements of transition probabilities are also being used to explore the non-conservation of parity predicted by the unified theories of the weak and electromagnetic interaction here, highly forbidden transitions must be tested for an admixture of allowed transition probabilities [8,9]. 

H. M. Quiney, H. Skaane, I. P. Grant, Relativistic calculation of electromagnetic interactions in molecules, J. Phys. B 30 (1997) L829-L834. 

A standard quantum chemical calculation, relativistic or not, typically calculates stationary states of a molecule at 0 K in vacuum. Focus is generally on electronic states in that the Bom-Oppenheimer approximation is invoked, whereby the nuclei are fixed in space and treated as sources of electrostatic potentials. In reality a molecule is subject to a variety of environments and thereby interactions and nuclei move and may possess spin. Attention is usually limited to electromagnetic interactions, but there are cases where one wishes to go beyond this model (see Chapter Nine). The additional degrees of freedom present in the more realistic models can be treated in a perturbational manner which... 

In this section we shall discuss the interactions that arise upon the introduction of electromagnetic fields in the relativistic electronic Hamiltonian, and we shall also consider the form of electromagnetic interactions in the non-relativistic limit. To simplify matters, we shall first limit our attention to one-electron systems. Consider the time-independent Dirac equation for a free particle... 

Grot, R.A. Relativistic continuum physics electromagnetic interactions—continuum physics. In Eringen, A.C. (ed.) Mixtures and EM Field Theories, vol. III. Academic, New York (1976)... 

In returning to the time-independent equation (1.1.1), it must be stressed that the Hamiltonian (1.1.2a) is still somewhat idealized, even for an isolated system. In writing it down, we have assumed that the nuclei are fixed in space and we have neglected all interactions between particles other than those which are purely electrostatic in origin. We shall consider the inclusion of terms corresponding to more general electromagnetic interactions and relativistic effects in Chsq)ter 11 here we comment only on the assumption of fixed nuclei. 

Let us assume that space is not filled with fluctuating electromagnetic modes, but with a gas of non-interacting (non-relativistic) fermions. 

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