Classical relativistic electrodynamics

The signs of spatial components are reversed in the corresponding contravariant 4-vectors, indicated by x etc. Dirac matrices are represented in a form appropriate to a 2-component fermion theory [38], in which helicity y5 is diagonal for zero-mass fermions, 

The covariant electromagnetic field is defined in terms of the electromagnetic 4-potential Afl by the antisymmetric form 

The Dirac and Maxwell fields are coupled through the covariant derivative 

QED theory is based on two distinct postulates. The first is the dynamical postulate that the integral of the Lagrangian density over a specified space-time region is stationary with respect to variations of the independent fields Atl and ijr, subject to fixed boundary values. The second postulate attributes algebraic commutation or anticommutation properties, respectively, to these two elementary fields. In the classical model considered here, the dynamical postulate is retained, but the algebraic postulate and its implications will not be developed in detail. 

The dynamical postulate implies the covariant field equations, 

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The second volume of the Landau-Lifshitz series on theoretical physics continues directly after volume I and covers the classical theory of fields. It starts with an introduction of Einstein s principle of relativity and a discussion of the special theory of relativity for mechanics. It follows a rather complete presentation of classical electrodynamics including radiation phenomena and scattering of waves of different energy. It concludes with an introduction of gravitational fields, the theory of general relativity and classical relativistic cosmology. 

In the last chapter the basic framework of classical nonrelativistic mechanics has been developed. This theory crucially relies on the Galilean principle of relativity (cf. section 2.1.2), which does not match experimental results for high velocities and therefore has to be replaced by the more general relativity principle of Einstein. It will directly lead to classical relativistic mechanics and electrodynamics, where again the term classical is used to distinguish this theory from the corresponding relativistic quantum theory to be presented in later chapters. 

How is physics, as it is currently practiced, deficient in its description of nature Certainly, as popularizations of physics frequently reniiiid us, theories such as Quantum Electrodynamics are successful to a reinarkiible degree in predicting the results of experiments. However, any reasonable measure of success requires that wc add the caveat, ...in the domain (or domains) for which the theory was developed. For example, classical Newtonian physics is perfectly correct in its description of slow-moving, macroscopic objects, but is fundamentally incorrect in its description of quantum and/or relativistic systems. 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. 

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>... 

In electromagnetic theory, we replace W 1 by GM the relativistic helicity of the field. Therefore, Eq. (770) forms a fundamental Lie algebra of classical electrodynamics within the Poincare group. From first principles of the Lie algebra of the Poincare group, the field B is nonzero. 

Thus, the most common assumption was that a material s properties are governed by quantum theory and that relativistic effects are mostly minor and of only secondary importance. Quantum electrodynamics and string theory offer some possible ways of combining quantum theory and the theory of relativity, but these theories have only very marginally found their way into applied quantum theory, where one seeks, from first principles, to calculate directly the properties of specific systems, i.e. atoms, molecules, solids, etc. The only place where Dirac s relativistic quantum theory is used in such calculations is the description of the existence of the spin quantum number. This quantum number is often assumed to be without a classical analogue (see, however, Dahl 1977), and its only practical consequence is that it allows us to have two electrons in each orbital. 

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. 

When an electron is accelerated, it produces electromagnetic radiation. The power radiated by an accelerated charge, P, whose speed is significantly less than the speed of light, is described in classical electrodynamics by the Larmor formula. In a synchrotron, the electrons are moving at a speed close to the speed of light (relativistic electrons). The power radiated by relativistic electrons is given by... 

The correct frame of description of interacting relativistic electrons is quantum electrodynamics (QED) where the matter field is the four-component operator-valued electron-positron field acting in the Fock space and depending on space-time = (ct, r) (x = (ct, —r)). Electron-electron interaction takes place via a photon field which is described by an operatorvalued four-potential A x ). Additionally, the system is subject to a static external classical (Bose condensed, c-number) field F , given by the four-potential (distinguished by the missing hat)... 

Most properties of synchrotron radiation may be derived starting from classical electrodynamics, where an oscillating dipole is subjected to a Lorentz transformation. Assuming that relativistic electrons move on curved trajectories in a bending magnet of the radius R, the radiated power... 

The refinement is based on classical electrodynamics and the usual quantum mechanical rules for forming operators (Chapter 1) or, alternatively, on the relativistic Breit Hamiltonian (p. 131). This is how we get the Hamiltonian (12.66) which contains the usual non-relativistic Hamiltonian (12.67) plus the perturbation (12.68) with a number of terms (p. 657). 

In this chapter basic concepts and formal structures of classical nonrelativistic mechanics and electrodynamics are presented. In this context the term classical is used in order to draw the distinction with respect to the corresponding quantum theories which will be dealt with in later chapters of this book. In this chapter we will exclusively cover classical nonrelativistic mechanics or Newtonian mechanics and will thus often apply the word nonrelativistic for the sake of brevity. The focus of the discussion is on those aspects and formal structures of the theory which will be modified by the transition to the relativistic formulation in chapter 3. 

Classical electrodynamics, i.e.. Maxwell s unquantized theory for time-dependent electric and magnetic fields is inherently a covariant relativistic theory— in the sense of Einstein and Lorentz not Newton and Galilei — fitting perfectly well to the theory of special relativity as we shall understand in chapter 3. In this section, only those basic aspects of elementary electrodynamics will be... 

In section 2.4 a brief summary of Maxwell s theory of electrodynamics has been presented in its classical, three-dimensional form. Since electrodynamics intrinsically is a relativistic gauge field theory, the structure and symmetry properties of this theory become much more apparent in its natural, explicitly... 

Having introduced the principles of special relativity in classical mechanics and electrodynamics as well as the foundations of quantum theory, we now discuss their unification in the relativistic, quantum mechanical description of the motion of a free electron. One might start right away with an appropriate ansatz for the basic equation of motion with arbitrary parameters to be chosen to fulfill boundary conditions posed by special relativity, which would lead us to the Dirac equation in standard notation. However, we proceed stepwise and derive the Klein-Gordon equation first so that the subsequent steps leading to Dirac s equation for a freely moving electron can be better understood. 

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