Dirac equation quantum electrodynamics

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

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]. 

There have been a number of recent reviews of hydrogenic systems and QED [9]-[12] these proceedings contain the most extensive and recent information. To calculate transition frequencies in hydrogen to an accuracy comparable with the experimental precision which has been achieved [3], it is necessary to take into account a large number of corrections to the values obtained using the Dirac equation. These include quantum electrodynamic (QED) corrections, pure and radiative recoil corrections arising from the finite nuclear mass, and a correction due to the non-zero volume of the nucleus. The evaluation of these corrections is an extremely challenging task. 

Theoretical calculations of two-electron ion energy levels have been the topic of much research since the discovery of quantum mechanics. The contribution of relativistic effects via the Dirac equation and QED contributions has been intensely studied in the last three decades [1]. Two-electron systems provide a test-bed for quantum electrodynamics and relativistic effects calculations, and also for many body formalisms [2]. 

In writing equation (4.6), we have assumed that the nuclei can be treated as Dirac particles, that is, particles which are described by the Dirac equation and behave in the same way as electrons. This is a fairly desperate assumption because it suggests, for example, that all nuclei have a spin of 1 /2. This is clearly not correct a wide range of values, integral and half-integral, is observed in practice. Furthermore, nuclei with integral spins are bosons and do not even obey Fermi Dirac statistics. Despite this, if we proceed on the basis that the nuclei are Dirac particles but that most of them have anomalous spins, the resultant theory is not in disagreement with experiment. If the problem is treated by quantum electrodynamics, the approach can be shown to be justified provided that only terms of order (nuclear mass) 1 are retained. 

Other electron nuclear interaction terms involving 7ra rather than Ia arise from this treatment. However, these terms have all been dealt with in the previous chapter and we do not repeat them here.) The terms in (4.23) are the same as those obtained previously starting from the Dirac equation. Equation (3.244) will yield both the electron and nuclear Zeeman terms and a Breit equation for two nuclei, reduced to non-relativistic form, would yield the nuclear-nuclear interaction terms. Although many nuclei have spins other than 1/2, and even the proton with spin 1 /2 has an anomalous magnetic moment which does not fit the simple Dirac theory, the approach outlined here is fully endorsed by quantum electrodynamics provided that only terms involving M l are retained (see equation (4.23)). The interested reader is referred to Bethe and Salpeter [11] for further details. In our present application we see that the expressions for both... 

Dirac s 1929 comment [227] The underlying physical laws necessary for the mathematical theory for a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too difficult to be soluble has become a part of the Delphic wisdom of our subject. To this confident statement Richard Feynman [228] added in 1985 a codicil But there was still the problem of the interaction of light and matter , and . .. the theory behind chemistry is quantum electrodynamics . He goes on to say that he is writing of non-covariant quantum electrodynamics, for the interaction of the radiation field with the slow-moving particles in atoms and molecules. 

According to Primas (1991, p. 163), "the philosophical literature on reductionism is teeming with scientific nonsense," and he quotes, among others, Kemeny and Oppenheim (1956), who said "a great part of classical chemistry has been reduced to atomic physics." Perhaps it was not philosophers who invented this story after all. Almost certainly, Oppenheim and other philosophers of science at the time were familiar with the influential statements of Dirac, Heisenberg, Reichenbach, and Jordan on this issue. " Notoriously, the physicist Dirac (1929, p. 721) said, the underlying laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that exact applications of these laws lead to equations which are too complicated to be soluble." Less famously, the philosopher of science Reichenbach (1978, p. 129) reiterated that "the problem of physics and chemistry appears finally to have been resolved today it is possible to say that chemistry is part of physics, just as much as thermodynamics or the theory of electricity." These views clearly stuck. For example, in a recent review of quantum electrodynamics (QED), to which Dirac made important contributions, the historian of science Schweber (1997, p. 177) says, "the laws of physics encompass in principle the phenomena and the laws of chemistry."... 

As the fundamental equation of relativistic quantum mechanics and quantum electrodynamics, the Dirac equation is perhaps the most important equation of modem physics. It is impossible to value the vast range of its applications in a single article and therfore we want to present an introduction to certain aspects only. This chapter has the character of a first overview and introduction to the Dirac operator. It covers material that is largely contained in my book [1], but, as I hope, in a more accessible form. Ref. [1] should be taken as a reference to more details and background information from a more mathematical point of view, and as a guide to the older literature on this subject. 

This second point of view can be illustrated by an example from the late 1940 s that will play an important role in this chapter. At that time the Schrodinger equation was well established, and its relativistic generalization, the Dirac equation, appeared to describe the spectrum of hydrogen perfectly, though the question of how to apply the Dirac equation to many-electron systems was still open. However, when more precise experiments were carried out, most notably by Lamb and Retherford [1], a small disagreement with theory was found. The attempt to understand this new physics stimulated theoretical efforts that led to the modern form of the first quantum field theory. Quantum Electrodynamics (QED). This small shift, which removes the Dirac degeneracy between the 2si/2 and states, known as the Lamb shift, is an example of a radiative correction. 

It is astonishing that the topic discussed in this subsection is hardly mentioned in textbooks on quantum mechanics. It also appears to be nearly unknown, that Levy-Leblond has not only derived the nrl of the Dirac equation as the Galilei invariant field theory for spin- particles, but also the nrl of electrodynamics as the Galilei inariant field theory for spin-1 particles with zero rest mass [16]. 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

In this chapter, we shall reassess some of the physical implications of the Dirac equation [5, 6], which were somehow overlooked in the sophisticated formal developments of quantum electrodynamics. We will conjecture that the internal structure of the electron should consist of a massless charge describing at light velocity an oscillatory motion (Zitterbewegung) in a small domain defined by the Compton wavelength, the observed spin momentum and rest mass being jointly generated by this very internal motion. 

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