Field-matter interaction, 0 electrodynamics

MSN.27.1. Prigogine and B. Leaf, On the field-matter interaction in classical electrodynamics, 1., Physica 25, 1067-1079 (1959). 

The equations of 0(3) electrodynamics can therefore be written in condensed form as Eqs. (730) and (731) in the vacuum. These equations can be written as a single conservation law under all conditions (vacuum and field-matter interaction) ... 

Within the framework of nonrelativistic quantum electrodynamics, the emission in electric-dipole transitions can be treated using two alternative Hamiltonians for field-matter interaction, i.e. a multipolar Hamiltonian and a minimal-coupling (p ) Hamiltonian, since the two are related by a canonical transformation . In what follows, the results concerning motional effects on the emission will be discussed and checked by showing that they are obtainable from both Hamiltonians. 

Hans Albrecht Bethe (1906-2005) was an American physicist, a professor at Cornell University, and a student of Arnold Sommerfeld. Bethe contributed to many branches of physics, such as crystal field theory, interaction of matter with radiation, quantum electrodynamics, and the structure and nuclear reactions of stars (for the latter achievement, he received the Nobel Prize in physics in 1967). 

As stated by Milonni [73], An arbitrarily large number n of photons may occupy the same state, and when this situation obtains, it is accurate to regard the photon wave function as defining a classical field distribution. Thus the quantum electrodynamic view of radiation for intense laser fields can be described classically. Overall, the light-matter interaction is treated semi-classically where the diatomic molecule is quantum mechanical and the laser pulse is classical in nature. The electric dipole approximation [74] is also used which reduces the form of the electric field due to the comparative size of the electric field wavelength compared to the molecule. The classical description of the laser field, E r,t), can be written in complex form according to... 

We now consider the effect of exposing a system to electromagnetic radiation. Our treatment will involve approximations beyond that of replacing (3.13) with (3.16). A proper treatment of the interaction of radiation with matter must treat both the atom and the radiation field quantum-mechanically this gives what is called quantum field theory (or quantum electrodynamics). However, the quantum theory of radiation is beyond the scope of this book. We will treat the atom quantum-mechanically, but will treat the radiation field as a classical wave, ignoring its photon aspect. Thus our treatment is semiclassical. 

As this concerns the nature of non-Abelian electrodynamics, we will pursue the matter of a GUT that incorporates non-Abelian electrodynamics. This GUT will be an 50(10) theory as outlined above. We have that an extended electro-weak theory that encompasses non-Abelian electrodynamics is spin(4) = 51/(2) x 517(2). This in turn can be embedded into a larger 50(10) algebra with spin(6) = 517(4). 50(10) may be decomposed into 517(2) x 517(2) x 517(4). This permits the embedding of the extended electro weak theory with 517(4), which may contain the nuclear interactions as 517(4) 51/(3) x 1/(1). In the following paragraphs we will discuss the nature of this gauge theory and illustrate some basic results and predictions on how nature should appear. We will also discuss the nature of fermion fields in an 517(2) x 51/(2) x 51/(4) theory. 

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. 

AN OVERVIEW OF QUANTUM ELECTRODYNAMICS AND MATTER-RADIATION FIELD INTERACTION... 

This review is concerned with the advances in our understanding of chemical problems that have occurred as a result of developments in computational electrodynamics, with an emphasis on problems involving the optical properties of nanoscale metal particles. In addition, in part of the review we describe theoretical methods that mix classical electrodynamics with molecular quantum mechanics, and which thereby enable one to describe the optical properties of molecules that interact with nanoparticles. Our focus will be on linear optical properties, and on the interaction of electromagnetic fields with materials that are large enough in size that the size of the wavelength matters. We will not consider intense laser fields, or the interaction of fields with atoms or small molecules. 

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)... 

As a second example for modifications of the quantum electrodynamical interaction between matter and the radiation field originating from controlled mode engineering in the following we discuss the spontaneous decay of a two-level system, such as an ion [24], in a half-open cavity with a... 

All science is based on a number of axioms (postulates). Quantum mechanics is based on a system of axioms that have been formulated to be as simple as possible and yet reproduce experimental results. Axioms are not supposed to be proved, their justification is efficiency. Quantum mechanics, the foundations of which date from 1925-26, still represents the basic theory of phenomena within atoms and molecules. This is the domain of chemistry, biochemistry, and atomic and nuclear physics. Further progress (quantum electrodynamics, quantum field theory, elementary particle theory) permitted deeper insights into the structure of the atomic nucleus, but did not produce any fundamental revision of our understanding of atoms and molecules. Matter as described at a non-relativistic quantum mechanics represents a system of electrons and nuclei, treated as point-like particles with a definite mass and electric charge, moving in three-dimensional space and interacting by electrostatic forces. This model of matter is at the core of quantum chemistry. Fig. 1.2. 

The covariant form of the Dirac equation of a freely moving particle, Eq. (5.54), allows us to incorporate arbitrary external electromagnetic fields. These fields can then be used to describe the interaction of electrons with light. But it must be noted that the treatment of light is then purely classical. A fully quantized description of light and matter on an equal footing is introduced only in quantum electrodynamics and discussed in chapter 7. 

The first term (jCrad) describes the electromagnetic degrees of freedom based on the 4-potential the second ( mat) the Dirac matter field ip, whose excitations may later, i.e., after quantization, be interpreted as electrons and positrons, and the last term ( int) accounts for the interaction between the former two. To write such a Lagrangian goes back to the early days of (the old) quantum electrodynamical theory [103]. 

The exact quantum theoretical treatment of the dispersion effect involves quantizing matter and electromagnetic fields as well. The coupled electron-photon system is to be treated on the basis of quantum electrodynamics. Using the method of second quantization, it is possible to build up the total Hamiltonian from an electron Hamiltonian H, a photon Hamiltonian and an electron-photon interaction operator Hin,. The dispersion energy between two particles now results in fourth order perturbation. Each contribution is due to the interaction of two electrons with, fwo photons. 

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