Quantum electrodynamics nonrelativistic

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

Nonrelativistic quantum electrodynamics (NRQED) [11] is an attempt to combine the simplicity of the quantum mechanical description with the power and rigor of field theory. The idea is to write ordinary relativistic quantum electrodynamics in the form of a nonrelativistic expansion with a Lagrangian containing vertices with arbitrary powers of fields. This is useful if we want to consider essentially nonrelativistic processes, like nonrelativistic bound states and threshold phenomena. In such a physical situation the dominant dynamics is nonrelativistic, and the calculations could be in principle simplified if 

Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill Book Co., NY, 1964. 

Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynamics, 2nd Edition, Pergamon Press, Oxford, 1982. 

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In this section we discuss the nonrelativistic 0(3) b quantum electrodynamics. This discussion covers the basic physics of f/(l) electrodynamics and leads into a discussion of nonrelativistic 0(3)h quantum electrodynamics. This discussion will introduce the quantum picture of the interaction between a fermion and the electromagnetic field with the magnetic field. Here it is demonstrated that the existence of the field implies photon-photon interactions. In nonrelativistic quantum electrodynamics this leads to nonlinear wave equations. Some presentation is given on relativistic quantum electrodynamics and the occurrence of Feynman diagrams that emerge from the B are demonstrated to lead to new subtle corrections. Numerical results with the interaction of a fermion, identical in form to a 2-state atom, with photons in a cavity are discussed. This concludes with a demonstration of the Lamb shift and renormalizability. 

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. 

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]. 

This result emerges self-consistently at all levels of physics, from the classical nonrelativistic to the quantum electrodynamic. On the nonrelativistic classical level, the technique of RFR is due to the interaction of B<3> with the Pauli matrix. One way of demonstrating this result, which has been observed empirically [37], is to extend the minimal prescription to complex A, starting... 

E. A. Power, T. Thirunamachandran, Quantum Electrodynamics With Nonrelativistic Sources. III. Intermolecular Interactions. Phys. Rev. A 28 (1983) 2671. 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

Our decision in favor of combining nonrelativistic quantum mechanics with the nrl of electrodynamics becomes very important, when we consider the interaction between moving electrons (section 7). In the nrl there is only a nonretarded (instantaneous) Coulomb interaction, while both the magnetic interaction and the retardation of the Coulomb interaction are relativistic corrections and are therefore neglected. One needs to consider them only if one also includes relativistic corrections to the kinematics. 

In nonrelativistic quantum mechanics the formulation of an n-electron Hamiltonian is rather straightforward, because in the nrl of electrodynamics the electrostatic potential satisfies the Poisson equation (95), which implies (in atomic units) that the potential due to a point charge is equal to 1/r and the interaction between a pair of electrons is simply... 

This paper reviews progress in the application of atomic isotope shift measurements, together with high precision atomic theory, to the determination of nuclear radii from the nuclear volume effect. The theory involves obtaining essentially exact solutions to the nonrelativistic three- and four-body problems for helium and lithium by variational methods. The calculation of relativistic and quantum electrodynamic corrections by perturbation theory is discussed, and in particular, methods for the accurate calculation of the Bethe logarithm part of the electron self energy are presented. The results are applied to the calculation of isotope shifts for the short-... 

Having left the framework of field theory outlined in chapter 7 and thus having avoided any need for subsequent renormalization procedures, the mass and charge of the electron are now the physically observable quantities, and therefore do not bear a tilde on top. In contrast to quantum electrodynamics, the radiation field is no longer a dynamical degree of freedom in a many-electron theory which closely follows nonrelativistic quantum mechanics. Vector potentials may only be incorporated as external perturbations in the many-electron Hamiltonian of Eq. (8.62). From the QED Eqs. (7.13), (7.19), and (7.20), the Hamiltonian of a system of N electrons and M nuclei is thus described by the many-particle Hamiltonian of Eq. (8.66). In addition, we refer to a common absolute time frame, although this will not matter in the following as we consider only the stationary case. 

T. Itoh. Derivation of Nonrelativistic Hamiltonian for Electrons from Quantum Electrodynamics. Rev. Mod. Phys., 37(1) (1965) 159-165. 

Milonni PW (1976) Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory. Phys Rep 25 1... 

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. 

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. 

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