Electrodynamics, quantum

Quantum electrodynamics is the fundamental physical theory which obeys the principles of special relativity and allows us to describe the mutual interactions of electrons and photons. It is intrinsically a many-particle theory, although much too complicated from a numerical point of view to be the basis for the theoretical framework of the molecular sciences. Nonetheless, it is the basic theory of chemistry and its essential concepts, and ingredients are introduced in this chapter. 

A crucial step motivated by experimental observations and theoretical considerations is the quantization of the radiation field, whereupon the electric and magnetic fields assume operator character. We first notice that the functions /(Z) and p(Z) defined for each (k, Ok) by 

These equations have the form of harmonic oscillator equations of motion for a coordinate q and momentum /). Indeed, Eqs (3.11) can be derived from the Hamiltonian 

This identifies a and as lowering and raising operators of the corresponding harmonic modes. Equation (3.6) is recognized as the Heisenberg representation of these operators 

In many applications we encounter such sums of contributions from different modes, and because in the limit Q oo the spectrum of modes is continuous, such sums are converted to integrals where the density of modes enters as a weight function. An important attribute of the radiation field is therefore the density of modes per unit volume in k-space, Pk, per unit frequency range, p, or per unit energy, ps (E = hoP). We find (see Appendix 3A) 

To see the physical significance of these results consider the Hamiltonian that describes the radiation field, a single two-level molecule located at the origin, and the interaction between them, using for the latter the fully quantum analog of Eq. (3.1) in the Schrodinger representation 

There is general agreement that QED provides the most satisfactory point of departure for the study of electronic structure of atoms and molecules. To simplify matters we treat the nuclei as classical charge-current distributions giving rise to a classical 4-potential a (jc) so that we can replace the 4-potential in (44) by 

The first term is the Lagrangian density for the free Maxwell field, Ffiy(x), 

Quantum electrodynamics requires the solution of the system (51) when a x), y/(x) and its adjoint (jc) are quantized fields. As a glance at any of the standard texts reveals, this involves many technical difficulties. Fortunately most of them do not affect the derivation of the DHFB method although they are unavoidable if we wish to go beyond self-consistent field approximations. 

Here we have invoked the Bom-Oppenheimer assumption and fixed the nuclear skeleton so that a x) = ( y(x),0) in this reference frame, where is an electrostatic potential, V (x) = -e j (x), and 

Note that expression (3.20a) is the same result as Eq. (2.95), obtained for the density of states of a free quantum particle except for the additional factor 2 in (3.20a) that reflects the existence of two polarization modes for a given k vector. Eq (3.20b) is obtained from (3.20a) by using co = k c where c = cl /sp to get Pay co) = [4jtPdk X Pkdk]k=co/c (compare to the derivation of Eq (2.97)). 

---

Juzeliunas G and Andrews D L 2000 Quantum electrodynamics of resonance energy transfer Adv. Chem. Rhys. 112 357-410... 

Craig D P and Thirunamachandran T 1984 Molecular Quantum Electrodynamics (New York Academic)... 

Figure 7.8 shows that the i 5i/2 state is shifted, but not split, when quantum electrodynamics is applied. It is, however, split into two components, 0.0457 cm apart, by the effects of nuclear spin I = for Ll). 

Cambridge, Mass.,) and R. P. Feynman (California Institute of Technology, Pasadena) fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles. 

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 the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

Schwinger, J., Selected papers on Quantum Electrodynamics, Dover Publications, New York, 1958 (see particularly Schwinger s preface). 

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. 

Strictly speaking the description of such local measurements can be carried out only within the framework of quantum electrodynamics, i.e, in a theory wherein photons can be exchanged between the measuring apparatus and the current distribution being measured. 

Thirring, W. E., Principles of Quantum Electrodynamics, Academic Press, New tfork, 1958. 

Quantum Electrodynamics in the Heisenberg Picture.— With the present section we begin our discussion of the quantum theoretical description of the interaction between the negaton-positon field and the radiation field i.e., of quantum electrodynamics proper. 

Starting from this formulation of quantum electrodynamics, the... 

It turns out that this choice is correct and that L and L" can be so chosen that (11-144) is satisfied. We shall return to this question in our discussion of the asymptotic condition in Section 11.5 of the present chapter. As preparation for these considerations, we turn in the next few seotions to a discussion of the invariance properties of quantum electrodynamics and their consequences. 

This section has outlined the current formulations of quantum electrodynamics. Since the theory assumes its local form when expressed in terms of potentials, these formulations were in terms of the... 

Invariance Properties.—Before delving into the mathematical formulation of the invariance properties of quantum electrodynamics, let us briefly state what is meant by an invariance principle in general. As we shall be primarily concerned with the formulation of invariance principles in the Heisenberg picture, it is useful to introduce the concept of the complete description of a physical system. By this is meant at the classical level a specification of the trajectories of all particles together with a full description of all fields at all points of space for all time. The equations of motion then allow one to determine whether the system could, in fact, have evolved in the way... 

Consider for example the theory under discussion, namely, quantum electrodynamics under the circumstance that the frame of reference with respect to which phenomena are described is changed from a right-handed to a left-handed coordinate system, i.e., from one in which the space-time coordinates are labeled by to one in which they are labeled by sc, with... 

The statement that quantum electrodynamics is invariant under such a spatial inversion (parity operation) can be taken as the statement that there exist new field operators >p (x ) and A x ) expressible in terms of tji(x) and Au(x) which satisfy the same commutation rules and equations of motion in terms of s as do ift(x) and A x) written in terms of x. In fact one readily verifies that the operators... 

We must next consider more precisely the connection between the description of bodily identical states by the two observers (the requirements of Postulate 1). Quite in general, in fact, a physical theory, and quantum electrodynamics in particular, is fully defined only if the connection between the description of bodily identical states by (equivalent) observers is known for every state of the system and for every pair of observers. Since the observers are equivalent every state which can be described by 0 can also be described by O. Given a bodily state of the same system, observer 0 will ascribe to it a state vector Y0> in his Hilbert space and observer O will attribute to it a state vector T0.) in his Hilbert space. The above formulation of invariance means that there exists a one-to-one correspondence between the vectors Y0> and Y0.) used by observers 0 and O to describe bodily the same state.3 This correspondence guarantees that the two Hilbert spaces are in fact isomorphic. It is, therefore, possible for the two observers to agree to describe states of the system by vectors in the same Hilbert space. A similar statement can be made for the observables there exists a one-to-one correspondence between the operators Q0 and Q0>, which observers 0 and O attribute to observables. The consistency of the theory (Postulate 2) demands, however, that the two observers make the same prediction as the outcome of the same experiment performed on bodily the same system. This requires the relation... 

The discussion at the beginning of this section, when coupled with the fact that the observers 0 and O agree to describe bodily the same state by the same state vector, has exhibited the invariance of quantum electrodynamics under space inversion in the Heisenberg-type description. 

Let us next adopt the Schrodinger-type description. The statement that quantum electrodynamics is invariant under space inversion can now be translated into the statement that there exists a unitary operator U(it) such that... 

Consider next the relativistic invariance of quantum electrodynamics. Again, loosely speaking, we say that quantum electrodynamics is relativistically invariant if its observable consequences are the same in all frames connected by an inhomogeneous Lorentz transformation a,A ... 

Invariance of Quantum Electrodynamics under Discrete Transformations.—In the present section we consider the invariance of quantum electrodynamics under discrete symmetry operations, such as space-inversion, time-inversion, and charge conjugation. 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

---