Quantization electromagnetic field

It should be stressed, however, that the introduction of the operator 2(k) in the present context is purely for mathematical convenience. All the subsequent development could also be carried out without its introduction. It is only when we consider the interaction of the quantized electromagnetic field with charged particles that the potentials assume new importance—at least in the usual formulation with its particular way of fixing the phase factors in the operators of the charged fields—since the potentials themselves then appear in the equations of motion of the interacting electromagnetic and matter fields. 

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. 

For a discussion of the quantized electromagnetic field interacting with a given (prescribed) external current, see ... 

We shall adopt these variables as the dynamical variables describing the quantized electromagnetic field. 

Strong fields introduce yet another set of phenomena allowing for the controlled manipulation of matter. Examples of light-induced potentials and the controlled focusing, alignment, and deposition of molecules are discussed in Chapter 12, after the introduction of the quantized electromagnetic field. 

As was shown in [13], to include the relativistic recoil corrections in calculations of the energy levels, we must add to the standard Hamiltonian of the electron-positron field interacting with the quantized electromagnetic field and with the Coulomb field of the nucleus Vc an additional term. In the Coulomb gauge, this term is given by... 

The difference in the energy of the 2 Sand 2 Pjy2 levels in hydrogenic atoms is a purely electrodynamic effect due to the interaction of the bound electron with the quantized electromagnetic field. The measurement of this splitting was a major stimulus for the development of renormalization theory and still provides an important test of Quantum Electrodynamics. The precise measurement of this split-ting is difficult because of the short radiative lifetime of the 2 P 2 state. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

Already to 0[c ) QED corrections, such as the Lamb shift, come into play, so beyond 0 c ) a theory based on a non-quantized electromagnetic field becomes somewhat meaningless. 

Reference [7] provides the detailed analysis of the peculiarities of excited nucleus spontaneous gamma decay both for various types of environment (screen and the system of excited nucleus atom electrons) and accounting for the multimode structure and phase characteristics of the quantized electromagnetic field. It was shown for the first time that the form, distance, and spectral characteristics of the screen most essentially influence the nuclei decay rate. It was also shown that the spontaneous decay character is considerably influenced by the degree of correlation and synchronization of the modes of electromagnetic field surrounding the excited nucleus. Such effect is possible only for the system of resonant Mossbauer nuclei. 

Here are correspondingly Hamiltonian operators of the nucleus, quantized electromagnetic field, the system... 

The quantized electromagnetic field in the considered case of a macroscopic system with linear sizes L y S> Xa = Inc/oia is a superposition of planar v/aves... 

The value Acoq + Acog determines the field shift of resonance level frequency (radiation shift or radiation correction) caused by the excited nucleus interaction with all the modes of quantized electromagnetic field and with melted nucleus... 

In countless studies, such terms have been investigated in detail for atoms (see, for examples. Refs. [238-250,250-252] and for overviews see Refs. [156, 253]). However, attempts have been made to include them also in molecular calculations in order to assess their magnitude compared to Breit interaction effects [254]. They turn out to be very small, even when compared to the Breit corrections, and therefore provide numerical evidence for the validity of the semi-classical theory of non-quantized electromagnetic fields developed in this book. 

---