Photon operators quantum electrodynamics

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

The energy operator considered above is an approximation, in which only the lowest terms of the correction for the retardation of the interaction are taken into account. More general is the formal quantum-electrodynamical interaction energy operator in the approximation of the exchange of one virtual photon [58]... 

This is what we were looking for. In quantum electrodynamics, the right-hand side of (60) is interpreted as the helicity operator, that is, the difference between the numbers of right-handed and left-handed photons. We can write the usual... 

The simplest way to show the principal difference between the representations of plane and multipole photons is to compare the number of independent quantum operators (degrees of freedom), describing the monochromatic radiation field. In the case of plane waves of photons with given wavevector k (energy and linear momentum), there are only two independent creation or annihilation operators of photons with different polarization [2,14,15]. It is well known that QED (quantum electrodynamics) interprets the polarization as given spin state of photons [4]. The spin of photon is known to be 1, so that there are three possible spin states. In the case of plane waves, projection of spin on the... 

The nature of media effects relates to the fact that, since the microscopic displacement field is the net field to which molecules of the medium are exposed, it corresponds to a fundamental electric field dynamically dressed by interaction with the surroundings. The quantized radiation is in consequence described in terms of dressed photons or polaritons. A full and rigorous theory of dressed optical interactions using noncovariant molecular quantum electrodynamics is now available [25-27], and its application to energy transfer processes has been delineated in detail [10]. In the present context its deployment leads to a modification of the quantum operators for the auxiliary fields d and h, which fully account for the influence of the medium—the fundamental fields of course remain unchanged. Expressions for the local displacement electric and the auxiliary magnetic field operators [27], correct for all microscopic interactions, are then as follows... 

In this review we have described some of the advances in the quantum electrodynamical formulation of theory for molecular photonics. We have shown how the framework described in an earlier review has now been extended to new areas of application, and reformulated for application to real dispersive media—as reflected in the new treatment of refractive, dissipative, and resonance properties. With all its conceptual splendor, conventional quantum optics has not generally been pursued at this level of detail on its dielectric host, and it is our hope that this work will help match its precepts with quantitative accuracy. Applications of the new theory have revealed new quantum optical features in two quite different aspects of the familiar process of second harmonic generation, one operating through local coherence within small particles and the other, a coherence between the quantum amplitudes for fundamental and harmonic excitation. Where the salient experiments have been performed, they exactly match the theoretical predictions. The theoretical foundation we have discussed therefore shows promise for the delivery of accurate insights into other optical processes yet to be characterized, and it should be well placed to facilitate the determination of meaningful data from the associated experiments. 

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)... 

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. 

It is one of the objectives of the quantum electrodynamic procedure to check the errors and limits of these semiclassical approaches. In particular there is the question of statistics. What is the reason for the dominant influence of Bose statistics in the final energy expression Another question is up to which order of the electron-photon coupling operator do the semiclassical approaches hold Is one of these approaches more powerful than the other or limited to certain molecules or separations ... 

The quantum electrodynamic procedure replaces electrons and photons by quasi-particles. The quasi-electrons are renormalized with respect to terms quadratic in the electron-photon coupling operator. They differ from Fermi statistics with respect to fourth order terms. Similarly, the quasi-photons are renormalized with respect to terms quadratic in the electron-photon coupling operator. They differ from Bose statistics with respect to fourth order terms. 

The present derivation of the scattering cross-section is based on a non-relativistic quantum electrodynamic approach. In this picture, the modes of the radiation field are quantized and the electric field is treated as a quantum-mechanical operator that annihilates or creates photons populating the various modes. The field operator is given by... 

Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

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