Transverse Photon Interaction

Although the transverse photon interaction (93) is needed also in high precision calculations for highly-ionized few electron atoms, it mainly influences the... 

The precise form of this correction depends upon the gauge condition used to describe the electromagnetic field. In the Coulomb gauge, which has been employed more often in relativistic atomic structure, the electron-electron interactions come from one-photon exchange process and is sum of instantaneous Coulomb interaction and the transverse photon interaction. 

To order e2 the Coulomb interaction term contributes to the first-order term in the 5-matrix expansion, i.e., to fd4xJffin(x), and the term contributes to second-order. Diagrammatically, we have illustrated these contributions in Fig. 11-1 ( stands for Coulomb, / for transverse photons). To the order indicated, the part of the 5-matrix contributing to the process is... 

It is sufficient to determine the quantity Rxk in the second order with respect to the CC photon interaction. We further assume that the optical preparation of the excited state by the applied field E is short compared to the emission process and, finally, we neglect anti-resonant contributions. When calculating F(u> t) we also have to perform a summation with respect to the transversal polarization and a solid angle integration. Introducing dm = dmem where eTO is the unit vector pointing in the direction of the transition dipole moment one gets... 

The relativistic correction to the fermion kinetic energy is represented as a potential. The Breit-Fermi interaction includes the effects of transverse photon exchange as well as relativistic corrections to Coulomb photon exchange. The potentials are given with the assumption that the states acted on are S states with total spin 1. 

Fig. 1. Two-electron Feynman diagrams representing electron interaction in the second order. Solid line corresponds to electrons, dotted line corresponds to the Coulomb photons, wavy line corresponds to transverse photons (Breit interaction). If A = A and B1 = B then the diagram is called direct , otherwise, if A = B and B = A the diagram is called exchange . Letters A, B denote one-electron state...

An alternative coupling scheme, that proves to be advantageous in many regards, and is now the preferred theoretical method, is the multipolar framework [32, 33] of molecular QED. In this formulation, molecules couple directly to the radiation fields through their molecular multipole moments. Because the Maxwell fields obey Einstein causality, interactions between species is properly retarded, which is mediated by the emission and absorption of transverse photons. It is convenient to partition the total many-body QED matter-field multipolar Hamiltonian into a sum of parficle, radiation field, and interaction terms as follows ... 

Within the above approximate relativistic CDFT scheme, the Breit interaction has been ignored. This radiative correction accounts for the retardation of the Coulomb interaction and exchange of transversal photons. A more complete version than that included in Equation (5.1) is given by the Hamiltonian (Bethe and Salpeter 1957 PyykkO 1978) ... 

In previous chapters we considered elementary crystal excitation taking into account only the Coulomb interaction between carriers. From the point of view of quantum electrodynamics (see, for example, (1)) such an interaction is conditioned by an exchange of virtual scalar and longitudinal photons, so that the potential energy, corresponding to this interaction, depends on the carrier positions and not on their velocity distribution. As is well-known, the exchange of virtual transverse photons leads to the so-called retarded interaction between charges. 

II being the crystal energy operator when only the Coulomb interaction is accounted for, H2 is the energy operator of transverse photons of the electromagnetic field in vacuum, and //mti is the operator of the interaction between all carriers with the field of the transverse photons. 

The operator describing the interaction of carriers with the transverse photon field in the nonrelativistic (as concerns the carrier motion) approximation is given... 

If the retarded interaction is ignored and the operator Hmt is removed from the total Hamiltonian (4.2), then the operator H becomes a sum of two independent Hamiltonians, one of them (Hi) describing the crystal elementary excitations - those which occur when the retardation effects are ignored and the second (H2) giving the elementary excitations - transverse photons in vacuum. The presence of the operator H3 leads to the interaction between carriers with the transverse electromagnetic field. In the case of an atomic gas this interaction causes, in particular, the so-called radiative width of energetic levels of excited states. In the case of an infinite crystal possessing translational symmetry the radiative width of excitonic states vanishes.29... 

Let us continue to consider in detail the operator characterizing the interaction of electrons with the field of transverse photons, given by the expression (4.4). The radius-vector of the i/th electron of the molecule na contained in this expression is given by... 

For the longitudinal excitons, p = py and T(j,k,/x) = 0 as follows from (4.11). In consequence, those excitons do not interact with the transverse photons and retardation effects for them are not essential. From (4.17) and (4.18) we obtain for the longitudinal excitons35... 

It follows from the above relation that the retarded interaction is important only in the vicinity of wavevectors k y/eoQ/c, i.e. in that part of the spectrum, where the frequencies of the Coulomb excitons are near to those of the transverse photons. When the retardation is ignored, the branches of the Coulomb excitons and the transverse photons intersect (Fig. 4.1a). This intersection is removed when the retardation is taken into account (Fig. 4.1b). In a similar way the dependence w(k) for polaritons can be found for crystals with different symmetries. 

A is the wavelength of light with frequency oj = t o /h, C phonons of this kind strongly interact with the transverse photons. As a result, in the region of long wavelengths, new elementary excitations - phonon-polaritons (see Ch. 4) - are formed instead of C phonons and transverse photons. 

To take the interaction between phonons and photons into consideration, it is necessary to add to the Hamiltonian (6.32), the Hamiltonian Ho(a) of the free field of transverse photons and the Hamiltonian Hint for the interaction of the field of transverse photons with phonons. The linear transformation from the operators a and C to the polariton creation and annihilation operators, i.e. to the operators t(k) and p(k), diagonalizes the quadratic part of the total Hamiltonian. The two-particle states of the crystal, corresponding to the excitation of two B phonons, usually have a small oscillator strength and the retardation for such states can be neglected. In view of the afore-said, the quadratic part of the total Hamiltonian with respect to the Bose operators can be written in the form of the sum H0(B) + where... 

Hence, taking eqn (6.31) into account, we find that Tp(k) = r p(k). However, now along with the operator (6.31), llmti also has to include an operator corresponding to direct interaction between the transverse photons and overtones. This operator corresponds to... 

To not leave the reader with the impression that these extensions are trivial, let us recall that a relativistic reformulation caimot ignore the virtual creation of electron-positron pairs nor the fact that the Breit interaction involves the exchange of transverse photons. 

The curve G corresponds to the first order Coulomb interelectron interaction, the curve C++ corresponds to the no pair approximation for the second-order Coulomb box interelectron interaction (Fig.7a). The curve B corresponds to the first order Breit interaction, the curve BC corresponds to the second-order Coulomb-Breit box interaction (exchange of the one Coulomb and one transverse photons) Fig.7c,d. The curve denoted by ( ) includes the contributions ( )= GC, BB, X, where GG is the negative-energy contribution to the Coulomb - Coulomb box interaction Fig.7a, BB is the Breit-Breit box interaction Fig.7g, X denotes all cross interactions Fig.7b,e,f,h. The order of magnitude of all ( ) corrections is defined by the high-energy intermediate electron state contributions. This means that the corresponding effective interelectron interaction potential does not depend on the ex-... 

The Breit interaction [29, p. 170] is that part of the interaction between electrons mediated by exchange of transverse photons. The lowest-order energy shift associated with the exchange of a transverse photon between two electrons in states a and b is... 

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