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In the present paper, the process of inverse double-Compton (IDC) scattering is considered in the context of astrophysical applications. It is assumed that the two hard X-ray photons emitted from an astrophysical source are scattered on a free electron and converted into a single soft photon of optical range. Using the QED S-matrix formalism for the derivation of a cross-section of direct double-Compton (DDC) scattering and assuming detailed balance conditions, an analytical expression for the cross-section of the IDC process is presented. It is shown that at fixed energies of incident photons, the inverse cross-section has no infrared divergences, and its behavior is completely defined by the spectral characteristics of the photon source itself, in particular by the finite interaction time of radiation with an electron. Thus, even for the direct process, the problem of resolving infrared divergence actually refers to a real physical source of radiation in which photons are never actually plane waves. As a result, the physical frequency profile of the scattered radiation for DDC as well as for IDC processes is a function of both the intensity and line shape of the incident photon field.

We investigate the low density limit of the Homogeneous Electron system, often called the Strictly Correlated regime. We begin with a systematic presentation of the expansion around infinite r S , based on the first quantized treatments suggested in the existing literature. We show that the expansion is asymptotic in the parameter r S 1 ∕ 4 and that the leading order result contains exponential corrections that are significant even for r S ∼ 100 . Thus, the systematic expansion is of limited utility. As a byproduct of this analysis, we find that there is no Wigner Crystal (WC) in one spatial dimension. This is an example of the Mermin–Wagner theorem, but was not appreciated in some earlier literature. More modern work (Schulz, 1993 [1] ) has come to conclusions identical to ours. Note that the long range Coulomb potential modifies the dispersion relation of phonons in one dimension, but still leads to the instability of the crystal, due to a very weak infrared divergence. We then propose a new approximation scheme based on renormalization group ideas. We show that the Wegner–Houghton–Wilson–Polchinski exact renormalization group equation reduces, in the low density limit, to a classical equation for scale dependent electron and plasmon fields. In principle, this should allow us to lower the wave number cutoff of the model to a point where Wigner’s intuitive argument for dominance of the classical Coulomb forces becomes rigorously correct.

The anomalous magnetic moment (AMM) for excited states of an electron in a constant magnetic field has been calculated within the framework of two-dimensional electrodynamics. The analytical results for the interaction energy of the anomalous magnetic moment with the external magnetic field are obtained in two limiting cases of nonrelativistic and relativistic energy values in a comparatively weak magnetic field. It is shown that the interaction energy of the spin with the external field does not contain infrared divergence and tends to zero as magnetic field decreases, while the electron's AMM increases logarithmically.

Phenomenological aspects of radiation by relativistic electrons in external field, in matter or the vicinity of matter are reviewed, among which: infrared divergence, coherence length effects, shadowing, enhancement in aligned crystals, quantum recoil and spin effects, electron side-slipping, photon impact parameter and the presence of tunnelling in the radiation process., 15 pages, 1 figure. Talk given at 6th International Conference - Channeling 2014 Charged & Neutral Particles Channeling Phenomena October 5-10, 2014 Capri (Naples) Italy

The problem of infrared divergence of the effective electromagnetic field produced by elementary charges is revisited using the model of an electron freely evolving in a photon bath. It is shown that for any finite travel time, the effective field of the electron is infrared-finite, and that in each order of perturbation theory the radiative contributions grow without bound in the large-time limit. Using the Schwinger-Keldysh formalism, factorization of divergent contributions in multi-loop diagrams is proved, and summation of the resulting infinite series is performed. It is demonstrated that the effective electromagnetic field of the electron vanishes in the large-time limit, and that this vanishing respects the total charge conservation and the Gauss law. It is concluded that the physical meaning of infrared singularity in the effective field is the existence of a peculiar irreversible spreading of electric charges, caused by their interaction with the photon field. This spreading exists in vacuum as well as at finite temperature, and shows itself in a damping of the off-diagonal elements of the momentum-space density matrix of electron, but does not affect its momentum probability distribution. It precludes preparation of spatially localized particle states at finite times by operating with free particle states in the remote past. Relationship of the obtained results to the Bloch-Nordsieck theorem is established, and discussed from the standpoint of measurability of the electromagnetic field. The effect of irreversible spreading on the electron diffraction in the classic two-slit experiment is determined, and is shown to be detectable in principle by modern devices already at room temperature., 26 pages, 11 figures

We calculate the anomalous magnetic moment of the electron in the Chern-Simons theory in 2+1 dimensions with and without a Maxwell term, both at zero temperature as well as at finite temperature. In the case of the Maxwell-Chern-Simons (MCS) theory, we find that there is an infrared divergence, both at zero as well as at finite temperature, when the tree level Chern-Simons term vanishes, which suggests that a Chern-Simons term is essential in such theories. At high temperature, the thermal correction in the MCS theory behaves as $\frac{1}{\beta} \ln \beta M$, where $\beta$ denotes the inverse temperature and $M$, the Chern-Simons coefficient. On the other hand, we find no thermal correction to the anomalous magnetic moment in the pure Chern-Simons (CS) theory., Comment: 13 pages, 2 figures

Radiative effects in the electroproduction of photons in polarized $ep$-scattering are calculated with the next-to-leading (NLO) accuracy. The contributions of loops and two photon emission were presented in analytical form. The covariant approach of Bardin and Shumeiko was used to extract the infrared divergence. All contributions to the radiative correction were presented in the form of the correction to the leptonic tensor thus allowing for further applications in other experiments, e.g., deep inelastic scattering. The radiative corrections (RC) to the cross sections and polarization asymmetries were analyzed numerically for kinematical conditions of the current measurement at Jefferson Lab. Specific attention was paid on analyzing kinematical conditions for the process with large radiative effect when momenta of two photons in the final state are collinear to momenta of initial and final electrons, respectively., Comment: 22 pages, 7 figures, 4 appendices

We have recently obtained cross-sections for Compton scattering by K-shell electrons at extreme-relativistic (ER) energies of the incoming photon. Our method is essentially analytical, and only at the end did it require a modest numerical computation. The results are valid for the Compton line of the scattered photon spectrum, but do not cover the infrared divergence at the soft-photon end of the spectrum. This case is treated in the present paper. The method we apply here is an adaptation of the one used for the Compton line. This allows us to make use of many previous results. The quadruply and doubly differential Compton cross-sections can be expressed in analytical forms which allow factorization of the ER photoeffect cross-sections (differential or total, respectively). This result is shown to be a manifestation of the soft-photon theorem connecting the Compton matrix element at low emitted photon frequencies to that of the photoeffect. The computation of ER Compton cross-sections with soft-photon emission is thereby reduced to that of the cross-sections for the ER photoeffect. We carry out a computation for the latter case and extend previously existing results.

This paper studies the model of the quantum electrodynamics (QED) of a single nonrelativistic electron due to W. Pauli and M. Fierz and studied further by P. Blanchard. This model exhibits infrared divergence in a very simple context. The infrared divergence is associated with the inequivalence of the Hilbert spaces associated with the free Hamiltonian and with the complete Hamiltonian. Infrared divergences that are visible in the perturbative description disappear in the space of the clothed electrons. In this model when the Hamiltonian is expressed in terms of the “physical” fields that create the electron together with its cloud of soft photons, the variational principle suggested earlier can be applied. At finite time the Heisenberg field of the model acts in the space of the perturbative electron together with a finite number of perturbative photons, while the “physical” field can be chosen to act in the space of the exact (“physical”) electron eigenstates together with a finite number of physical photons. The space of the physical (or clothed) electron states can be chosen to be a Fock space.

It is shown that the Coulomb interaction between f electrons and broad band electrons has a profound influence on the dynamical properties of an f hole in mixed valence and heavy fermion materials. At zero temperature the dynamics of the screening process contains an infrared divergence. The broadening of this divergence by temperature causes the motion of the f electrons to crossover from wave propagation to diffusion. This mechanism explains the observed dual nature of the f electrons, namely that at low temperatures they behave like a Fermi liquid, while at high temperatures they evolve into a lattice of localized magnetic moments.

The problem of infrared divergence of the effective electromagnetic field produced by elementary particles is revisited using the non-equilibrium model of an electron interacting with low-temperature photons. It is argued that the infrared singularity of the effective field can be interpreted as a thermalization of the electron. It is shown that this thermalization is negligible in actual field measurements as it is completely dominated by the usual quantum spreading., Comment: 8 pages, 3 figures

We have developed an exact second-order S-matrix code for the relativistic numerical calculation of cross sections for Compton scattering of photons by bound electrons within the independent-particle approximation. We find good agreement with less exact treatments in regions where such theories are valid, recovering the impulse approximation and the soft-photon infrared divergence. We do not find agreement with an earlier calculation of Whittingham. We compare with recent scattering experiments in the regime Z\ensuremath{\alpha}(\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{i}}$)\ensuremath{\sim}${\mathit{E}}_{\mathit{B}}$, where we find that the impulse approximation is not adequate.

Interaction of a tunneling particle with fluctuations of the barrier induced by phonon or electron excitations is analyzed. Special role of nonadiabatic excitations with the energy smaller than the reciprocal time a particle spends under the barrier is discussed. In an insulator the interaction with barrier fluctuations can play a dominant role leading to qualitatively alteration of the pattern of the quantum diffusion. At that, the increase of the tunneling transition amplitude (coherent bandwidth) instead of the polaronic narrowing and significant change of the temperature dependence of incoherent tunneling motion take place. Analysis of recent experimental results on the quantum diffusion of muonium in KCl crystal revealing the decisive role of the barrier fluctuations is given. – The infrared divergence accompanying the rescattering of the electron-hole excitations near the Fermi-surface causes the increase of the interaction with the barrier fluctuations in a metal. Rigorous analysis for a two-well system with a completely arbitrary interaction of the electrons with a tunneling particle is carried out. It shows that there are no parameter values for which the interaction with barrier fluctuations overweight the intrawell interaction though the inelastic transition rate sharply enhances. The general problem of the validity of substitution of a phonon heat bath for an electron one is discussed.

The problem of infrared divergence of the effective electromagnetic field produced by elementary particles is revisited using the non-equilibrium model of an electron interacting with low-temperature photons. An infrared regularization of the real-time Feynman integrals is proposed which allows factorization of the infrared contributions to the effective field in multi-loop diagrams. The result of summation of the corresponding infinite series suggests that the infrared singularity in the effective field signifies existence of an irreversible spreading of the charge, which can be interpreted as a thermalization of the electron through its interaction with photons. It is shown that this thermalization is negligible in actual field measurements as it is completely dominated by the usual quantum spreading, but may become important in sufficiently slow processes. A non-perturbative aspect of the problem is discussed on the basis of the Ward identities.

The behavior of very slow hollow atoms traveling inside a solid is presented. The radiative electron capture of the target conduction electrons by these ions has been observed owing to the infrared divergence cross section of this process. The very specific properties of the hollow atoms are used to study their interaction in the first mono-atomic layers below the surface.

A detailed discussion of the temperature dependence of 1/f noise based on the model of diffusion transport with infrared divergence of localized electrons in metals is given. Theoretical results agree with experiments fairly well.

Coincidence techniques have been employed in a detailed study of Compton scattering of 279.2keV y rays by K-shell electrons of tin and gold. Double-differential cross sections at four angles have been determined from about 30 keV to the kinematic limit. In the case of gold, clear evidence has been obtained for the infrared divergence effect predicted by Gavrila, and for Cornpton defects towards lower energies. The data obtained with tin do not show significant infrared divergence or Compton defects. The widths of the fully developed quasi-Compton peaks are about 85 and 55 keV in the case of gold and tin, respectively. A detailed investigation has been made of possible false coincidences arising from secondary processes. Energy-integrated single-differential cross sections have also been determined. Comparisons with other experimental data and with results of theoretical calculations have been presented. The experimental cross sections in the regime of low outgoing photon energies are much larger than nonrelativistic and relativistic theoretical predictions. The experimental data above about 100 keV are in much better agreement with the relativistic second-order Smatrix calculations of Whittingham, although at some angles deviations of about 30%%uo are noticeable. The desirability of additional relativistic calculations and measurements with extremely thin targets has been pointed out.

The author reports a study of the infrared properties of the electron propagator in a framework which uses a gauge-invariant electromagnetic potential. The quantisation of the electromagnetic potential is performed in the Zwanziger formulation (1979) of quantum electrodynamics to avoid the usual infrared divergence. The Kallen-Lehmann representation of the infrared electron propagator in the photon vacuum is shown to be well defined and convergent in this framework. Furthermore, he shows that the probability density of an electron wavepacket propagating in the vacuum decreases like (time)-3, a well known property which is recovered here in quantum electrodynamics.

We consider heavy particles placed in the sea of metallic electrons. Both systems are represented by plane waves and interaction between them is considered. We pay special attention to the limit where the mass of the heavy particles tends to infinity. Even in this limit, the heavy particles are not equivalent to fixed particles, but represent a system with internal degrees of freedom. As a result of it, infrared divergences occur in physical parameters. Scattering matrix of the electrons on the heavy particles involves a factor (D/kT)g, where D is the cut-off energy for the electron and g is a coupling constant. The RKKY-type effective interaction between two heavy particles involves the square of the above factor. Furthermore, a new effective interaction which is always negative results from higher-order terms.

Energy distributions of photons after Compton scattering of 279.2 keV gamma rays by K shell electrons of gold and tin have been determined by coincidence techniques involving the detection of K X-rays and the scattered gamma rays. The measured energy distributions extend from about 25 keV to higher energies including the quasi-Compton peak. A detailed study of possible secondary events has been made. Evidence for infrared divergence contribution in the case of gold has been presented.

correlation function are studied in the Born approximation. A new formulation is developed which relates the hole correlation function to the retarded polarization propagator of the electron gas. Previously obtained results for the free-electron gas at finite temperature are recovered. The Migdal approximation is employed to find an expression for the imaginary part of the polarization propagator and it is seen that although the strength of the singularity is potentially modified by a constant factor the time dependence is not changed. Inclusion of the electron-phonon interaction via the Migdal form for the electron spectral weight func­ tion has no effect on the hole correlation function if the one-electron density of states varies slowly on the scale of the width of the spectral weight function. Similarity of this behavior to that of the nuclear spin relaxation in electron-phonon system is pointed out.

Because of the astrophysical importance of beta-decay lifetimes in stellar interiors, we have calculated the rate of photon-induced beta decay ("photobeta" decay). In the photobeta process the photon can be considered to decay virtually into an electron-positron pair, with the positron being absorbed by the nucleus. The photobeta process is in competition with normal beta decay (if energetically possible), excited-state beta decay, and free-positron capture. An infrared divergence problem for exothermic photobeta decay is discussed, although the exothermic photobeta rate is too small to enhance most spontaneous beta decays. Applications are discussed in the driven decay of stable nuclei. As an example, the photobeta lifetime of a nucleus stable by 200 keV drops from 3\ifmmode\times\else\texttimes\fi{}${10}^{10}$yr at 3\ifmmode\times\else\texttimes\fi{}${10}^{8}$ \ifmmode^\circ\else\textdegree\fi{}K, to ${10}^{5}$yr at 1.2\ifmmode\times\else\texttimes\fi{}${10}^{9}$ \ifmmode^\circ\else\textdegree\fi{}K for a nuclear transition matrix element characterized by log ft=6. The competition between the photobeta process and excited-state beta decay is discussed.

This chapter describes asymptotic formulae of quantum electrodynamics. The asymptotic constancy of the total cross-section is a characteristic property of scattering processes whose diagrams can be cut across internal photon lines. It occurs even when more than two particles are present in the final state of the reaction. The double-logarithmic corrections occur in cases of two kinds. One kind includes scattering through a fixed finite angle. The cross-sections decrease in the asymptotic high-energy range. In such cases, the double-logarithmic corrections are associated with the infrared divergence. These cases include elastic scattering of electrons in an external Coulomb field. The other class of cases includes reaction cross-sections, which decrease with increasing energy for a given square of the momentum transfer, that is, for scattering angles, which asymptotically approach zero or π.

There is increased recognition 1,2 of the importance in studying the insulator phase on the insulating side of the transition. For example, the donor susceptibility is found to diverge as the transition is approached from the insulator side and can be related to a diverging localization length. Coulomb interactions are expected to play a significant role in the insulator phase. Efros3 have pointed out a Coulomb gap at T=0. The ground state is complicated and it is recognized to resemble 4,5 that of a glass or spin glass. The analogy with the glassy state arises from competition between the 1/r long-range Coulomb interactions and the randomness of site energies. For these reasons, the system on the localized side is called an electron glass. 4,5 Experimentally the slow relaxation behavior in the ac conductivity5 bears a strong resemblance to magnetization relaxation in spin glasses and dielectric or mechanical relaxations in glasses.6,7 In the critical region and at low temperatures, a conductance dependence σ(ω)=Kωs) with s=0.9 has been found5 from 105 to 102 Hz. The measured donor capacitance C has the frequency dependence ωs−1. Hence the dielectric relaxation function e* ≡e′−ie″ exhibit the dependence ωs−1 (a negative small-power law) in both its real and imaginary parts as ω→0. This “infrared divergence” behavior5,6 of e’ and e” is commonly observed in glasses. In spin-glasses, the spin susceptibility behaves in a similar way. 7 This study of low frequency relaxation in the electron glass ruled out the noninteracting Fermi glass model, and confirmed the importance of Coulomb interactions.5 Calculation of the ac conductivity of the electron glass at zero on finite temperature are restricted to the two-site or pair approximation.3 In our view if ħω≪kT, as is the case in all2,5 low-frequency measurements, a relaxation description of the entire assembly is more appropriate. This is particularly relevant in the critical region.

As all of you know, the metal electrons constitute a degenerate Fermi system. Its excitation energy ranges from zero to several electron volts. Then, one may ask what is the energy scale of the metal electrons. In the case of static perturbations acting on the electrons, it is of the order of the Fermi energy ɛF. For example, suppose that an impurity potential V(r) is placed in the jellium of the electrons. The energy shift due to this perturbation may be expanded in V: $$\Delta {\rm{E = }}{{\rm{c}}_1}{v_0}\; + \;{c_2}{v_0}^2{\rm{\rho + }}\;{{\rm{c}}_3}{v_0}^3{{\rm{\rho }}^2}\; + \;,$$ where V0 is the matrix element of V(r), which is assumed to be independent of the wave numbers, ρ is the density of the electron states. Thus, the expansion parameter is V0ρ, which is about V0/ɛF. This means that the energy scale of the electrons is the Fermi energy. On the other hand, when the perturbation is dynamical and local, such as the s-d exchange model, excitation modes of low energy come into play and give rise to an infrared divergence. We call this fact the Fermi surface effect.

Publisher Summary This chapter discusses the experimental facts that reflect the several theoretical results and aspects and underlines the process of quantum scattering. The physical processes involved with the quantum modulation are of two kinds: a bremsstrahlung process and a coherence property. The first is a single-particle description (particle picture) while the coherence properties at interference of excited or modified states of the wave field of the electron with its initial state being a basic property of wave picture of the electrons. The experiment for the interaction of an electron beam with a crossing laser beam using gaseous argon atoms or other molecules within the crossing area for providing the necessary recoil for the interaction is shown schematically. Other than the Schwarz–Hora effect, which arises from the stimulated radiation processes only, the Handel effect is an infrared divergence phenomenon and arises exclusively from the spontaneous bremsstrahlung process.

In this paper, we suggest a new physical mechanism for the in 1/f noise metals phenomena, which have a history of experimental observation for more than forty years. We propose that the 1/f noise is caused by the diffusion transportation with infrared divergence of the local electrons in metals. Based on the general theory of the low-frenquency fluctuation, dissipation and relaxation, a non-Markovian diffusion equation is established. The noise spectra deduced from this equation can explain the four basic characteristics of the 1/f noise. With reasonable choice of parameters, our noise spectra agree fairly well with experimental results. Some further discussions concerning the current theory and related problems are also given.