aa r X i v : . [ phy s i c s . g e n - ph ] O c t Fundamental formulation of light-matter interactions revisited
H. R. Reiss
Max Born Institute, Berlin, Germany andAmerican University, Washington, DC, USA ∗ (12 October 2019) Abstract
The basic physics disciplines of Maxwell’s electrodynamics and Newton’s mechanics have beenthoroughly tested in the laboratory, but they can nevertheless also support nonphysical solutions.The unphysical nature of some dynamical predictions is demonstrated by the violation of symmetryprinciples. Symmetries are fundamental in physics since they establish conservation principles. Theprocedures explored here involve gauge transformations that alter basic symmetries, and these alter-ations are possible because gauge transformations are not necessarily unitary despite the widespreadassumption that they are. That gauge transformations can change the fundamental physical mean-ing of a problem despite the preservation of electric and magnetic fields is a universal proof thatpotentials are more basic than fields. These conclusions go to the heart of physics. Problems arenot evident when fields are perturbatively weak, but the properties demonstrated here can be crit-ical in strong-field physics where the electromagnetic potential becomes the dominant influence ininteractions with matter. ∗ Electronic address: [email protected] . INTRODUCTION The unfamiliar properties of electromagnetism to be described here can be overlookedwhen the electromagnetic field is no more than a perturbative influence in physical pro-cesses. However, when the electromagnetic field is the dominant influence, then these prop-erties become profoundly important. The ever-expanding use of powerful lasers imparts afundamental significance to these unfamiliar properties.Current beliefs about electromagnetism that are challenged here come from the demon-strations that: electromagnetic potentials convey more physical information than electricand magnetic fields; reliance on electric and magnetic fields can introduce basic errors;gauge transformations alter the properties of a physical system; gauge transformations arenot unitary; concepts such as the adiabaticity property in laser phenomena are false andwasteful; a proposed nondipole correction is unphysical; and predictions that follow fromthe solutions of Maxwell’s equations can be unphysical. From these results, it is a corollarythat Newton’s mechanics can also support unphysical solutions.An ancillary matter is the objection to the widespread use of an intensity parameter thatlacks Lorentz invariance, but is held to be descriptive of otherwise covariantly-describedphenomena.It is emphasized that neglect of basic electrodynamic principles in applications to strong-field laser processes has caused important hindrances to the development of the discipline.These hindrances continue, and can lead to needless delays in the development of this largeand expanding field of study.
II. GAUGE TRANSFORMATIONS ALTER PHYSICAL PROPERTIES
The fact that gauge transformations can fundamentally alter the physical identity of asystem is evident even in the elementary problem of an electron immersed in a uniformconstant electric field E .A possible set of potentials to describe the field is φ = − r · E , A = 0 . (1)The Lagrangian that describes the electron in the field is independent of time. By Noether’sTheorem [1], this means that energy is conserved. Another possible gauge for the description2f the constant field is φ = 0 , A = − c E t. (2)The Lagrangian for an electron with the field described by Eq. (2) has time dependence,but is independent of the spatial coordinate r . In this case, energy is not conserved butmomentum is conserved.The potentials (1) and (2) have different symmetries, and represent different physicalsituations. These differences are produced by the gauge transformation.An important case that possesses only one gauge that satisfies all relevant symmetrieswas examined in Ref. [2]. The electromagnetic field examined is a plane-wave field, suchas that of a laser beam. The symmetry that is present in that case is the propagationproperty, which requires that the field can depend on the spacetime 4-vector x µ only as ascalar product with the propagation 4-vector k µ : ϕ ≡ k µ x µ = ωt − k · r , (3)where ω is the field frequency and k is the propagation 3-vector. When a scalar potential φ such as that from a Coulomb potential is also present, then the sole possible gauge satisfyingthe necessary symmetry is the radiation gauge (also called Coulomb gauge), where the 3-vector component A is descriptive of the plane-wave field and the scalar potential φ describesthe binding potential, so that the total 4-potential is A µ : (cid:0) φ scalar , A planewave (cid:1) . (4) III. GAUGE TRANSFORMATIONS ARE NOT NECESSARILY UNITARY
The starting point here is the property known as form invariance, where the Schrödingerequation has the same form when expressed in terms of the gauge-transformed potentialsas it does in the original gauge. See, for example, Ref. [3]. Form invariance under a gaugetransformation generated by the operator U can be written as e H − i ℏ ∂ t = U ( H − i ℏ ∂ t ) U − , (5)where e H is the transformed Hamiltonian. This gives the gauge-transformed Hamiltonian e H = U HU − − i ℏ U (cid:0) ∂ t U − (cid:1) . (6)3his means that the gauge transformation cannot be a unitary transformation if U istime-dependent.For laser-related problems, the time dependence of the field imparts time dependence toany gauge transformation employed. Such transformations are not, in general, unitary. IV. ELECTROMAGNETIC POTENTIALS ARE MORE FUNDAMENTAL THANELECTRIC AND MAGNETIC FIELDS
The primacy of potentials over fields was first established by the Aharonov-Bohm effect[4, 5]. This relates to a specific example: the deflection of an electron beam as it moves inthe field-free region around a solenoid. It is the potential that causes the deflection, sincethere is a potential but no field outside the solenoid. That quantum result stood for manyyears as the sole example of the fundamental role of electromagnetic potentials. A moregeneral case is the demonstration [2] that there exists an unphysical solution of the Maxwellequations for a plane-wave field propagating in the vacuum. This has consequences that areboth quantum and classical.Furthermore, as shown in the following Section, when a solution of the Maxwell equationsis unphysical, then the properties of the potentials are necessary to distinguish physical fromunphysical solutions. This is a universal proof that potentials are more fundamental thanfields.
V. SOLUTIONS OF MAXWELL EQUATIONS ARE NOT NECESSARILY PHYS-ICAL
A single unphysical solution of Maxwell’s equations is sufficient to demonstrate that suchunphysical solutions can exist. The example selected here is significant since it has beenproposed or employed for practical laser-induced processes.The symmetry condition that applies to all plane-wave fields, such as laser fields, comesfrom the Einstein Principle [6] that the speed of light in vacuum is the same in all inertialframes of reference. This was referred to above as the propagation property. Its mathe-matical statement is that the spacetime 4-vector x µ can occur only as the scalar product ϕ A µ ( ϕ ) .A gauge transformation of the electromagnetic field is generated by the function Λ : A µ → e A µ = A µ + ∂ µ Λ . (7)The only constraints on Λ are that it be a scalar function and that it satisfies the homoge-neous wave equation ∂ µ ∂ µ Λ = 0 . (8)This is sufficient to preserve the electric and magnetic fields. If A µ satisfies the Lorenzcondition ∂ µ A µ = 0 , the same will be true of e A µ . Now consider the generating function [7] Λ = − A µ x µ , (9)which leads to the gauge-transformed potential e A µ = − k µ ( x ν A ′ ν ) , (10)where A ′ ν is the total derivative of A ν with respect to ϕ : A ′ ν = ( d/dϕ ) A ν . Equation (10)takes a familiar form if the initial gauge for A µ is the radiation gauge. A pure plane-wavefield is described in the radiation gauge by the 4-vector A µ ( ϕ ) : (0 , A ( ϕ )) . (11)The gauge-transformed 4-vector is then e A µ = − b k µ r · E ( ϕ ) , b k µ ≡ k µ ω/c , (12)where b k µ is the unit propagation 4-vector that lies on the light cone.The form (12) resembles the dipole-approximation scalar potential r · E ( t ) that is soubiquitous in length-gauge Atomic, Molecular, and Optical (AMO) physics. This is thereason why it was examined in Ref. [7] in an attempt to provide a rigorous basis for theKeldysh approximation [8] of strong-field atomic physics. It was rejected in Ref. [7] onmultiple grounds, the most obvious of which is that it violates the Einstein Principle. Theviolation is evident in Eq. (10) from the appearance of x ν in isolation from the propagation4-vector, and the presence of the 3-vector r in Eq. (12) that requires an origin for a fixed5patial coordinate system that is contrary to the nature of a freely propagating plane-wavefield. Nevertheless, the fields are preserved by the gauge transformation (9), and so are theLorenz condition ∂ µ A µ = 0 and the transversality condition k µ A µ = 0 [7].The 4-potential in Eq. (10) or (12) has the curious feature that it lies on the lightcone. A plane-wave field is described by a spacelike 4-potential, not one that is lightlike.Furthermore, a fundamental property of a charged particle in interaction with a plane-wavefield is the ponderomotive energy [9–11] U p , which is proportional to A µ A µ . However, since k µ is self-orthogonal, k µ k µ = 0 , (13)the e A µ of Eq. (10) or (12) predicts a zero ponderomotive energy for any charged particle.For all of these reasons, Eqs. (10) and (12) are unphysical. Nevertheless, they are arrivedat by a valid gauge transformation from a proper plane-wave 4-potential, meaning that theypredict the same electric and magnetic fields, and hence they satisfy the same Maxwellequations, since the Maxwell equations depend only on the fields, not on the potentials.This is proof that Maxwell’s equations can support unphysical solutions.Equations (10) and (12) were first proposed and discussed in Ref. [7], where the above-mentioned problems were noted, and Eqs. (10) and (12) were rejected as unphysical. How-ever, a Heidelberg group [12] took note of these equations and applied them to practicalproblems on the grounds that they described correctly the electric and magnetic fields oflaser beams. Also, a Norwegian group [13], apparently without knowledge of Ref. [7],proposed these equations as a way of introducing nondipole corrections into the study oflaser-induced reactions.While Eq. (10) or (12) is not acceptable for a properly formulated theory, it is possiblethat qualitative information can be attainable from it. It was used in Ref. [14] to estimatethe onset of magnetic effects, which it established correctly.6 I. SOLUTIONS OF NEWTON’S EQUATIONS ARE NOT NECESSARILY PHYS-ICAL
Newtonian physics is based on forces, and electromagnetic forces are dependent on electricand magnetic fields, as given by the Lorentz force expression F = q (cid:16) E + v c × B (cid:17) . (14)Hence, the reasoning applied to show the possibility of unphysical solutions of the Maxwellequations applies as well to Newton’s equations.Alternative formulations of classical mechanics, such the Lagrangian, Hamiltonian,Hamilton-Jacobi, ... are based on potentials, and hence they convey more information thana force-based theory like Newton’s mechanics. This explains the common practice in me-chanics textbooks to show that potential-based formalisms imply the Newtonian formalism,but the reverse is never shown. VII. PRACTICAL CONSEQUENCES
When approximations are employed in the study of a physical process, results can beinefficient and possibly erroneous if basic symmetries are not observed. An example fromstrong-field physics is the phenomenon known as Above-Threshold Ionization (ATI), whichrefers to the observation [15] that ionization by an intense laser beam can exhibit processes ofphoton number in addition to, or in place of, the lowest allowed order predicted by perturba-tion theory. AMO physics has experienced accurate and reliable results from perturbationtheory, and the observation of ATI came as a shock to the AMO community. A recentassessment by prominent researchers [16] of this unexpected result can be paraphrased inabbreviated form as “... multiphoton ionization experiments using intense infrared pulsesfound the then-amazing result that an ionizing electron often absorbed substantially morephotons than the minimum needed for ionization. This puzzling behavior led to the term ...ATI ... The problem was ultimately solved by computer simulations and the semiclassicalrecollision model.” Citations to the relevant theory place the date for eventual understandingof the 1979 experiment at 1993, a span of 14 years.The important fact here is that both analytical and numerical studies employed thedipole approximation, which has the effect of replacing the propagating laser field by an7scillatory electric field. This loses the propagation symmetry that is at the heart of strong-field processes described above.From the point of view of propagating fields, the significant contribution of many photonorders at high field intensities is obvious, and noted long before the 1979 experiment. Forexample, in bound-bound transitions, there is the 1970 statement [17] “...as the intensity getsvery high ... higher order processes become increasingly important.” For photon-multiphotonpair production in 1971 [18]: “ ... an extremely high-order process can ... dominate the lowestorder ...”. For interband transitions in band-gap solids in 1977 [19]: “...high-order processescan be more probable than lower-order processes when the intensity is sufficiently high.”The 1980 ionization paper [20], written before the ATI experiment, describes ATI in detail.Other high-intensity phenomena, such as channel closing and stabilization, are also discussedin the early papers just cited.
A. Nondipole corrections
The difficulty of Eqs. (10) or (12) for the introduction of nondipole corrections have beendiscussed above. A valuable laboratory project would be to determine the limitations onsuch an approach.A fully relativistic propagating strong-field theory is certainly applicable for all nondipole,magnetic field, and relativistic studies. The construction of such a theory was elaborated inRef. [21] for the Klein-Gordon case, and implemented in detail in Ref. [22] for the Diraccase.
B. Local Constant Field Approximation (LCFA)
The LCFA is an example of how field-based criteria can differ from potentials-basedcriteria. One justification of the LCFA follows from the field-based observation that the twoLorentz invariants of plane-wave fields E − B = 0 , E · B = 0 , (15)can be satisfied by constant crossed fields [23, 24].8hen viewed from the standpoint of potentials, the potentials that describe constantcrossed fields E , B are φ = − r · E , A = − r × B (16) | E | = | B | , E ⊥ B . (17)These potentials are unrelated to the A µ ( ϕ ) requirement for propagating fields. C. Low frequency limit of a plane wave
Plane waves are characterized by the fact that they propagate in vacuum at the speedof light. This feature is independent of frequency. There is a line of reasoning, adopted formany years in the strong-field community, that there exists a zero-frequency limit of planewaves, and this limit is simply a constant electric field. This is inferred from the dipoleapproximation, so that there is no magnetic field present, distinguishing if from the LCFA.There is no such thing as a zero frequency plane wave. Plane waves propagate at the speedof light, independently of frequency. An example of a plane wave phenomenon of extremelylow frequency is the Schumann resonance [25]. This is a naturally occurring phenomenon inwhich powerful lightning strikes generate extremely low frequency radio waves that resonatein the cavity formed by the Earth’s surface and the ionosphere. The lowest mode of thiscavity is . Hz , corresponding to a wavelength about equal to the circumference of theEarth. On a laboratory scale, a plane wave with a wavelength equal to the circumferenceof the Earth would appear to be a constant field. Yet neither a constant crossed field nor aconstant electric field can spread its influence over the entire planet.A pernicious consequence of the concept of a low frequency limit of a laser field asbeing a constant electric field was its use as a criterion for judging the worth of analyticalapproximations. For many years, a zero-frequency limit equivalent to a constant electricfield was regarded as a feature of sufficient importance to reject any theory that did notpossess that property, See, for example, Ref. [26]. This limit was regarded as an adiabaticlimit, and the qualitative stance was adopted that low frequency fields should exhibit thisadiabaticity. In actuality, the ω → limit of plane waves is relativistic [27], not adiabatic.It is the relativistic property of propagation at the speed of light that distinguishes theSchumann resonance from a constant field phenomenon.9t is impossible to estimate the cost in valuable research resources of the long-term ap-plication of the adiabaticity test as a basic criterion, but it is undoubtedly considerable. VIII. CENTRAL ROLE OF A µ The basic properties of a propagating field can be described entirely by the 4-vectorpotential. This makes possible a covariant statement of those properties, including theidentity of the coupling constant of strong-field physics.The 4-vector potential enters the description of propagating fields in the three fundamen-tal expressions: ∂ µ A µ = 0 , (18) k µ A µ = 0 , (19) z f ∼ A µ A µ . (20)The first is the Lorenz condition, second is the transversality condition, and the third entersinto the definition of the strong-field coupling constant z f . The implications of Eq. (20)seem to be little-known, but they are perhaps the most direct expressions of the ascendancyof potentials over fields.The Lorenz condition can be expanded into ∂ µ A µ = ∂c∂t φ − ∇ · A = 0 . (21)In the radiation (or Coulomb) gauge, where the scalar potential φ applies only to longitudinalpotentials, the Lorenz condition for the propagating field reduces to ∇ · A = 0 , which is oftenused as the identifying condition for the radiation gauge.The expression (19) is the covariant transversality condition. This is readily shown toinfer geometrical transversality: k · E = 0 and k · B = 0 . The coupling constant of strong-field physics was identified [9, 10] long ago. Strong-fieldphysics as a separate discipline was established [28] as a consequence of the demonstrationby Dyson [29] that standard QED (Quantum Electrodynamics) does not possess a conver-gent perturbation expansion. This raised the question of the convergence properties of anexternal-field theory, which represents a strong-field situation where the number of photons10resent during an interaction is large. The expansion parameter of standard QED is the fine-structure constant α . A convergence study of the external-field theory revealed the fact thatevery appearance of α involved the same intensity-dependent factor. That is, the expansionparameter is not α, but rather the product of α with that factor. This product was labeled z in the original studies [9, 10], since an expansion parameter must be extended into thecomplex plane to find the singularities that limit convergence, and z is often used to label acomplex number. In more recent work z was re-labeled z f to indicate that it is the intensityparameter for free electrons as opposed to two new parameters z (nonperturbative intensityparameter) and z (bound-state intensity parameter) that arise when scalar potentials existthrough interactions of the electron with binding potentials in addition to the plane-wavefield. See Section 1.3 in Ref. [30] for further discussion.In current terminology, the coupling constant is written z f = 2 U p /mc , (22)where U p is the ponderomotive energy, defined as U p = e mc h| A µ A µ |i . (23)The angle brackets denote an average over a full cycle of the field, and the absolute value istaken because A µ is a spacelike 4-vector.The quantity z f just identified as the coupling constant for strong laser fields is alreadyknown as an intensity parameter for strong fields, but its additional role as the couplingconstant seems to have escaped general attention.From Eq. (23) the ponderomotive energy and hence z f are Lorentz invariants. If z f is tobe a proper coupling constant it must also be gauge-invariant, and this is not apparent in(23). However, when A µ describes a propagating field, then U p has been shown [11, 31] tobe gauge-invariant.An objection is raised here to an intensity parameter that has found acceptance in therelativistic strong-field literature. The quantity A µ A µ is rendered as | E | /ω , and then,since the square seems unnecessary, the parameter is commonly written as proportional to E/ω . The intent apparently is to introduce the electric field in the belief that it is morefundamental than the 4-vector potential. In addition to its inappropriate emphasis on theelectric field rather than on the 4-vector potential, this convention is objectionable in a11heory that is founded on covariant expressions. Lorentz-invariance is lost because Lorentztransformation properties of the electric field E are not the same as for the frequency ω .The quantity z f of Eq. (22) is Lorentz-invariant, gauge-invariant, and covariant.It is not possible to express A µ A µ in terms of fields. When a quantity can be stated withfields it is always possible to convert it to potentials because potentials are found from fieldsby differentiation; a local procedure. To convert a quantity stated entirely with potentials,any attempt to convert the expression to fields will fail because such a procedure requiresintegration, which is nonlocal.The z f parameter also occurs in the intensity-dependent mass-shell condition for theelectron in a strong field. The usual mass shell of QED is p µ p µ = ( mc ) . (24)However, all the early studies of strong-field interactions [9, 10, 32–35] found the shifted-massequation p µ p µ = ( mc ) (1 + z f ) . (25)Sarachik and Schappert find [36] that the altered mass shell expression (25) also exists withclassical strong fields.A further implication of z f becomes clear when it is expressed in terms of the photondensity ρ . This expression has the form [11] z f = αρV, (26)with the fine structure constant α multiplied by the number of photons contained in aneffective interaction volume V . This volume is approximately a cylinder of radius given bythe electron Compton wavelength and a length λ given by the wavelength of the plane-wavefield. The Compton wavelength is the expected interaction length for a free electron, but λ is a macroscopic length in most laboratory applications. This is a way to understand why ω → (or λ → ∞ ) leads to relativistic behavior and not adiabatic behavior.The fundamental quantity A µ A µ is expressed directly in terms of the 4-vector potential A µ . There is no equivalent expression in terms of the electric field. This simple compellingfact supports the primacy of potentials over fields.12 X. THE PATH AHEAD
As laser intensities increase and as low-frequency capabilities improve, the lessons con-tain herein are basic. In brief, one must consider the true electromagnetic properties of verystrong fields, including especially the requirements that follow from the propagation prop-erty. The dipole approximation, long a reliable feature of AMO physics, is not to be trusted,and new criteria must be adopted that are consonant with relativistic behavior. The penaltyin waste of research resources that was mentioned in connection with the explanation of ATIby a transverse-field method 14 years earlier than in terms of the dipole approximation, andthe fallacious adiabaticity demand that also delayed progress by many years, is a cautionthat also applies to the LCFA model. Guidance in research activities allow for some investi-gation of the limits of applicability of the proposed nondipole correction of Eq. (10) or (12),but always with the knowledge that it lacks support as a reliable method.Perhaps most important of all is the need to be aware that electromagnetic potentials arethe essential determinants of the nature of electromagnetic phenomena, and that dependenceon electric and magnetic fields carries existential risks. [1] E. Noether,
Invariante Variationsprobleme , Nachr. d. König. Gesellsch. D. Wiss. zu Göttingen,Math-phys. Klasse , 235-257 (1918).[2] H. R. Reiss, Physical restrictions on the choice of electromagnetic gauge and their practicalconsequences , J. Phys. B , 075003 (2017).[3] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977).[4] W. Ehrenberg and R. E. Siday,
The refractive index in electron optics and the principles ofdynamics,
Proc. R. Soc. B , 8-21 (1949).[5] Y. Aharonov and D. Bohm, Significance of potentials in the quantum theory,
Phys. Rev. ,485-491 (1959).[6] A. Einstein,
Zur Elektrodynamik bewegter Körper , Ann. Phys. , 891-921 (1905).[7] H. R. Reiss, Field intensity and relativistic considerations in the choice of gauge in electrody-namics , Phys. Rev. A , 1140-1150 (1979).[8] L. V. Keldysh, Ionization in the field of a strong electromagnetic wave , Sov. Phys. JETP , , 1945-1957 (1964)].[9] H. R. Reiss, PhD Dissertation, Univ. of Maryland (1958).[10] H. R. Reiss, Absorption of light by light , J. Math. Phys. 3, 59 (1962).[11] H. R. Reiss,
Mass shell of strong-field quantum electrodynamics , Phys. Rev. A , 022116(2014).[12] M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, Gauge-invariant relativistic strong-fieldapproximation , Phys. Rev. A , 053411 (2006).[13] S. Selstø and M. Førre, Alternative descriptions of the light-matter interaction beyond the dipoleapproximation , Phys. Rev. , 023427 (2007).[14] H. R. Reiss, Dipole-approximation magnetic fields in strong laser beams , Phys. Rev. A ,013409 (2000).[15] P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Free-free transitions followingsix-photon ionization of xenon atoms , Phys. Rev. Lett. , 1127-1130 (1979).[16] S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, Photon-momentum transfer in multiphotonionization and in time-resolved holography with photoelectrons , Phys. Rev. A , 051401(R)(2015).[17] H. R. Reiss, Atomic transitions in intense fields and the breakdown of perturbation calculations ,Phys. Rev. Lett. , 1149-1151 (1970).[18] H. R. Reiss, Production of electron pairs from a zero-mass state , Phys. Rev. Lett. , 1072-1075(1971).[19] H. D. Jones and H. R. Reiss, Intense field effects in solids , Phys. Rev. B , 2466-2473 (1977).[20] H. R. Reiss, Effect of an intense electromagnetic field on a weakly bound system , Phys. Rev. A , 1786-1813 (1980).[21] H. R. Reiss, Complete Keldysh theory and its limiting cases , Phys. Rev. A , 1476-1486(1990).[22] H. R. Reiss, Relativistic strong-field photoionization , J. Opt. Soc. Am. B , 574-586 (1990).[23] A. I. Nikishov and V. I. Ritus, Ionization of systems bound by short-range forces by the fieldof an electromagnetic wave , Sov. Phys. JETP , 168-177 (1966) [Zh. Eksp. Teor. Fiz. ,255-270 (1966)].[24] V. I. Ritus, Quantum effects of the interaction of elementary particles with an intense electro-magnetic field , J. Sov. Laser Res. , 497-617 (1985) [Lebedeva Akad. Nauk SSSR , 5-151 Schumann resonances. [26] C. J. Joachain, N. J. Kylstra, and R. M. Potvliege,
Atoms in Intense Laser Fields (CambridgeUniv. Press, Cambridge, 2012).[27] H. R. Reiss,
Limits on tunneling theories of strong-field ionization , Phys. Rev. Lett. ,043002 (2008);
Erratum , Phys. Rev. Lett. , 155901(E) (2008).[28] H. R. Reiss,
Special analytical properties of ultrastrong coherent fields,
Eur. Phys. J. D ,365–374 (2009).[29] F. J. Dyson, Divergence of perturbation theory in quantum electrodynamics , Phys. Rev . ,631-632 (1952).[30] H. R. Reiss, Theoretical methods in quantum optics: S-matrix and Keldysh techniques forstrong-field processes,
Prog. Quant. Electr. , 1-71 (1992).[31] H. R. Reiss, On a modified electrodynamics , J. Mod. Optics , 1371–1383 (2012); Corrigen-dum , J. Mod. Optics , 687 (2013).[32] N. D. Sengupta, On the scattering of electromagnetic waves by a free electron , II-Wave me-chanical theory , Bull. Math. Soc. (Calcutta) , 175-180 (1952).[33] L. S. Brown and T. W. B. Kibble, Interaction of intense laser beams with electrons , Phys. Rev. , A705-A719 (1964).[34] A. I. Nikishov and V. I. Ritus,
Quantum processes in the field of a plane electromagnetic waveand in a constant field, I , Sov. Phys. JETP , 529-541 (1964) [Zh. Eksp. Teor. Fiz. , 776-796(1964)].[35] I. I. Goldman, Intensity effects in Compton scattering , Phys. Lett. , 103-106 (1964).[36] E. S. Sarachik and G. T. Schappert, Classical theory of the scattering of intense laser radiationby free electrons , Phys. Rev. D , 2738-2753 (1970)., 2738-2753 (1970).