Fundamental formulation of electrodynamics revisited, and the precision of quantum electrodynamics
aa r X i v : . [ phy s i c s . g e n - ph ] M a r Fundamental formulation of electrodynamics revisited, and theprecision of quantum electrodynamics
H. R. Reiss
Max Born Institute, Berlin, Germany andAmerican University, Washington, DC, USA ∗ (16 March 2020) Abstract
It was shown recently that unambiguous description of electromagnetic environments requireselectromagnetic potentials; knowledge only of electric and magnetic fields is insufficient and canlead to error. Consequences of that demonstration are here applied to propagating fields, suchas laser fields. Gauge invariance is replaced by symmetry preservation. This alteration makesit possible to understand how the known failure of the convergence of perturbation expansions inquantum electrodynamics (QED) follows from the fact that QED is incomplete; it does not containits strong-field limit. Inherent in that demonstration are the strong-field coupling constant and thestrong-field alteration of the mass shell of a charged particle. A variety of physically importantconsequences ensue, including the loss of guidance from Feynman diagrams. The meaning of testsfor the precision of QED is questioned since such evaluations apply only to perturbative QED, butnot to extensions required for complete QED. ∗ Electronic address: [email protected] . INTRODUCTION Gauge transformations are designed to find alternative sets of electromagnetic potentialsassociated with a specific configuration of electric and magnetic fields. The conventional be-lief has long been that electric and magnetic fields define the electromagnetic environment,so that the existence of alternative sets of potential functions identifies them as auxiliaryquantities. That situation is inverted in Ref. [1], where gauge transformations are shown toalter the fundamental symmetries associated with a physical system. That opens the possi-bility of finding spurious solutions in Maxwellian electrodynamics and Newtonian mechanics.It is necessary to select an appropriate set of potentials to define fully the physical problem.This identifies potentials as more fundamental than electric and magnetic fields. It is alsoshown in Ref. [1] that gauge transformations are not, in general, unitary transformations.The symmetries appropriate to a physical environment are determined by properties ofthe Lagrangian describing the system [2]. The quantum theory of fields is based on theproperties of Lagrangians, and so that discipline is free of the ambiguities explored hereand in Ref. [1]. Gauge invariance is a foundation principle when employed in the quantumtheory of fields, but it has important limitations in the context of how electric and magneticfields are related to potential functions,The conclusions of Ref. [1] are applied to the important case where the electromagneticenvironment includes a propagating field (also known as a transverse field, a plane-wavefield, a sourceless field, or a photon field). For example, these results apply to all laser-fieldproblems.The usual meaning of quantum electrodynamics (QED) is in the sense of Feynman-Dysonperturbation theory, with the primary computational method being the use of Feynmandiagrams. The subjects examined here become very important when strong fields exist,with the accompanying concepts of intensity-dependent coupling constant and mass shell forcharged particles. These concepts are not part of standard QED. It is known that Feynman-Dyson perturbation theory is not convergent. That lack is ascribed to the incompletenessof QED, which does not include strong field phenomena.The significance of studies of the precision of QED is questioned, since such studies focuson the value of the fine-structure constant. That has meaning only for perturbative QED, butnot if the meaning of QED is extended to include strong-fields. Laboratory experiments with2trong fields interacting with charged particles have for many years exceeded perturbativelimits. Importantly different conditions for extended QED are proposed.The background for results presented here spans the entire history of strong-field physics.It combines recent results, such as Ref. [1], with early work on strong fields from the 1950sand 1960s, whose significance could not be fully appreciated at the time.Gaussian units are used for electromagnetic quantities.
II. GAUGE INVARIANCE
In the quantum theory of fields, gauge invariance has profound importance, but that the-ory is entirely in terms of potentials. See, for example, Ref. [3]. The sense in which gaugeinvariance is discussed here is in the more mundane connection between potentials and elec-tric and magnetic fields. This semiclassical connection has not had a detailed examinationequivalent to that for quantized fields.The principle of gauge invariance in its classical sense has, as its origin, the notion thatelectric and magnetic fields determine all dynamical consequences in electromagnetism, andthat scalar and vector potentials have only an auxiliary function. This type of gauge invari-ance is upset as a basic principle when potentials are shown to be necessary to define anelectromagnetic environment. The propagation property constitutes a symmetry that hasnot been adequately considered, but it is fundamental in showing that there exist nominallyvalid gauge transformations that violate that symmetry. Gauge invariance is replaced bysymmetry preservation as a requirement for equivalence.
A. Equations of quantum mechanics
Acting upon the long-standing assumption that fields are basic and potentials are sec-ondary, many investigators attempted to express the Schr¨odinger equation directly in termsof fields [4–9]. All such attempts resulted in making the Schr¨odinger equation nonlocaland thus unsatisfactory. This conclusion applies to relativistic equations of motion (Klein-Gordon, Dirac, Proca) in addition to the nonrelativistic Schr¨odinger and Pauli equations.The implication, not realized at the time, is that potentials are more fundamental thanfields. 3 . Aharonov-Bohm effect
The Aharonov-Bohm effect [10, 11] is a direct demonstration that potentials are morefundamental than fields in that the deflection of an electron beam passing over a solenoidtakes place in a region that is free of fields, but has a potential that explains the deflection.This is a quantum effect, and it is discussed in textbooks as being exclusively a quantumanomaly that represents a departure from the notion that fields are primary and potentialsare secondary. Even that limited role has been questioned [12].The Aharonov-Bohm effect involves a magnetic field, so it has no direct significance forthe study of propagating fields.
C. Altered symmetries
Reference [1] demonstrates that a gauge transformation can alter the basic symmetriesthat characterize a problem in electromagnetism. Symmetries determine conservation prop-erties [2], so that a change in symmetries represents a change to a different problem inelectrodynamics. This finding is quite general, and it occurs in both classical and quantumphysics. Different gauges for a given field configuration need not be equivalent, and potentialfunctions are required to define the appropriate electromagnetic environment.A practical example presented in Ref. [1] has important implications. A propagatingelectromagnetic field, such as a laser field, must satisfy the Einstein Principle [13] that thespeed of light in vacuum is the same in all inertial frames of reference. The formal statementof this principle is that the spacetime 4-vector x µ can occur only as a scalar product withthe propagation 4-vector k µ . That scalar product: ϕ ≡ k µ x µ = ωt − k · r , (1)is the phase of a propagating field. This will be referred as the propagation condition . Whenthe field is a propagating field, the 4-vector potential must be expressible as A µ ( ϕ ).A gauge transformation in electrodynamics is expressed as A µ −→ e A µ = A µ + ∂ µ Λ , (2)where Λ, the generating function for the transformation, is a scalar function that satisfies4he homogeneous wave equation: ∂ µ ∂ µ Λ = 0 . (3)When those conditions are satisfied, the electric and magnetic fields are unchanged by thetransformation.A valid gauge transformation that produces an invalid 4-vector potential for a transversefield has the generating function [1, 14]Λ = − A µ x µ . (4)This is a scalar function that satisfies the condition (3) required of a generating functionand it is also stated covariantly. However, it is clear that this will introduce a violation ofthe propagation condition because x µ appears in isolation from k µ . The gauge-transformed4-vector potential is e A µ = − k µ ( x ν A ′ ν ) , where A ′ ν ≡ ddϕ A ν ( ϕ ) . (5)The propagation condition is violated, but the electric and magnetic fields are unchanged bythis gauge transformation. This violation of a basic property of a propagating field leads tothe general conclusion that potentials are more fundamental than fields, since preservationof the fields can nevertheless lead to an invalid representation of a propagating wave.An alternative form of the gauge-transformed potential in Eq. (5) is [1, 14] e A µ = − (cid:18) k µ ω/c (cid:19) r · E ( ϕ ) . (6)Although this gauge is unacceptable, it has found some favor [15] because of its resemblanceto the ubiquitous length-gauge potential − r · E ( t ). D. Gauge invariance versus symmetry preservation
The historically accepted conditions required to maintain electric and magnetic fieldsin a gauge transformation are insufficient to characterize an electromagnetic environment.For maintenance of the propagation property of a plane-wave field, a necessary additionalrequirement is preservation of the propagation condition of Eq. (1).When an electromagnetic environment has both a propagating field and a scalar field,then only the radiation gauge (also known as Coulomb gauge) can be compatible with an5rigin of coordinates, typifying a scalar potential such as a Coulomb field and, simultane-ously, the absence of an origin of coordinates necessary to describe a propagating field suchas a laser field [16]. In the radiation gauge, the time component of the 4-vector potential rep-resents the scalar field, and the 3-vector component represents the propagating field. Whenonly the propagating field is present, the Lorenz condition ∂ µ A µ = 0 reduces to ∇ · A = 0 .The 3-vector gradient condition is often stated to be the defining condition for the radiationgauge.The general expression for a gauge transformation is given by Eq. (2). The conditionthat A µ must be a function only of ϕ imposes the same constraint on Λ, leading to e A µ = A µ + k µ Λ ′ , (7)where Λ ′ = d Λ /dϕ . That is, the only alteration of the potential that is possible differs from A µ by a component that lies on the light cone. An important consequence is that e A µ e A µ = A µ A µ . (8)This follows from the transversality condition k µ A µ = 0 as well as the fact that a 4-vectoron the light cone is self-orthogonal: k µ k µ = 0 . Gauge invariance employed in classical and semiclassical electrodynamics as a generalprinciple cannot be correct since a gauge change will normally alter symmetry conditions,meaning that the physical problem is changed. For radiation fields in interaction withmatter, gauge invariance is replaced by symmetry preservation.There is some flexibility possible even when the propagation condition is enforced. Thatpossibility arises when the generator of the gauge transformation itself satisfies the propa-gation condition, as expressed in Eq. (7). A further differentiation gives ∂ µ e A µ = ∂ µ A µ + k µ k µ Λ ′′ = ∂ µ A µ , (9)where Λ ′′ = d Λ /d ϕ and k µ k µ = 0. The Lorenz condition ∂ µ A µ = 0 and the condition ofEq. (3) are automatically satisfied. E. Length gauge aberration
Of the several gauges in use in strong-field physics, there is one that stands out for theinsupportable claims made for it. 6he “length gauge” is a name used to refer to the − r · E ( t ) scalar function to representthe interaction of radiation fields with matter. There is an influential body of literaturedevoted to the claim that the length gauge is the only proper gauge to be used, and if adifferent gauge is to be employed, then it must always carry with it the gauge transformationfactor. This hypothesis is here termed an “aberration” since it is not possible for a scalarpotential to be fundamental for treatment of a vector field like a laser field.Two of the most ambitious efforts advocating the primacy of the length gauge are citedhere: [17, 18]. Both of these papers assume that the interaction Hamiltonian behaves uni-tarily in a gauge transformation. This is untrue [1]. It is plainly a contradiction since U ( r · E ) U − remains just r · E in any attempted gauge transformation because that inter-action Hamiltonian contains no operators. This means that, if a gauge transformation ismade into the length gauge, then it is impossible to do the inverse transformation back tothe initial gauge. This is untenable. III. QUANTUM ELECTRODYNAMICS IS INCOMPLETE
In 1952, Dyson showed [19] that the perturbation expansion of QED is not convergent.This continues to pose a dilemma since increasing accuracy in both experiments and com-putation have failed to show any discrepancies. Dyson conjectured that QED was somehowincomplete.That Dyson’s conjecture is correct was demonstrated long ago [20, 21]. Standard QEDhas the basic defect that it does not contain its strong-field limit. That is, QED existsas a perturbation expansion without knowledge of the complete theory to which it is anapproximation.The identification of the defect followed from an effort to find the convergence proper-ties of relativistic quantum mechanics (RQM) as obtained from a problem using the Volkovsolution [22, 23]. This is an exact solution of the Klein-Gordon equation found by Gordon[22] for a scalar charged particle in a plane-wave field, and an exact solution of the Diracequation found by Volkov [23] for a spin- particle in a plane-wave field. It has become con-ventional to refer to both solutions as the Volkov solution. In the context of the applicationto Breit-Wheeler pair production [24] for arbitrarily high intensities, an analysis showedthat RQM possesses a convergent perturbation expansion with the radius of convergence7imited by intensity-dependent singularities in the complex coupling-constant plane. Twovery important and unexpected features arose in the investigation: an altered mass shell,and an intensity-dependent coupling constant. A. Strong-field mass shell
The quantity referred to as the “mass shell” is the expression p µ p µ = ( mc ) , (10)where p µ is the 4-momentum vector of the particle of mass m . It was discovered indepen-dently by Sengupta [25] and by the present author [20, 21] that, in strong fields, the massshell is altered to p µ p µ = ( mc ) + 2 mU p , (11)where U p is the ponderomotive potential of a particle of charge q in a transverse field, definedas U p = q mc h| A µ A µ |i . (12)The ponderomotive potential is plainly Lorentz invariant and, from Eq. (8) it is also gauge-invariant for a propagating field. The absolute value | A µ A µ | is employed because A µ is aspacelike 4-vector and it is best to use a positive number as a basic measure. The anglebrackets in Eq. (12) refer to an average over a full cycle of the field. That is employedbecause, within any cycle of oscillation in a periodic field, it is known [26] that there is acontinuous exchange between kinetic and potential energies even though, in any completecycle, there can be no net energy transfer between a transverse field and a free chargedparticle.The difference between the mass shell expressions in Eqs. (11) and (12) is minor inexperiments with low-power laser beams, but it is a major factor with modern high-powerpulsed laser beams. B. Strong-field coupling constant
The coupling constant in standard QED between a plane-wave field and a particle ofcharge e is the fine-structure constant α = e / ℏ c . In the convergence investigation of Refs.820, 21] it was found that the coupling parameter of strong-field physics is given by thedimensionless intensity-dependent quantity z f = 2 U p /mc . (13)Expressed in terms of z f , the mass shell of Eq. (11) is p µ p µ = ( mc ) (1 + z f ) , (14)so that z f is a direct measure of the distinction between strong-field and standard electro-dynamics.There is another way to express z f that is very informative. If α is extracted as amultiplier, then z f can be written in the form [27, 28] z f = αρV = αρ (cid:0) λ C λ (cid:1) , (15)where ρ is the density of photons and V = 2 λ c λ is the volume that supplies photons toa strong-field process. This volume is essentially a cylinder of radius λ C and length λ ,where λ C is the Compton wavelength and λ is the wavelength of the propagating field. TheCompton wavelength is the usual measure of the interaction radius of a charged particle ina propagating field. The wavelength λ is a macroscopic quantity. That is, all the photons inthis cylinder contribute to the interaction even though λ might be many orders of magnitudelarger than the size of a target that is subjected to the strong field.It is the presence of a macroscopic quantity that can be said to characterize a strong field. C. Dressed electrons
When an electron (or any charged particle) is immersed in a strong propagating field, itpossesses a field-caused potential energy of U p . This energy comes from the propagating field,which is a relativistic phenomenon, so that the electron must also acquire the momentum U p /c of the “dressing” field. Photons from the background field have a 4-momentum on thelight cone. That is, the electron must acquire the 4-momentum U µ = U p (cid:18) k µ ω/c (cid:19) . (16)9he dressed electron can be regarded as a free particle with the 4-momentum p µ + U µ /c, and satisfy the usual mass shell condition of Eq. (10), which becomes (cid:18) p µ + 1 c U µ (cid:19) (cid:18) p µ + 1 c U µ (cid:19) = p µ p µ + 2 c p µ U µ , (17)since U µ U µ ∼ k µ k µ = 0. The Lorentz invariant quantity on right-hand side of this equationcan be evaluated in the rest frame of the electron, so that Eq. (17) becomes (cid:18) p µ + 1 c U µ (cid:19) (cid:18) p µ + 1 c U µ (cid:19) = ( mc ) + 2 mU p , (18)which is exactly Eq. (11). That is, Eq. (11) can be regarded as the mass shell of a freeelectron dressed by the propagating field. D. Summed Feynman diagrams
Fried and Eberly [29] showed that it was possible to sum the Feynman diagrams of QEDto all orders in a Compton scattering problem in which the spinor electron is replaced by ascalar particle. With one important revision, the result they found exactly duplicates whatis obtained by using a Volkov solution. The only difference is that z f does not appear, andthe mass shell of Eq. (10) is obtained.The Fried and Eberly calculation verifies that QED is incomplete.A subsequent investigation by Eberly and Reiss [30] examined a class of diagrams, each ofwhich is divergent, that was omitted from the Fried and Eberly calculation on the groundsthat the divergent diagrams are unphysical. The Eberly and Reiss paper showed that thesediagrams can be summed exactly, and the sum is finite. The importance of this demonstra-tion is that the summed divergent diagrams introduce exactly the strong-field mass-shellexpression of Eq. (11) or (14).The Eberly and Reiss calculation shows the manner in which QED fails to be complete. E. Radius of convergence of perturbation theory and the failure of Feynman dia-grams
The convergence investigation of Refs. [20] and [21] shows that an extreme upper limitfor perturbation theory is marked by the first channel closing that can occur. A feature of10trong-field processes is that the field must supply the basic energy required for a transitionas well as the potential energy U p of any created charged particle immersed in a strong field.For example, in the strong-field Breit-Wheeler pair production process, multiple photonsare needed to supply the rest energy 2 mc of the pair produced, but also the ponderomotiveenergy of the two charged particles created. As the field intensity increases, the lowest-orderprocess that can occur must index upward to supply the required ponderomotive energyof the electron pair. That indexing is referred to as a “channel closing”, and perturbationtheory will fail at or before the intensity for the first channel closing.Reference [30] shows that an infinite number of Feynman diagrams must be summedin order to explain the strong-field mass shell. The strong-field mass shell is therefore anonperturbative effect. When the intensity is high enough for perturbation theory to fail,then Feynman diagrams lose all meaning since they provide a pictorial representation onlyof perturbative processes.Figures 7 and 8 of Ref. [32] show graphically the change from perturbative to nonper-turbative behavior. At low field intensity the lowest allowed order of interaction is the solecontributor to a transition. At high field intensity the superposition of many different photonorders is necessary to describe a quantum transition. F. Green’s function of the Volkov solution
The properties of the Green’s function of the Volkov solution provide a clear pictureof the differences between a standard propagator in QED and that for strong fields. It isconventional to examine the behavior of the Green’s function in a complex p space (i.e.complex energy space). This is instructive because the path employed for an integrationin this space establishes whether the Green’s function represents an advanced solution,a retarded solution, or a Feynman solution in which positive energy solutions propagateforward in time and negative energy solutions propagate backward.In QED, the mass shell of Eq.(10) applies, and the only poles in the complex energyspace occur at p = ± p p + mc . In the Volkov Green’s function, the mass shell of Eq.(11) applies, and families of sideband singularities appear in addition to the two QED poles.Each QED pole has its own set of sideband states, but they do not overlap. See Ref. [31]for details. 11hese Green’s function properties make explicit the differences found between the QEDcalculation of Ref. [29] and the strong field calculation of Ref. [30]. QED has but twosingularities on the complex p space, whereas the strong-field case has infinite families ofsidebands. This phenomenon illustrates the failure of Feynman diagrams, which cannotrepresent an infinity of singularities. IV. STRONG-FIELD INFERENCES
Two important strong-field matters will be mentioned here, in addition to the strong-fieldfeatures discussed in preceding Sections.
A. Fixed origin for energy measure
The ponderomotive energy U p is the fundamental measure applicable to a charged particlein a strong field. This is reflected in the essential properties of z f as the coupling constant forcharged particle interactions with strong fields, as well as its role in the strong-field alteredmass shell. When interactions of the field with bound-state particles are considered, anotherdimensionless intensity parameter arises, which is the ratio of the ponderomotive potentialto the binding energy E B [27, 32] z = 2 U p /E B . (19)A third dimensionless parameter is z = U p / ℏ ω, (20)which is of universal applicability in strong-field problems since it is a measure of the mini-mum number of photons that enter into a strong-field-induced interaction. As U p increases,this minimum indexes upward, illustrating channel closing. As mentioned above, the firstsuch channel closing identifies an upper limit for the convergence of perturbation theory[20, 21, 27, 32].A novel feature that follows from the basic importance of U p is that it fixes an absoluteorigin for energy measures. As Eq. (12) shows, the zero of energy is established by the zeroof the 4-vector A µ . In the dipole approximation as employed in atomic physics, the zero ofenergy can be arbitrary as long as it is applied universally.12his fixed origin of energy measure is an important feature distinguishing the two varietiesof the Strong-Field Approximation (SFA). B. Ambiguity in the SFA
The SFA is regarded as the standard analytical approximation for the interaction ofstrong laser fields with matter. There is an existential problem with this appraisal in thatthe SFA exists in two incompatible forms.
1. SEFA
The dipole approximation as used in atomic physics neglects entirely the magnetic com-ponent of laser fields. When the dipole approximation is imposed from the outset, the SFAis a theory of oscillatory electric fields. It is not a theory of propagating fields like those oflasers. Despite the similarities in some ranges of parameters, the differences are fundamental,and of major importance in other ranges of parameters.The first strong-field analytical approximation employed for laser-induced processesis that of Keldysh [33], who employs the dipole approximation. This and subsequentdipole-approximation methods will be termed the Strong Electric-Field approximation(SEFA).The dipole approximation is also employed in numerical solution of the time-dependentSchr¨odinger equation (TDSE), so that it is also of the SEFA character. It is not an exactcalculation of laser-induced transitions, as is often claimed.
2. SPFA
A different analytical approximation follows from taking the nonrelativistic limit of atheory based on propagating fields, which will be referred to as the Strong Propagating-Field Approximation (SPFA). The genesis of the SFA of Ref. [32] from relativistic origins isdemonstrated in Refs. [34, 35]. When a laser field is very strong, it is the dominant influencein interactions of the field with matter. Laser fields propagate at the speed of light, so that13 relativistic treatment is necessary. When such a theory is reduced to its nonrelativisticlong-wavelength form [32], its provenance from a relativistic formalism remains importanteven though the general appearance of the SPFA resembles that of the SEFA.
3. SEFA/SPFA differences
When field frequencies are relatively high, SEFA and SPFA theories coalesce. This isshown in detail in Ref. [36], for example. The authors do numerical integration of the time-dependent Schr¨odinger equation, employing the dipole approximation, so it corresponds toa SEFA. The method they cite as SFA is the SPFA.When field frequencies are low, then SEFA and SPFA predictions become profoundly dif-ferent [37, 38]. The SEFA trends toward what has been labeled as the “asymptotic limit”,where the field becomes a constant electric field. By contrast, as the field frequency de-creases, the SPFA increasingly manifests the effects of the magnetic component of a laserfield, trending towards relativistic behavior. The location of the transition from high andlow frequency domains has yet to be established, but it corresponds approximately to wave-lengths in the few- µm range.A very important matter is that the SPFA of Refs. [32, 34, 35] is the only strong-fieldapproximation method that can be categorized as SPFA. Everything else, including TDSE,is SEFA. V. VARIETIES OF ELECTRODYNAMICS
Electrodynamics can be viewed from the standpoint of quantum field theory (QFT),or of relativistic quantum mechanics, or as a purely classical phenomenon. Each of theseviewpoints overlaps the adjacent one, generating a unified view of electrodynamics.
A. Electrodynamics as a quantum field theory
QFT has become a highly developed formalism based on symmetry principles, leading tothe modern “Standard Model”. Electrodynamics has a place in this scheme as the gaugeparticle of the electromagnetic field. For purposes of this article, the discipline that earned a14obel prize for Feynman, Schwinger, and Tomonaga is sufficient. The salient point in QFTis the existence of a number operator whose eigenvalues count the number of photons thatparticipate in an interaction.The practical application of quantum electrodynamics to laboratory phenomena is ac-complished according to the graphic means of the Feynman diagrams that follow from per-turbation theory.
B. Electrodynamics in relativistic quantum mechanics
In RQM, the field is not quantized. That is, there is no number operator. Nevertheless,RQM employs what is called the Floquet property, in which the periodicity of an electro-magnetic plane wave leads to transfer of energy in integer packets of ℏ ω . Using standardS-matrix methods in a relativistic formulation, a set of computational rules can be evolvedthat are identical to the Feynman rules of QFT. A clear representation of the equality ofthe Feynman rules in QED and in RQM is given by the two textbooks of Bjorken and Drell[39, 40]. The first volume uses RQM methods to produce the Feynman rules, followed bythe QFT volume that produces exactly the same rules.The novel feature of RQM is the existence of the Volkov solution, which makes possiblethe construction of a nonperturbative domain of electrodynamics. The essential distinctionsbetween the QFT of electrodynamics and electrodynamics within RQM is elucidated byRefs. [29] and [30]. C. Classical electrodynamics
Classical electrodynamics does not employ a quantized version of the electromagneticfield, but it is nevertheless possible to define the photon density of a monochromatic field byusing the classical energy density of a plane-wave field divided by ℏ ω . This is the methodused in Eq. (15) to evaluate photon density.The application of classical electrodynamics to such practical matters as antenna theoryor the properties of transmission lines seems to have no correspondences with RQM orQFT, but there is nevertheless an important connection with nonperturbative RQM. Thisconnection arises through the wavelength dependence of the coupling constant of RQM in15he form given by Eq. (15). The salient question arises from the possibility that λ can besuch a large quantity that its connection to the microscopic world of quantum mechanicsbecomes difficult to understand.Some insight into this question comes from a situation in which classical phenomena atextreme wavelengths is also difficult to understand.There is a practical application of extremely long wavelengths to the problem of communi-cation with deeply submerged submarines. Several countries have devised systems operatingat very long wavelengths to take advantage of the fact that the skin depth of a conductingmedium such as seawater varies a λ / . The system employed by the U.S. Navy [41] operatedat a frequency of 76 Hz , corresponding to a wavelength of 4 × m , which is almost equalto the Earth’s radius of 6 . × m . The receiving antenna can be regarded as the length ofthe submarine, of the order of 10 m , or one part in 40 ,
000 of the wavelength of the radiosignal. On the scale of the submarine, the electric and magnetic fields of the radio wave areconstant. Nevertheless, the radio wave carries a coded signal that is intelligible.This is related to the problem of constant crossed fields. The two relativistic invariantsof a propagating electromagnetic field have zero value: E − B = 0 , E · B = 0 . (21)It is also possible to generate constant E and B fields that satisfy the conditions (21). If theelectric and magnetic fields are the governing quantities that identify fields, then constantcrossed fields and propagating fields of very long wavelengths should be equivalent. Theyare not. Transverse fields propagate in vacuum at the speed of light. Constant crossedfields “propagate” at zero speed. When identified by potentials, constant crossed fields andpropagating fields are unrelated. This is direct proof of the primacy of potentials over fields.The problem of how an atom can respond to a propagating field many orders of magnitudegreater than the size of the atom is analogous to the problem of how a submarine can deciphera radio signal with a wavelength 40 ,
000 times the length of the submarine. In each case thetarget of the plane wave can respond to the information carried by the potential functionsof the propagating field. Properties of the E and B fields are secondary.The critical strong-field parameter z f varies as the square of the wavelength. It is possibleto achieve z f = O (1) with commercial radio-frequency equipment. That is, familiar classicalenvironments can exhibit certain strong-field effects of powerful lasers.16 . Summary of the varieties of electrodynamics Descriptions of the effects of transverse fields are equivalent within QFT and RQM withinthe domain of the validity of perturbation theory of QFT. RQM makes available an analyticalcontinuation of the effects of the transverse field into a domain where QFT fails. Classicalelectromagnetism as applied to macroscopic problems has no relevance to the microscopicworld of quantum systems, but nonperturbative RQM shares some of the important behaviorof macroscopic transverse fields at very long wavelengths.
VI. PRECISION OF QED
Appraisals of the precision of QED are based on the accuracy to which the value of α can be determined [42]. That is, the premise is accepted that α measures the couplingof transverse fields to charged particles. One intent of the present article is to show thatFeynman-Dyson perturbation theory applies only to a subset of electrodynamics. Strongfields are neglected, and the coupling parameter of strong fields is z f , not α .The proposition is now made that the failure of QED to be convergent is governed bythe inability of QED to explain strong field phenomena. QED is not a subset of strong-fieldphysics, but rather it is an approximation to strong-field physics. The importance of strongfields is measured by z f = 2 U p /mc , and perturbation theory in the context of strong fieldshas an absolute limit z <
1, in terms of the intensity parameter z of Eq. (20). That is QEDis subject to the limit z f = 2 U p mc , z = U p ℏ ω < , so that z f < ℏ ωmc . (22)For a typical laser photon energy of 1 . eV , the limit given in Eq. (22) is z f < × − . Thisis an extreme upper limit, and nonperturbative behavior is known to exist at much smaller z f values. The best-known manifestation of nonperturbative behavior is the above-thresholdionization (ATI) effect, first observed by Agostini, et al. [43]. The first successful match of anonperturbative theory to experiment, reported in Ref. [44], was for a case where the peak17 f was z f = 8 × − , and this was clearly well into the nonperturbative domain. [1] H. R. Reiss, Fundamental formulation of light-matter interactions revisited, Phys. Rev. A , 052105 (2019).[2] E. Noether, Invariante Variationsprobleme, Nachr. d. K¨onig. Gesellsch. D. Wiss. zu G¨ottingen,Math-phys. Klasse , 235-257 (1918).[3] S. Weinberg, The Quantum Theory of Fields, Volume 1: Foundations (Cambridge UniversityPress, Cambridge, 1995).[4] S. Mandelstam, Quantum electrodynamics without potentials, Ann. Phys. (N.Y.) , 1 (1962).[5] B. S. DeWitt, Quantum theory without electromagnetic potentials, Phys. Rev. , 2189(1962).[6] F. J. Belinfante, Consequences of the postulate of a complete commuting set of observablesin quantum electrodynamics, Phys. Rev. , 2832 (1962).[7] M. L´evy, Non-local quantum electrodynamics, Nucl. Phys. , 152 (1964).[8] F. Rohrlich and F. Strocchi, Gauge independence and path independence, Phys. Rev. ,B476 (1965).[9] M. Priou, Quasi local electrodynamics, Nucl. Phys. , 641 (1966).[10] W. Ehrenberg and R. E. Siday, The refractive index in electron optics and the principles ofdynamics , Proc. R. Soc. B , 8 (1949).[11] Y. Aharonov and D. Bohm, Significance of potentials in the quantum theory, Phys. Rev. ,485 (1959).[12] J. D. Jackson and L. B. Okun, Historical roots of gauge invariance, Rev. Mod. Phys. , 663(2001).[13] A. Einstein, Zur Elektrodynamik bewegter K¨orper, Ann. Phys. , 891 (1905).[14] H. R. Reiss, Field intensity and relativistic considerations in the choice of gauge in electrody-namics, Phys. Rev. A , 1140 (1979).[15] M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, Gauge-invariant relativistic strong-fieldapproximation, Phys. Rev. A , 053411 (2006).[16] H. R. Reiss, Physical restrictions on the choice of electromagnetic gauge and their practicalconsequences, J. Phys. B , 075003 (2017).
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