The Abraham-Lorentz force and electrodynamics at the classical electron radius
TThe Abraham-Lorentz force and electrodynamics at the classicalelectron radius
Janos Polonyi
Strasbourg University, High Energy Theory Group, CNRS-IPHC,23 rue du Loess, BP28 67037 Strasbourg Cedex 2 France (Dated: March 29, 2018)The Abraham-Lorentz force is a finite remnant of the UV singular structure ofthe self interaction of a point charge with its own field. The satisfactory descriptionof such interaction needs a relativistic regulator. This turns out to be a problem-atic point because the energy of regulated relativistic cutoff theories is unboundedfrom below. However one can construct point splitting regulators which keep theAbraham-Lorentz force stable. The classical language can be reconciled with QEDby pointing out that the effective quantum theory for the electric charge supports asaddle point producing the classical radiation reaction forces.
I. INTRODUCTION
The radiation reaction problem, the intrinsic instability of the interaction of a pointcharge with its own field, has been clearly stated since more than a century, however thediscovery of quantum mechanics somehow deflected the interest of the majority of the physicscommunity. Nevertheless number of methods have been developed to tackle the problemand several solution have been proposed in the meantime. The present work is based on thepoint of view that a field theory of point particles displays singular short distance dynamicsand needs a cutoff, a minimal distance down to which the theory is applicable, both in theclassical and the quantum domain. The UV divergences of quantum field theories are awell known problem but one should not loose sight of the singular classical Coulomb fieldof point charges, one of the tumbling stones of the radiation reaction problem. The mainpoint of this work is to show that electrodynamics supports stable self interaction for pointcharges when supplied an appropriate regulator.Let us approach the radiation reaction problem, the last open chapter of classical elec- a r X i v : . [ h e p - t h ] M a r trodynamics, in three steps. First concerns the origin of the phenomenon. It is known thatan accelerated charge looses part of its energy by radiation hence there is a radiation reac-tion force. However the Lagrangian do not contain higher than first order time derivativeshence can not give account the energy, lost to the radiation. The solution of the problem inthe jargon of field theory of today is that the radiation reaction is an effective force and itappears only in the closed equation of motion for the charge. This equation can be obtainedby solving the Maxwell equation for a given particle world line and inserting the solutionback into the mechanical equation of motion. This is the construction of the effective theoryfor the charge and the effective dynamics, generated by the feedback of the electromagneticfield contains the radiation reaction force. The calculation of the force of a point charge ishindered by the divergence of the near field and the Abraham-Lorentz force was found bychecking the energy-momentum balance equation for the radiating charge [1, 2].The second step is about the attempts of reconciliation of an unacceptable feature ofthe Abraham-Lorentz force, namely it is an O (... x ) dissipative force which generates self-accelerating, runaway trajectories. Such a state of affairs questions the applicability of fieldtheory to point particles and a wide range of remedies have been proposed. (i) The problemis related to the third auxiliary condition, needed to integrate the equation of motion. Byimposing an additional final condition one trades the instability into acausality [3], an ideawhich can be realized by assuming a complete absorption of the electromagnetic radiationat the final state [4]. (ii) One can expand the solution in the retardation and the Abraham-Lorentz force can be approximated by making an iterative step, using the second orderequations. The result is an O (¨ x ) equation which remains stable as long as the resolutionis worse than r [5, 6]. The same equation can be recovered by restricting the trajectoriesof the Abraham-Lorentz force to the stable manifold [7]. (iii) A more flexible family ofmodifications results from giving up the local nature of the reaction force in time. One wayto achieve this is to retain a memory term in the reaction force and the resulting integro-differential equation offers important improvements [8]. Another possibility is to assumeand extended charge distribution [9] of non-electromagnetic origin [10, 11] or the presenceof a polarizable medium [12]. Yet another approach is to assume a suitable chosen formfactor [13]. A particular non-locality has been evoked by assuming a discrete structure intime [14]. (iv) Finally, one can step back and seek a change of the effective equation ofmotion, based on physical intuition [15], on magnetic moment charge [16], or on quantumeffects [17]. So far no generally accepted solution has been found to cure the instability.The point of view, developed below, is based on a non-local dynamics hence is closest tothe group (iii) with the difference of being minimalistic, namely we stay within QED in animaginary world without other particles and interactions. The non-locality arises during theunavoidable, conventional regularization of the theory and is treated in the usual manner,known from the renormalization of quantum field theories.The third step is to embed the radiation reaction problem into QED by considering it assaddle point physics. The Abraham-Lorentz force can be identified in the (cid:126) → O ( (cid:126) ), tree-level graphs. But this holds for elementary, closed theories only. If there are unobserveddegrees of freedom, an environment, and we are looking for a closed system of equations forthe observed quantities, then the elimination of the unobserved coordinates generates loopdiagrams. The iterative solution of the classical equation of motion of a charge, movingon the background of a fixed electromagnetic field, can be represented by an infinite seriesof tree-level graphs where the electromagnetic field is attached to the world line as anexternal leg. However the elimination of the electromagnetic field by the help of the Maxwellequations, using the world line as a source, couples the pairs of the external legs and formsloops. The loop integral is O ( (cid:126) ), a power of (cid:126) is lost compared to the usual counting inQED because the line, representing the charge is classical, O ( (cid:126) ). The loop integrals ofclassical effective theories have already been spotted as the on-shell contributions to theloop integrals of the full quantum theory [21]. The self-interaction of a point charge withits own field is UV singular due to the O ( r − ) near field, requiring the usual regularizationand renormalization procedure of quantum filed theories, applied already in the classicaldomain.The traditional strategy to derive the effective equation of motion without touchingthe issue of divergences is to exploit the energy-momentum conservation for the energy-momentum flux, calculated for a tube around the point particle world line [3]. Howeverenergy-momentum is modified by the regulator, making the conserved energy unboundedfrom below in relativistic theories, thereby removing a sufficient condition of stability [22].The regulator, used in this work is the point splitting of the interaction, consisting of theuse of a smeared electromagnetic field in the interaction term. The smearing is chosen insuch a manner that superluminal charges do not interact. The resulting effective equationof motion provides stable dynamics since a runaway charge must acquire velocities beyondthe speed of light. The appearance of the cutoff in the classical effective equation of motionmakes certain concepts of the renormalization group method important for classical fieldtheories.First we consider the linearized effective equation of motion [23] in section II A whichdescribes an stable dynamics for low cutoff. If the resolution in the space-time is betterthan the classical electron radius then the usual O (... x ) Abraham-Lorentz force appears andinduces self accelerating, unstable particle trajectories. Next we turn to the full, non-linear,integro-differential equation of motion in section II B. Its numerical solution displays twodistinct scale regimes, separated by the classical electron radius. The usual linear Abraham-Lorentz force is displayed in the IR domain and the non-linear effects prevent the run awayand keep the motion stable in the UV regime. The issue of embedding this scenario in QEDis taken up briefly in section III where it is argued that the radiation reaction problem is atree-level saddle point effect governed by an O ( (cid:126) ) effective equation of motion and hidingdeeply within the quantum regime of the theory. II. CLASSICAL EFFECTIVE DYNAMICS OF A POINT CHARGE
The effective dynamics of a point charge is found by solving Maxwell’s equation forthe electromagnetic field, A µ ( x ), induced by a point charge which follows a given worldline, x µ ( s ), and substituting the result into the equation of motion of the charge. Theelectromagnetic field, A µ ( x ) = e (cid:90) dsD rMµν ( x − x ( s )) ˙ x ν ( s ) , (1)is given by the retarded Green’s function, D rMµν ( x ) = − π ( g µν − ∂ µ ∂ ν / (cid:3) ) D r ( x ), where D r ( x ) = − Θ( x ) δ ( x ) / π is the free, massless Green’s function, and ˙ x ( s ) = dx ( s ) /ds . Themechanical equation of motion, m B c ¨ x µ = ec [ ∂ µ A ν ( x ) − ∂ ν A µ ( x )] ˙ x ν − k µ , (2)contains the bare mass, m B , and an external source, k µ ( s ), to diagnose the dynamics. Weavoid the UV singularities of the near field self-interaction by introducing a cutoff, realizedby the smearing of the Dirac-delta, δ ( x ) → δ B ( x ), in the retarded Green’s function.The regulated Dirac-delta should satisfy three conditions: (i) To preserve the flux of theradiated field we require the normalization condition (cid:90) ∞−∞ dzδ B ( z ) = 1 . (3)(ii) The Lorentz symmetry prevents us to smear the factor Θ( x ) of the retarded Green’sfunction, we impose δ B (0) = 0 to separate the singular points in the product of two gener-alized functions in the Green’s function. (iii) It will be argued below that the stability ofthe dynamics requires the suppression of the interactions for charges moving faster than thelight, expressed by the condition δ B ( z ) = 0 for z <
0. The simplest possibility is to displaceits retarded Green’s function slightly off the light-cone, δ B ( x ) = δ ( x − (cid:96) ) , (4)It leads to oscillations in the momentum space which can be avoided by smearing the sin-gularity, δ B ( x ) = Θ( x )12 (cid:96) x e − √ x (cid:96) . (5)The effective equation of motion with the regulated self interaction is¨ x = 4 r B (cid:90) s −∞ ds (cid:48) δ (cid:48) B (( x − x (cid:48) ) ) { ( x − x (cid:48) )( ˙ x ˙ x (cid:48) ) − [ ˙ x ( x − x (cid:48) )] ˙ x (cid:48) ] , (6)where x = x ( s ) and x (cid:48) = x ( s (cid:48) ), and r B = e /m B c stands for the bare classical electronradius and δ (cid:48) ( z ) = dδ ( z ) /dz . The characteristic scale of the radiation reaction problem isprovided by the only dimensional constant of this equation, the classical electron radius,playing the role of the coupling constant. A. Linearized equation of motion
The linearized the equation of motion,¨ x = 4 r B (cid:90) −∞ duδ (cid:48) ( u )( x − x (cid:48) + u ˙ x (cid:48) ) , (7)contains a regular and a singular force on the right hand side, F r = 2 r B (cid:90) −∞ duu δ ( u ) (cid:20) x − x (cid:48) + u ˙ x (cid:48) − u (cid:18) ¨ x (cid:48) −
12 ¨ x (cid:19) + 2 u x (cid:21) ,F s = − r B (cid:90) −∞ duδ ( u ) (cid:18) ¨ x + 4 u x (cid:19) (8)where the expression in the square bracket of the integrand of F r contains the regular, O ( u )terms making up a uniformly convergent integral. The rest, F s , can be written as F s = − r B x (cid:90) ∞ dz √ z δ B ( z ) + 23 r B ... x , (9)giving rise to a mass renormalization, m = m B + δm , with δm = e c (cid:90) ∞ dz √ z δ B ( z ) . (10)and to the Abraham-Lorentz force.It is illuminating to check the order of magnitude of the two forces as the cutoff is removed.We assume that Λ = 1 /(cid:96) is large enough to approximate the square bracket in the integrandof the uniformly convergent part by u /(cid:96) x where (cid:96) x is the length scale of the world line andfind F r ≈ r B (cid:96)(cid:96) x (cid:90) d ˜ u ˜ δ B (˜ u )˜ u , (11)in terms of the dimensionless variable ˜ u = u/(cid:96) and the regular function ˜ δ B (˜ u ) = (cid:96) δ B ( u ).The regular force is suppressed during the renormalization owing to the smallness of theimportant integration region, u = O ( (cid:96) ) of an δ ( u ) u = O ( (cid:96) ) integrand. The importantintegration region is u = O ( (cid:96) ) in the non-uniformly convergent part, too. The linearlydivergent mass renormalization comes from an O ( (cid:96) − ) integrand and the reduced O ( (cid:96) − )divergence of the integrand in the Abraham-Lorentz force yields a cutoff independent result.Such a cutoff-independent cutoff-scale contribution owes its existence to the non-uniformconvergence and is the hallmark of anomalies. In fact, the coefficient of ... x is an “accidentallyfinite” loop-integral as in the case of the chiral anomaly for massless fermions.The retarded world line Green’s function, F r , is defined by˙ x µ ( s ) = (cid:90) ∞−∞ ds (cid:48) F r ( s − s (cid:48) ) k µ ( s (cid:48) ) , (12)and its Fourier transform, F rω = (cid:90) ∞−∞ dse iωs F r ( s ) , (13)can be written as F rω = 1 / [( ω + i(cid:15) ) χ rω ], in terms of the susceptibility χ rω = 1 + r (cid:20) i ω (14) − ω (cid:90) −∞ duδ B ( u ) (1 + iωu − u ω ) e − iuω − u ω − i u ω u (cid:21) , and the long memory tail when the regulator (5) is used requires Im ω >
0. The harmoniceffective dynamics is stable and causal if the susceptibility is analytic and pole-free on theupper part of the complex ω plane.The rational function, multiplying the Dirac-delta in the integrand is ω u (1 + O ( ωu ))hence χ rω = 1 + r ω (cid:20) i + O ( ω(cid:96) ) (cid:21) . (15)As the cutoff is removed with a fixed frequency the Abraham-Lorentz force is left behindand the infamous self-acceleration is recovered. The unstable dynamics generates high fre-quency components which make the particular details of the regulator influence the motion.The numerical monitoring of the (cid:96) -dependence shows that both regulators lead to a suscep-tibility which develops a pole with positive imaginary part, destroying the stability whenthe cutoff is comparable or larger than the classical electron radius, as predicted by thecutoff-independent acausal pole of the susceptibility (15).One would think that the appearance of the instability at (cid:96) ∼ r implies that the sourceof the instability is related to the dynamics around the scale r . But the dynamics isshaped both by the O ( ω ) kinetic energy and the self interaction. The O ( ω ) friction termis forbidden by Lorentz symmetry, the impossibility of measuring absolute velocity, thus theradiation energy loss is represented by the O ( ω ) dissipative force. This latter which is weakfor slow motions turns out however to be dominant at high frequencies, the crossover beingaround ω ∼ /(cid:96) and makes the linearized dynamics unstable in the UV regime. B. Full effective dynamics
The regulator was introduced in section II in such a manner that the interaction becomessuppressed if the velocity of the particle exceed the speed of light, making the instability,arising from the self interaction, unable to drive the particle to arbitrarily high velocity.Therefore it is natural to inquire whether the non-linear terms of the effective equation ofmotion (6) can stabilize the dynamics. The traditional derivation of the effective equationof motion, based on the energy-momentum conservation, leads to an equation where theonly non-linearity arises from the projection of the reaction force onto the linear subspace,orthogonal to the four-velocity [24]. The cutoff-dependence of the instability suggests thatif the non-linearities stabilize the dynamics they should come from another source than thiscutoff-independent projection operator.The shifted Dirac-delta, (4), produces the equation of motion with a finite delay,¨ x = r mm B
1[ ˙ x (cid:48) ( x − x (cid:48) )] (cid:20) ¨ x (cid:48) ( x (cid:48) − x ) + 1˙ x (cid:48) ( x − x (cid:48) ) { ( x − x (cid:48) )( ˙ x ˙ x (cid:48) ) − [ ˙ x ( x − x (cid:48) )] ˙ x (cid:48) } +( x − x (cid:48) )( ˙ x ¨ x (cid:48) ) + [ ˙ x ( x (cid:48) − x )]¨ x (cid:48) (cid:21) , (16)where the retarded source point, x (cid:48) , is found by the condition (cid:96) = ( x − x (cid:48) ) . The equationof motion, found by the help of the smeared Dirac-delta, (5), has infinitely long memoryand contains the velocities only in the right hand side,¨ x = r (cid:96) mm B (cid:90) −∞ du (cid:32) − (cid:112) ( x − x (cid:48) ) (cid:96) (cid:33) e − √ ( x − x (cid:48) )2 (cid:96) ×{ ( x − x (cid:48) )( ˙ x ˙ x (cid:48) ) + [ ˙ x ( x (cid:48) − x )] ˙ x (cid:48) ] } . (17)We impose the initial condition that the charge is at rest, x ( t ) = ( t, ), for t < t and thecharge follows a prescribed trajectory, x i ( t ) for t < t < t + t i . A certain external source, k i ( s ), is supposed to generate this motion which is turned off after this initial phase andthe invariant length, s , of the world line is measured from the time t + t i . The numericalsolution of the equation of motion becomes straightforward with such initial conditions: Oneintroduces a small but finite ∆ t step size and writes eqs. (16) or (17) as differential equationand finds the retarded time, u , or calculates the integral of the memory term numerically,respectively at each step, t → t + ∆ t .The equation of motion has two free, adjustable parameters, the cutoff, (cid:96) , and the baremass, m B . Hence we need a renormalization condition to fix the theory, it is chosen to be χ rω = 1 + 23 ir ω, (18)cf. eq. (15).The numerical solution of the equation of motion indicates stable dynamics for sufficientlyweak force, ie. small | m/m B | . The acceleration changes in a monotonous, exponential
50 100 150 200 250 300s (cid:144) r r (cid:144) r (cid:45) (cid:45) (cid:45) r (a) (b)FIG. 1: A component of the spatial acceleration, ar = | ¨ x | r , plotted against the proper time, s/r for the smeared Dirac-delta regularization, r /(cid:96) = 3, (a): m/m B = 1 .
95 (dashed line), m/m B = 1 .
98 (solid line) and m/m B = 2 (dotted line) and (b): m/m B = − . m/m B = − .
91 (solid line) and m/m B = − . manner after some transient period, depending on the initial conditions, if m B > s . When m B < m/m B at the stabilityedge is found to be slightly dependent on the initial, prescribed trajectory. This might comefrom the finite ∆ s resolution of the finite difference equation, solved numerically because theunstable, runaway trajectories support no fixed, finite ∆ s . The existence of stable regionssuggests that despite the unboundedness of the energy in regulated electrodynamics thereare energy barriers which stabilize the charge.The phase structure of the effective theory is shown in Fig. 2. The stability region nar-rows as the cutoff is removed, (cid:96) →
0, since the regulator subjects the trajectory to somedeformation within the length scale ∆ s ∼ (cid:96) , inducing a larger value of the loop integralin the effective equation of motion, (6), and requiring smaller coefficient, r B . The non-monotonous behavior of the renormalized trajectory indicates the presence of an IR andan UV scaling regime, separated by the intrinsic length scale, r . There are two solutionsof the renormalization condition, one with m B > m B <
0. The latteris qualitatively consistent with the linearized equation of motion and displays “Zitterbewe-gung”, fast oscillations. The non-linear terms of the equation of motion play an important0 r (cid:144) l (cid:45) (cid:45) m (cid:144) m B r (cid:144) l (cid:45) (cid:45) m (cid:144) m B (a) (b)FIG. 2: The phase structure of the effective theory with (a): shifted and (b): smeared regulatedDirac-delta on the plane ( r /(cid:96), m/m B ). The dynamics is stable within the shaded region and thesolid lines indicates the solution of the renormalization condition, (18). The dotted line belongs tothe linearized theory, fixed by the counterterm (10). role at any value of the cutoff since the linearized theory is unstable despite having bareparameters within the stability region of the full equation. There is no numerical evidenceof a Landau-pole, an obstruction of the limit (cid:96) → x generates strongly localized minima in | ¨ x ( s ) | in a periodic mannerthe envelope follows the prediction of the Abraham-Lorentz force with a remarkable precisiondespite the non-linearity of the equation of motion. The zoom into Fig. 3 (a), shown inFig. 3 (b), supports the expectation that the length of an oscillation scales with the cutoff.Similar behavior can be found for m B > III. QED
The physics around the classical electron radius is deeply within the quantum domainand we turn to the question of placing the classical considerations, presented above, into thecontext of QED.1
10 20 30 40 sr (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) a sr (cid:45) a (a) (b)FIG. 3: (a): The spatial acceleration, | ¨ x | r , plotted against the proper time, s/r for the smearedDirac-delta regularization at r /(cid:96) = 3 (fat line) and r /(cid:96) = 15 (thin line) with m B <
0, plottedtogether with the prediction of the renormalization condition (dotted line). (b): The zoom is intoa more restricted scale region.
A. Scale hierarchy and subclassical physics
Weakly coupled quantum field theories have an intrinsic hierarchy of scales, assured by thesmallness of the dimensionless strength of interaction. In the case of QED with electronsfour of the scales r n = α n r , α = e / (cid:126) c , have already been identified, the Bohr radius, r − = a = (cid:126) /me , the Compton wavelength, r − = λ C = αa = (cid:126) /mc , the classicalelectron radius, r = α a = e /mc and finally the Lamb shift scale, r = α a = e / (cid:126) mc .The scale dependence of the dynamics is driven by different elementary processes at differentscale regimes. This can easily be seen at the first three scales by identifying the fundamentalconstant which is missing from the expression of the scales. In fact, the driving force comesfrom the non-relativistic quantum mechanics at the Bohr radius (absence of c ), from paircreation at the Compton wavelength, independently of the specific nature of the underlyinginteractions (absence of e ) and from classical electrodynamics at the classical electron radius(absence of (cid:126) ). The physics of the Lamb shift at r is driven by involved vacuum polarizationeffects. The scales with n ≥ n ≤ − λ , appearing in QED when the rescaling, (cid:126) → λ (cid:126) which shiftsthe quantum-classical transition scales, is performed. This induces the change α → α/λ ,showing that the saddle point and the usual weak coupling expansion represent two oppositeextrema. In fact, we have r n → λ − n r n , the gradual turning on of the quantum fluctuationsby moving λ from 0 to 1 reshuffles the scale hierarchy: The imaginary world with weakquantum fluctuations and large fine structure constant, a < λ C < r < r n , n > r n < r < λ C < a , are separated by a strongly coupled regime without scaleseparation, r n ∼ r ∼ λ C ∼ a at λ ∼ / O ( (cid:126) ) saddle point contribution around r , embedded deeplywithin the quantum domain.But the classical limit of a quantum system is more than the recovery of some classicalequations of motion. It is instructive in this respect to consider an extension of this problem,the expectation value of local operators in quantum field theory. These expectation valuesdefine space-time dependent functions which satisfy integro-differential equation of motionand thus can superficially be viewed as classical fields of a classical effective dynamics as areminiscent of Ehrenfest’s theorem. However the local field variable is classical only if itsreduced density matrix is strongly peaked on the diagonal matrix elements. The off-diagonalvalues characterize the importance of the linear superposition in the averages and must benegligible in the classical domain. Actually it is better to call the expectation values of localoperators subclassical fields [25], the difference between them and the classical fields beingthe worse space-time resolution, i.e., lower UV cutoff, and the strong decoherence in thelatter case. B. Effective saddle point dynamics
The cutoff theory supports an open dynamics owing to the unobserved UV degrees offreedom hence the regularization of quantum field theories is to be performed in a frameworkdesigned for open systems [26], namely within the Closed Time Path (CTP) formalism [27].This is a CQCO scheme, it handles classical, quantum, closed and open systems on equalfooting and treats initial rather than boundary value problems. The redoubling of the degrees3of freedom, the distinguishing feature of this scheme, allows the extension of the variationalprinciple of classical mechanics for dissipative forces in open systems [28] and the quantumeffects arise as an O ( √ (cid:126) ) separation of the two coordinates, describing the same degree offreedom. Furthermore we obviously have to rely on initial rather than boundary conditionsin problems, related to the radiation.There is yet another reason to use the CTP scheme: The radiation reaction provides aneffective force which appears only by eliminating the electromagnetic field, by consideringit as the unobserved environment of the charge. To pick up the effects of the outgoingradiation, the friction forces, the electromagnetic field must be allowed to occupy any excitedfinal state. This condition requires the CTP formalism.The motion of a charged particle can be reconstructed from the expectation value of thecharge density, to be extended in a relativistic treatment to the expectation value of theelectric current. Hence we seek the effective theory for the electric current. The effectiveaction is constructed from the generator functional for the connected Green’s functions ofthe electric current [25], e i (cid:126) W [ˆ a ] = Tr (cid:2) U [ a + , ¯ η + , η + ] | (cid:105)(cid:104) | U † [ a − , ¯ η − , η − ] (cid:3) (19)where the source a µ , coupled linearly to A µ generates the Green’s functions and ¯ η , and η are coupled linearly to ψ and ¯ ψ , respectively to produce a coherent initial state, describingan electron. In the path integral representation of this functional, e i (cid:126) W [ˆ a ] = (cid:90) D [ ˆ ψ ] D [ ˆ¯ ψ ] D [ ˆ A ] exp i (cid:126) (cid:20) S M [ ˆ A ] + S D [ ˆ¯ ψ, ˆ ψ ]+ S i [ ˆ¯ ψ B , ˆ ψ B , ˆ A B + ˆ a ] + (cid:90) dx [¯ˆ η ( x ) ˆ ψ ( x ) + ˆ¯ ψ ( x )ˆ η ( x )] (cid:21) , (20)the integration is over the CTP doublet fields, ˆ A = ( A + , A − ), ¯ ψ = ( ψ + , ψ − ) and ˆ¯ ψ =( ¯ ψ + , ¯ ψ − ). One uses a similar notation for the sources, ˆ a = ( a + , a − ), ˆ η = ( η + , η − ) andˆ¯ η = (¯ η + , ¯ η − ), as well. The first two contribution to the action are the Maxwell action, S M [ ˆ A ] = 12 c (cid:90) dxdy ˆ A ( x ) ˆ D − Cl ( x, y ) ˆ A ( y ) , (21)with a relativistic gauge fixing term,ˆ D − µνCl = − π ( T µν + ξL µν ) ˆ D (22)4with L µν = ∂ µ ∂ ν / (cid:3) and T µν = g µν − L µν ,ˆ D m ( p ) = D n ( p ) + iD i ( p ) − D f ( p ) + iD i ( p ) D f ( p ) + iD i ( p ) − D n ( p ) + iD i ( p ) = p − m + i(cid:15) − πiδ ( p − m )Θ( − p ) − πiδ ( p − m )Θ( p ) − p − m − i(cid:15) (23)standing for the propagator of a scalar particle of mass m . The CTP propagator containsthe Feynman propagator, D ++ , and the retarded and advanced Green’s functions, D r a = D n ± D f . The free Dirac action, S D [ ˆ¯ ψ, ˆ ψ ] = 1 c (cid:90) dxdy ˆ¯ ψ ( x ) ˆ G − ( x, y ) ˆ ψ ( y ) , (24)contains the inverse of the electron propagator ˆ G m ( p ) = ( p/ + m ) ˆ D m ( p ). The interaction isdescribed by the action S i [ ˆ¯ ψ, ˆ ψ, ˆ A ] = e (cid:88) σ σ (cid:90) dx ¯ ψ σ ( x ) A/ σ ( x ) ψ σ ( x ) . (25)The CTP symmetry of the action, S [ φ + , φ − ] = − S ∗ [ φ − , φ + ] , (26) φ ± denoting the CTP doublet pair of a generic field variable, follows from the definition ofthe generator functional.The point splitting has already been used for gauge theories [29], and the particularregularization, implemented here is the replacement of the local fields with smeared ones,ˆ A B = ˆ κ ˆ σ ˆ A , ˆ ψ B = ˆ χ [ ˆ A ]ˆ σ ˆ ψ and ˆ¯ ψ B = ˆ¯ ψ ˆ σ ˆ¯ χ − [ ˆ A ] with ¯ χ = γ χ † γ , in the interaction whereˆ σ = Diag(1 , −
1) denotes the simplectic metric tensor of the CTP scheme [30]. The smearedphoton field contains the transverse component only, ˆ κ µν = T µν ˆ κ T , the longitudinal compo-nents being suppressed in the interactions. The action is kept invariant under gauge trans-formations, A → A + ∂α and ψ → e − ieα ψ , by applying the replacement, ∂ µ → ∂ µ + ieLA µ ,within the smearing function, ˆ χ . The physical, gauge invariant components of the electro-magnetic field do not appear in the smearing function of the charged field and the poten-tially dangerous regulator vertices of the higher order derivative scheme [31] are avoided. Itis advantageous to perform the change of integral variable, ˆ ψ B → ˆ ψ , ˆ¯ ψ B → ˆ¯ ψ , ˆ A B → ˆ A ,5¯ˆ ηχ − → ¯ˆ η and ¯ˆ χ − ˆ η → ˆ η , in the generator functional, (20), which amounts to the re-placement ˆ D Cl → ˆ D ClB = ˆ κ ˆ D Cl ˆ κ and ˆ G → ˆ G B = χ ˆ G B ¯ˆ χ of the propagators. The action,expressed in terms of the smeared, bare fields, displays local interaction and modified freedispersion relations [22].The regularization of the retarded Green’s function, used in section II, can be ex-tended to the whole CTP Green’s function. Owing to the positivity of the energy of theexcitations the support of the spectral function, iD − + , is over positive negative energy, iD i ( q )sign( q ) D f ( q ) = sign( q ) i Im D r ( q ), resulting the Feynman propagator D ++ ( q , q ) = D r ( | q | , q ). For instance, the retarded Green’s function, defined by eq. (4), D rB ( q ) = − | q | (cid:90) ∞ dr rr (cid:96) e ir (cid:96) ( q + i(cid:15) ) sin | q | r, (27)where r (cid:96) = √ (cid:96) + r , yields D rB (0 , q ) = − q (cid:90) ∞ du u (cid:112) q (cid:96) + u e − (cid:15) | q | √ q (cid:96) + u sin u,D rB ( q , ) = − q (cid:90) ∞ du u (cid:112) q (cid:96) + u e ( i − (cid:15)q ) √ q (cid:96) + u . (28)Thus the free Green’s functions is O (cid:16) | p | − (cid:17) in the UV regime and renders the Feyn-man graphs with internal photon lines finite. The CTP matrix, ˆ κ , is assumed tohave the block structure of the propagator, (23), with κ ++ ( q ) = (cid:112) D ++ B ( q ) /D ++ ( q ) and κ r ( q ) = (cid:112) D rB ( q ) /D r ( q ). The choice ˆ η = (cid:112) − Λ ˆ G Λ , with Λ = 1 /(cid:96) and the replacement ∂ µ → ∂ µ + ieLA µ , preserves causality [22].The perturbation series is generated by the formal expression e i (cid:126) W [ˆ a ] = e i (cid:126) e c (cid:82) dxdy δδ ˆ a ( x ) ( x ) ˆ D Cl δδ ˆ a ( y ) e i (cid:126) W [ˆ a ] , (29)where the free generator functional is given by e i (cid:126) W [ˆ a ] = (cid:90) D [ ˆ¯ ψ ] D [ ˆ ψ ] e i (cid:126) c (cid:82) dx ˆ¯ ψ [ ˆ G − +ˆ σ ˆ a/ ] ˆ ψ + i (cid:126) ˆ¯ η ˆ ψ + i (cid:126) ˆ¯ ψ ˆ η . (30)The Gaussian integration leads to W [ˆ a ] = − ˆ¯ η c ( ˆ G − − ˆ σ ˆ a/ ) η − i (cid:126) Tr ln[ ˆ G − − ˆ σ ˆ a/ ] , (31)the sum of a tree-level and a quantum fluctuation contribution. If one uses the powers of (cid:126) totrace the weight of the quantum fluctuations then the Compton wavelength of the electron6 mc/ (cid:126) , the mass term in ˆ G must be considered as a fixed, (cid:126) -independent number. Theresulting (cid:126) -independence of the first term is due to the charge conservation. The effectiveaction for the electric current is given by the Legendre transformation,Γ[ˆ j ] = W [ˆ a ] − ˆ a ˆ j, ˆ j = δW [ˆ a ] δ ˆ a (32)and the Euler-Lagrange equation, δ Γ[ˆ j ] δ ˆ j = ˆ a (33)is satisfied by the subclassical fields.A Gaussian integral can be reproduced by solving the saddle point equation for thevariable, (cid:90) ∞−∞ dAe i D A + iJA = e − i J D (cid:90) ∞−∞ dAe i D A , (34)the linear equation of motion can be used as an exact operator equation. Thus there is tree-level saddle point contribution to the generator function of the connected Green’s functions, W [ˆ a ] = W [ˆ a ] − e c (cid:90) dxdy ˆ j ( x ) ˆ D A ( x − y )ˆ j ( y ) + O ( (cid:126) ) , (35)where the quantum corrections correspond to the free Dirac see. The effective action for thecurrent is thereforeΓ[ˆ j ] = Γ [ˆ j ] − e c (cid:90) dxdy ˆ j ( x ) ˆ D A ( x − y )ˆ j ( y ) + O ( (cid:126) ) , (36)where Γ [ˆ j ] = W [ˆ a ] − ˆ a ˆ j (37)stands for the effective action in the free Dirac-see.The free effective action is highly involved, displaying a non-local, non-polynomial struc-ture without a small parameter to organize an expansion [32]. Rather than seeking anapproximate solution we assume that it approaches the form,Γ [ˆ j ] = − m B c (cid:90) ds ( √ ˙ x +2 − √ ˙ x − ) (38)in the point-like limit, j σµ ( x ) = δW [ˆ a ] δ ˆ a ( x ) → (cid:90) dsδ ( x − x σ ( s )) ˙ x σµ ( s ) , (39)7in the absence of pair creation, ˙ x >
0. The action (36) is the CTP extension of theaction-at-a-distance theory [4, 33], including the retarded radiation field [23]. The tree-leveleffective action,Γ[ˆ x ] = − m B c (cid:90) ds ( √ ˙ x +2 − √ ˙ x − ) − e c (cid:88) σσ (cid:48) σσ (cid:48) (cid:90) dsds (cid:48) ˙ x σ,µ ( s ) ˙ x σ (cid:48) µ (cid:48) ( s (cid:48) ) D σσ (cid:48) Cl ( x σ ( s ) − x σ (cid:48) ( s (cid:48) )) , (40)contains the one-loop electron self interaction. This latter is of O ( (cid:126) ) because the electronline of the corresponding Feynman graph describes a coherent state and is O ( (cid:126) ).The form Γ[ˆ x ] = Γ [ x + ] − Γ [ x − ] + Γ [ˆ x ], of the effective action with real Γ andΓ [ x + , x − ] = − Γ ∗ [ x − , x + ] is consistent with the symmetry (26). The unitarity of thetime evolution in the full QED implies W [ a, a ] = 0 in eq. (19) and consequentlyΓ[ x, x ] = Γ [ x, x ] = 0. The expectation value (cid:104) x (cid:105) is identical when calculated by the helpof U or U † in using the generator functional and the solution of the equations of motionproduces x + = x − . The equation of motion for x + , if x + = x − ,0 = δ Γ [ x ] δx + δ Γ [ x, x (cid:48) ] δx | x (cid:48) = x , (41)is identical of eq. (6) and shows the dissipative nature of the radiation reaction force [34]. IV. CONCLUSIONS
The scaling laws of an electron around the classical electron radius, r = 2 . f m , deeplywithin the quantum regime, are governed by the classical equation of motion, the dominantforce arises from the interaction of the electron with its own field and the standard regular-ization procedure must be employed. The effective dynamics of a point charge is derived inthis work by the help of a point splitting regularization which smears the electromagneticfield over the invariant distance ds = (cid:96) . The linearized equation of motion describes a sta-ble, causal dynamics for (cid:96) (cid:29) r and cutoff-dependent instability arises if (cid:96) (cid:28) r . Howeverthe non-linear terms of the equation of motion owing their existence to the cutoff stabilizethe dynamics. Two different renormalized trajectories are found and one of them fits qual-itatively to the dynamics, described by the unstable, linearized equation. The removal ofthe cutoff seems to be numerically possible, there is no evidence of a Landau pole withinthe tree-level renormalization. The radiation reaction can be fit into QED by realizing it asa tree-level saddle point effect.8The historical name of the scale r expresses expectations that the classical electrody-namics of a point charge is ill-defined at shorter distances [6]. One should at this pointdistinguish two different inquiries. Are we looking into the physics of an imaginary worldwith classical physics only, (cid:126) = 0, or into a physical phenomenon of our world where (cid:126) (cid:54) = 0?The latter scenarios is followed here by adopting the point of view that classical physics issupposed to be derived from the quantum level and classical electrodynamics should joinsmoothly to QED at the quantum-classical crossover. Regarding the radiation reaction prob-lem from this point of view one encounters two remarkable features of the Abraham-Lorentzforce which are prone to lead to misunderstanding, namely its (cid:126) - and cutoff-independence.While the radiation reaction can be identified in classical electrodynamics and is thereforea purely classical phenomenon however there are three considerations indicating that it isnot a typical classical physics problem. First, the radiation reaction force originates froma scale region which is deeply quantum and the correspondence principle, a guiding ruleof our intuition, is strongly violated. Second, the tree-level effective equation of motionapplies to the expectation value of the world line only, leaving a necessary condition of theclassical limit, the decoherence, an open issue. The decoherence, being an IR effect [34], isnot generated at the scale r and the charge maintains its coherent quantum state at thisscale, in other words, the Abraham-Lorentz force is a subclassical effect [25]. The third pointconcerns the origin and the features of the radiation reaction which bear the fingerprint ofquantum field theory, namely being generated by a loop-integral. This integral is divergentand needs a regulator, implying the techniques and concepts of the renormalization group,developed in quantum field theory.The radiation reaction force of a point particle is obviously an UV cutoff-effect, thevelocity of a massive particle is bounded by the speed of light hence the world line of a pointparticle can not cut through the light cones of its own radiation field. Among the severalcutoff-dependent terms in the effective equation of motion the Abraham-Lorentz force isdistinguished by being cutoff-independent. It is generated by the cutoff but its strength isindependent of the cutoff scale. This is a well known phenomenon in quantum field theory,has the somehow unfortunate name of anomaly, and reflects the non-uniform convergenceof the loop-integrals of the perturbative solutions when the cutoff is removed [34].The puzzle of the radiation reaction force, the apparent instability of the Abraham-Lorentz force, can be resolved by bearing in mind that the effective classical dynamics9contains a one-loop integrals which needs regularization. The introduction of the cutoffmakes the parameters of the equation of motion non-physical and forces us to follow thepainstaking limit (cid:96) → Acknowledgement
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