stochastic theory of relativistic particles moving in a quantum field

7/29/2019 Stochastic Theory of Relativistic Particles Moving in a Quantum Field

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arXiv:qu

ant-ph/0101001v2

1Jun2001

Stochastic Theory of Relativistic Particles Moving in a Quantum Field: II. Scalar

Abraham-Lorentz-Dirac-Langevin Equation, Radiation Reaction and Vacuum

Fluctuations

Philip R. Johnson and B. L. Hu

Department of Physics, University of Maryland,College Park, Maryland 20742-4111

(April 26, 2000)

We apply the open systems concept and the influencefunctional formalism introduced in Paper I to establish astochastic theory of relativistic moving spinless particles in aquantum scalar field. The stochastic regime resting betweenthe quantum and semi-classical captures the statistical me-chanical attributes of the full theory. Applying the particle-centric world-line quantization formulation to the quantumfield theory of scalar QED we derive a time-dependent (scalar)Abraham-Lorentz-Dirac (ALD) equation and show that it isthe correct semiclassical limit for nonlinear particle-field sys-tems without the need of making the dipole or non-relativisticapproximations. Progressing to the stochastic regime, wederive multiparticle time-dependent ALD-Langevin equationsfor nonlinearly coupled particle-field systems. With theseequations we show how to address time-dependent dissipa-tion/noise/renormalization in the semiclassical and stochasticlimits of QED. We clarify the relation of radiation reaction,quantum dissipation and vacuum fluctuations and the role thatinitial conditions may play in producing non-Lorentz invariantnoise. We emphasize the fundamental role of decoherence inreaching the semiclassical limit, which also suggests the correctway to think about the issues of runaway solutions and preac-celeration from the presence of third derivative terms in theALD equation. We show that the semiclassical self-consistentsolutions obtained in this way are paradox and pathology freeboth technically and conceptually. This self-consistent treat-ment serves as a new platform for investigations into problemsrelated to relativistic moving charges.

I. INTRODUCTION

This is the second in a series of three papers [1,2] ex-ploring the regime of stochastic behavior manifested byrelativistic particles moving through quantum fields. Thisseries highlights the intimate connections between back-

reaction, dissipation, noise, and correlation, focusing onthe necessity (and the consequences) of a self-consistenttreatment of nonlinearly interacting quantum dynamicalparticle-field systems.

Electronic address: [email protected] address: [email protected]

In Paper I [1], we have set up the basic frameworkbuilt on the influence functional and worldline quantiza-tion formalism. In this paper we apply the results obtainedthere to spinless relativistic moving particles in a quan-tum scalar field. The interaction is chosen to be the scalaranalog of QED coupling so that we can avoid the com-plications of photon polarizations and gauge invariance inan electromagnetic field. The results here should describecorrectly particle motion when spin and photon polariza-

tion are unimportant, and when the particle is sufficientlydecohered such that its quantum fluctuations effectivelyproduce stochastic dynamics.

The main results of this investigation are

1. First principle derivation of a time-dependent (modi-fied) Abraham-Lorentz-Dirac (ALD) equation as theconsistent semiclassical limit for nonlinear particle-scalar-field systems without making the dipole ornon-relativistic approximations.

2. Consistent resolution of the paradoxes of the ALDequations, including the problems of runaway andacausal (e.g. pre-accelerating) solutions, and otherpathologies. We show how the non-Markovian na-ture of the quantum particle open-system enforcescausality in the equations of motion. We also discussthe crucial conceptual role that decoherence plays inunderstanding these problems.

3. Derivation of multiparticle Abraham-Lorentz-Dirac-Langevin (ALDL) equations describing the quantumstochastic dynamics of relativistic particles. Thefamiliar classical Abraham-Lorentz-Dirac (ALD)equation is reached as its noise-averaged form. Thestochastic regime, characterized by balanced noiseand dissipation, plays a crucial role in bridging the

gap between quantum and (emergent) classical be-haviors. The N-particle-irreducible (NPI) and mas-ter effective action [36] provide a route for gener-alizing our treatment to the self-consistent inclusionof higher order quantum corrections.

We divide our introduction into three parts: First, adiscussion of the pathologies and paradoxes of the ALDequation from the conventional approach and our sug-gested cures based on proper treatment of causal and non-Markovian behavior and self-consistent backreaction. Sec-

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ond, misconceptions in the relation between radiation re-action, quantum dissipation and vacuum fluctuations andthe misplaced nature and role of fluctuation-dissipationrelations. Finally we give a brief summary of previouswork and describe their shortcomings which justify a newapproach as detailed here.

A. Pathologies, Paradoxes, Remedies and Resolutions

The classical theory of moving charges interacting witha classical electromagnetic field has controversial difficul-ties associated with backreaction [7,8]. The generally ac-cepted classical equation of motion in a covariant formfor charged, spinless point particles, including radiation-reaction, is the Abraham-Lorentz-Dirac (ALD) equation[9]:

z + 2e2/3m (z

z2+...z

) = (e/m)zFext(z). (1.1)

The timescale 0 =

2e2/3mc3

determines the relativeimportance of the radiation-reaction term. For electrons,0 1024 secs, which is roughly the time it takes lightto cross the electron classical radius r0 1015m. TheALD equation has been derived in a variety of ways, ofteninvolving some regularization procedure that renormalizesthe particles mass [10]. Feynman and Wheeler have de-rived this result from their Absorber theory which sym-metrically treats both advanced and retarded radiation onthe same footing [11].

The ALD equation has strange features, whose statusare still debated. Because it is a third order differentialequation, it requires the specification of extra initial data(e.g. the initial acceleration) in addition to the usual posi-tion and velocity required by first order Hamiltonian sys-tems. This leads to the existence of runaway solutions.Physical (e.g. non-runaway) solutions may be enforced bytransforming (1.1) to a second order integral equation withboundary condition such that the final energy of the par-ticle is finite and consistent with the total work done on itby external forces. But the removal of runaway solutionscomes at a price, because solutions to the integral equa-tion exhibit the acausal phenomena of pre-acceleration ontimescales c. This is the source of lingering questions onwhether the classical theory of point particles and fields iscausal1.

1Another mystery of the ALD equation is the problem of theconstant force solution, where the charge uniformly accelerateswithout radiation reaction, despite it being well-establishedthat there is radiation. This is a case where local notions ofenergy must be carefully considered in theories with local in-teractions (between point particles) mediated by fields [12,13].The classical resolution of this problem involves recognizing

There have been notable efforts to understand chargedparticle radiation-reaction in the classical and quantumtheory (for a review see [8]), of which extended chargetheories [14] and quantum Langevin equations are impor-tant examples [1517] (See Subsection C below).

B. Radiation Reaction and Vacuum Fluctuations

The stochastic regime is characterized by competitionbetween quantum and statistical processes: particularly,quantum correlations versus decoherence, and fluctuationsversus dissipation. The close association of vacuum fluctu-ations with radiation-reaction is well-known, but the pre-cise relationship between them requires clarification. It iscommonly asserted that vacuum fluctuations (VF) are bal-anced by radiation reaction (RR) through a fluctuation-dissipation relation (FDR). This can be misleading. Ra-diation reaction is a classical process (i.e., = 0) while

vacuum fluctuations are quantum in nature. A counter-example to the claim that there is a direct link betweenRR and VF is the uniformly accelerated charged parti-cle (UAP) for which the classical radiation reaction forcevanishes, but vacuum fluctuations do not. However, at thestochastic level variations in the radiation reaction forceaway from the classical averaged value exist it is thisquantum dissipation effect which is related to vacuum fluc-tuations by a FDR.

From discussions in Paper I we see how decoherenceenters in the quantum to classical transition. Under rea-sonable physical conditions [18] the influence action or de-coherence functional is dominated by the classical solution(via the stationary phase approximation). Also from thenon-Markovian fluctuation-dissipation relations found inPaper I, we see that the relationship between noise and dis-sipation is extremely complicated. Here too, decoherenceplays a role in the emergence of the usual type fluctuation-dissipation kernel, just as it plays a role in the emergenceof classicality. When there is sufficient decoherence, theFDR kernel (see Eq. (5.23) in Paper I) is dominated bythe self-consistent semiclassical (decoherent) trajectories.An approximate, linear FDR relation may be obtained(see Eq. (5.24) in Paper I) except that the kernel is self-consistently determined in terms of the average (mean)system history, and the FDR describes the balance of fluc-tuations and dissipation about those mean trajectories.

These observations have important ramifications on theappropriateness of assigning a FDR for radiation reaction

that one must consider all parts of the total particle-field sys-tem with regard to energy conservation: particle energy, radia-tion energy, and non-radiant field energy such as resides in theso-called acceleration (or Shott) fields [13].

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and vacuum fluctuations. Ordinary (e.g. classical) radia-tion reaction is consistent with the mean particle trajecto-ries determined by including the backreaction force fromthe field. But at the classical level, there are no fluctua-tions, and hence no fluctuation-dissipation relation. Evenregarding radiation reaction as a damping mechanism is

misleading it is not dissipation in the true statisticalmechanical sense. Unlike a particle moving in some vis-cous medium whose velocity is damped, radiation reactiongenerally vanishes for inertial (e.g. constant velocity) par-ticles moving in vacuum fields (e.g. QED). Furthermore,radiation reaction can both damp and anti-damp parti-cle motion, though the average effect is usually a dampingone. On the other hand, we show that the fluctuationsaround the mean-trajectory are damped, as described bya FDR. At the quantum stochastic level as discussed here,quantum dissipation effect results from the changes in theradiation reaction force that are associated with fluctu-ations in the particle trajectory around the mean, and

therefore it is not the same force as the classical (i.e. av-erage) radiation reaction force. Only these quantum pro-cesses obey a FDR. This is a subtle noteworthy point.

C. Prior work in relation to ours

1. There are many works on nonrelativistic quantum(and semiclassical) radiation reaction (for atoms aswell as charges moving in quantum fields) includingthose by Rohrlich, Moniz-Sharp, Cohen-Tannoudjiet. al., Milonni, and others [8,13,1921]. Kampen,Moniz, Sharp, and others [14] suggested that theproblems of causality and runaways can be resolvedin both classical and quantum theory by consideringextended charge models. However, this approach,while quite interesting, misses the point that pointparticles (like local quantum field theories) obey agood low energy effective theory in their own right.While it is true that extended objects can cure in-finities (e.g., extended charge models, string the-ory), it is nonetheless important to recognize thatat sufficiently low energy QED as an effective fieldtheory is consistent without recourse to its high-energy limit. Wilson, Weinberg and others [22] haveshown how effective theory description is sufficientto understand low energy physics because compli-

cated, and often irrelevant, high energy details of thefine structure are not being probed at the physicalenergy scales of interest. The effective theory ap-proach has also provided a new perspective on thephysical meaning of renormalization and the sourceof divergences, showing why nonrenormalizable in-teractions are not a disaster, why the apparent largeshift in bare-parameters resulting from divergentloop diagrams (i.e. renormalization) doesnt inval-idate perturbation theory, and clarifying how and

when high-energy structure does effect low-energyphysics. Therefore one should not need to invokeextended charge theory to understand low energyparticle dynamics.

Our work is a relativistic treatment which includescausal Non-Markovian behavior and self-consistentbackreaction. We think this is a better approacheven in a nonrelativistic context because the regular-ized relativistic theory never breaks down. In ourapproach we adopt the ideas and methods of effec-tive field theory with proper treatment of backreac-tion to account for the effects of high-energy (short-distance) structure on low-energy behavior, and todemonstrate the self-consistency of the semiclassicalparticle dynamics. We consider the consequencesof coarse-graining the irrelevant (environmental) de-grees of freedom, and the nature of vacuum fluc-tuations and quantum dissipation in the radiationreaction problem. We emphasize the crucial role of

decoherence due to noise in resolving the pathologiesand paradoxes, and in seeing how causal QED leadsto a (short-time modified) causal ALD semiclassicallimit.

2. By examining the time-dependence of operatorcanonical commutation relations Milonni showed thenecessity of electromagnetic field vacuum fluctua-tions for radiating nonrelativistic charges [21]. Theconservation of the canonical commutation relationsis a fundamental requirement of the quantum theory.Yet, if a quantum particle is coupled to a classicalelectromagnetic field, radiative losses (dissipation)lead to a contraction in phase-space for the ex-pected values of the particle position and momen-tum, which violates the commutation relations. It isthe vacuum field that balances the dissipation effect,preserving the commutation relations [x, p] = ias a consequence of a fluctuation-dissipation rela-tion (FDR). The particle-centric worldline frame-work allows us to consider similar issues for rela-tivistic particles. Interestingly, our covariant frame-work shows how quantum fluctuations (manifestingas noise) appear in both the time and space coor-dinates of a particle. This is not surprising sincerelativity requires that a physical particle satisfieszz = t

2 ()

z2 () = 1, and therefore any vari-

ation in spatial velocities must be balanced by achange in t0 keeping the particle on-shell.

3. The derivation and use of quantum Langevin equa-tions (QLEs) to describe fluctuations of a systemin contact with a quantum environment has a longhistory. Typically, QLEs are assumed to describefluctuations in the linear response regime for a sys-tem around equilibrium, but its validity does notneed be so restrictive. Nonequilibrium conditionscan be treated with the Feynman-Vernon influence

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functional. Caldeira and Leggetts study of quantumBrownian motion (QBM) has led to an extensive lit-erature [23], particularly in regard to decoherenceissues [24]. Barone and Caldeira [25] have appliedthis method to the question of whether nonrelativis-tic, dipole coupled electrons decohere in a quantum

electromagnetic field. An advantage of Barone andCaldeiras work is that it is not limited to initiallyfactorized states; they use the preparation functionmethod which allows the inclusion of initial par-ticle field correlations. Despite this, Romero andPaz have pointed out that the preparation functionmethod still suffers from an (implicit) unphysical de-piction of instantaneous measurement characterizingthe initial state preparation [26].

4. Ford, Lewis, and OConnell have extensively dis-cussed the electromagnetic field as a thermal bathin the linear, dipole coupled regime [17], and pio-neered the application of QLEs to nonrelativisticparticle motion in QED. They have detailed the con-ditions for causality in the thermodynamic, equilib-rium limit described by the late-time linear quantumLangevin equation. A crucial point of their analy-sis is that particle motion can be (depending on thecutoff of the field spectral density) runaway free andcausal in the late-time limit [27]. In [28], they sug-gest a form of the equations of motion that givesfluctuations without dissipation for a free electron,but this result is special to the particular choice of afield cutoff made by the authors, namely, one imply-ing that the bare mass of the particle exactly van-ishes. When one does not assume a special value

for the cutoff one generally does find both fluc-tuations and dissipation, though Ford and Lewisscounter-example shows how careful one must be inhandling FDRs. Another counter-example againstmaking hasty generalizations is the vanishing of theaveraged (classical) radiation reaction for uniformlyaccelerated particles despite quantum fluctuationsthat make the (suitably coarse-grained) trajectoriesstochastic.

We do agree with the result that the particle motionis only causal and consistent when the field cutoff isbelow a certain critical value2 (determined from theparticles classical radius), but we find no compelling

justification for the claim that the cutoff should takeexactly that special value.

2A noted example is in Caldeira and Leggetts original work[16]. Their master equation governs a reduced density matrixwhich is not positive definite because they choose an (infinite)cutoff greater than the inverse temperature (see Hu, Paz, andZhang [23] for details).

It is certainly interesting that with the special field-cutoff chosen by Ford and OConnell higher time-derivatives vanish from the equations of motion.However, even without such a specially chosen cut-off, the influence of higher derivative terms arestrongly suppressed at low energies. On this issue

we take the effective theory point of view which em-phasizes the generic insensitivity of low-energy phe-nomena to unobserved high-energy structures. Sincehigher-time derivatives in the equations of motiondo not (necessarily) violate causality, there is noreason to rule them out. It is perhaps instructiveto review the meaning of renormalizable field theo-ries in the light of effective theories. Effective the-ories generically have higher derivative interactionsthat are again strongly suppressed at low energies.What makes renormalizable terms special is thatthese are the interactions that remain relevant (ormarginal) at low energies, while non-renormalizable

terms (such as higher derivative terms) are expo-nentially suppressed, hence, it is no longer believedthat non-renormalizable terms are fundamentallyexcluded so long as one recognizes that physicaltheories are mostly effective theories. Since QEDis certainly an effective theory, these observations(together with the suppression of the higher time-derivative terms) weaken the argument that the spe-cial cutoff theory of FOL is fundamental. Ford andOConnell also propose a relativistic generalizationof their modified equations of motion for the aver-age trajectory derived from the nonrelativistic QLE[29]. Our derivation of the stochastic limit from arelativistic quantum mechanics (i.e. worldline for-mulation) and field theory goes well beyond this byallowing the treatment of fully quantum relativis-tic processes, and by deriving a relativistic Langevinequation.

5. A perturbative expansion of the ALD equation upto order e3 has been derived from QED field the-ory by Krivitskii and Tsytovich [30], including theadditional forces arising from particle spin. Theirwork shows that the ALD equation may also be un-derstood from field theory, but the authors have notaddressed the role of fluctuations, correlation, deco-herence, time-dependent renormalization, nor self-

consistent backreaction. Our derivation yields thefull ALD equation (all orders in e for the perturba-tion expansion employed by [30]) through the loop(semiclassical) expansion; this method automati-cally includes all tree-level diagrammatic effects.

6. Low [31] showed that runaway solutions apparentlydo not occur in spin 1/2 QED. But Low does notderive the ALD equation, nor address the semiclassi-cal/stochastic limit. We also emphasize, again, thatit does not, and should not, matter whether parti-

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cles are spin 1/2, spin 0, or have some other internalstructure in regard to the causality of the low energyeffective theory for center of mass particle motion.

7. Using the influence functional, Diosi [32] derivesa Markovian master equation in non-relativistic

quantum mechanics. In contrast, it is our intentto emphasize the non-Markovian and nonequilib-rium regimes with special attention paid to self-consistency. The work of [32] differs from ours in thetreatment of the influence functional as a functionalof particle trajectories in the relativistic worldlinequantization framework. Ford has considered theloss of electron coherence from vacuum fluctuationinduced noise with the same noise kernel that weemploy [33]. However, his application concerns thecase of fixed or predetermined trajectories.

D. Organization

In Section II we obtain the influence functional (IF) forspinless relativistic particles. We then define the stochas-tic effective action (Sstoch) for this model, using our re-sults from Paper I, and use it to derive nonlinear integralLangevin equations for multiparticle spacetime motion. InSection III we consider the single particle case. We derivethe (scalar-field) Abraham-Lorentz-Dirac (ALD) equationas the self-consistent semiclassical limit, and show how thislimit emerges free of all pathologies. In Section IV we de-rive a Langevin equation for the stochastic fluctuations ofthe particle spacetime coordinates about the semiclassicallimit for both the one particle and multiparticle cases. We

also show how a stochastic version of the Ward-Takahashiidentities preserves the mass-shell constraint. In SectionV, we give a simple example of these equations for a singlefree particle in a scalar field. We find that in this particu-lar case the quantum field induced noise vanishes, thoughthis result is special to the overly-simple scalar field, anddoes not hold for the electromagnetic field [2]. In the finalsection, we summarize our main results and mention areasof applications, of both theoretical and practical interest.

II. SPINLESS RELATIVISTIC PARTICLES

MOVING IN A SCALAR FIELD

A. The model

Relativistic quantum theories are usually focused pri-marily on quantized fields, the notion of particles followingtrajectories is somewhat secondary. As explained in PaperI [1], we employ a hybrid model in which the environ-ment is a field, but the system is the collection of particle

spacetime coordinates zn (n) (i.e., worldlines) where n in-dicates the nth particle coordinate, with worldline param-eter n, charge en, and mass mn. The free particle actionis

SA[z] = ndn(zn)2 [mn + V(zn(n))] . (2.1)

From SA follow the relativistic equations of motion:

mzn/

z2n = V (zn) . (2.2)For generality, we include a possible background potentialV, in addition to the quantum scalar field environment.The scalar current is

j[z, x) =n

dnenun(n)(x zn(n)), (2.3)

where

un (n) =

zn (n) zn, (n). (2.4)

In Paper III [2], we treat a vector current coupled to theelectromagnetic vector potential A. In both cases weassume spinless particles. The inclusion of spin or coloris important to making full use of these methods in QEDand QCD.

In the full quantum theory reparametrization invari-ance of the relativistic particle plays a crucial role, andthe particle-field interaction must respect this symmetry.When dealing with the path integral for the quantizedparticle worldline, a quadratic form of the action is oftenmore convenient than the square-root action in (2.1). The

worldline quantum theory is a gauge theory because of theconstraint that follows from reparametrization invarianceunder () = + () , making this perhaps thesimplest physical example of a generally covariant theory.(We may therefore view this work as a toy model for gen-eral relativity, and for string theory, in the semiclassicaland stochastic regime.) The path integral must thereforebe gauge-fixed to prevent summing over gauge-equivalenthistories. In [34] we treat the full quantum theory, in de-tail, and derive the worldline influence functional afterfixing the gauge to the so-called proper-time gauge. Thisgive the worldline path integral in terms of a sum overparticle trajectories z () satisfying z2 = N2, but where

the path integral still involves an integration over possibleN (the range of this integration depends on the bound-ary conditions for the problem at hand). However, apply-ing the semiclassical (i.e., loop) expansion in [34] we findthat the N = 1 trajectory gives the stationary phase solu-tion; hence, the semiclassical limit3 corresponds to setting

3Note however that in the stochastic limit there can poten-tially be fluctuations in the value of the constraint.

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N = 1. In the following we will therefore assume the con-straint z2 = N2, or even z2 = 1, when we are discussingthe semiclassical limit for trajectories. We show that thisis self-consistent in that it is preserved by the equationsof motion including noise-induced fluctuations in the par-ticle trajectory. Since we are addressing the semiclassi-

cal/stochastic limit in this paper there is no need to adoptthe quadratic form of action for a relativistic particle; itis adequate, and it turns out to be simpler in this paper,to just work with the scalar functions un defined in (2.4),and the square-root form of the relativistic particle actionin (2.1).

While we emphasize the microscopic quantum origins,we might also view our model in analogy to the treat-ment of quantum fields in curved spacetime. There, onetakes the gravitational field (spacetime) as a classical sys-tem coupled to quantum fields. One important class ofproblems is then the backreaction of quantum fields onthe classical spacetime. The backreaction of their mean

yields the semiclassical Einstein equation which forms thebasis of semiclassical gravity. The inclusion of fluctuationsof the stress-energy of the quantum fields and the inducedmetric fluctuations yields the Einstein-Langevin equations[35] which forms the basis of this stochastic semiclassicalgravity [36]. In our work here, the particle coordinatesare analogous to the gravitational field (metric tensor).Whereas the Einstein-Langevin equations have not beenderived from first principles for the lack of a quantum the-ory of gravity, we shall take advantage of the existence ofa full quantum theory of particles and fields to examinehow, and when, a regime of stochastic behavior emerges.

B. The influence functional and stochastic effectiveaction

The interaction between the particles and a scalar fieldis given by the monopole coupling term

Sint =

dx j[z, x) (x) (2.5)

= en

dnun(n)(zn(n)).

This is the general type of interaction treated in PaperI. In the second line we have used the expression for the

current (2.3). The expression for the influence functional(Eq. (3.16) of Paper I) [37] is,

F[j] = exp{ i

dx

dx[2j(x) GR(x, x)j+(x)

ij(x)GH(x, x)j(x)]}, (2.6)where

j (j[z, x)j[z, x)) (2.7)j+ (j[z, x) + j[z, x)) /2

and {z, z} are a pair of particle histories. GR and GHare the scalar field retarded and Hadamard Greens func-tions, respectively. Substitution of (2.3) then gives themultiparticle influence functional

F[

{z

},

{z

}] = exp

{e2

nm

dn dm (2.8) (T z0n)(z0n z0m) [unGH(zn, zm)um unGH(zn, zm)um unGH(zn, zm)um + unGH(zn, zm)um+ unG

R(zn, zm)um unGR(zn, zm)um+ unG

R(zn, zm)u

m unGR(zn, zm)um]}

where un = un(zn). The influence functional may be ex-

pressed more compactly by using a matrix notation whereuTn = (un, u

n)

u1n, u

2n

, giving

F[za] = exp{ e2nm

dndm (2.9)

uTnGRnmum + uTnGHnmum}= exp

e

2

uTnG

Rnmum + u

TnG

Hnmum

(2.10)

= exp

i

SIF[z

a]

. (2.11)

The superscript T denotes the transpose of the columnvector u. In (2.10), and below, we leave the sum nm andintegrations dndm implicit for brevity. The matricesGHnm

,GRnm

are given by

GHnm = (T z0n)(z0n z0m) (2.12)

GH(11)

z1n, z1m

GH(12) z1n, z2mGH(21)

z2n, z

1m

GH(22)

z2n, z

2m

and

GRnm = (T z0n)(z0n z0m) (2.13)

GR(11)

z1n, z1m

GR(12)

z1n, z

2m

GR(21)

z2n, z

1m

GR(22) z2n, z2m

.

In Eq. (2.11), we have defined the influence action SIF.

GH,R are the scalar-field Hadamard/Retarded Greensfunctions evaluated at various combinations of spacetimepoints z1,2n,m.

The stochastic effective action is defined by (see [1])

S[z] = SA[z

] +

dxj[z, x)((x) (2.14)

+ 2

dxGR(x, x)j+[z, x))

= SA[za] + uTnG

Rnmum + u

Tn n,

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where

uTn n = un (zn) (zn) un (zn) (zn) , (2.15)and (zn) is a (classical) stochastic field (see Section 3 ofPaper I) evaluated at the spacetime position of the nth

particle. The stochastic field has vanishing mean and au-tocorrelation function given by

(x) (x)s = { (x) , (x)} = GH (x, x) . (2.16)

Hence, (x) s statistics encodes those of the quantum

field (x).The stochastic effective action provides an efficient

means for deriving self-consistent Langevin equations thatdescribe the effects of quantum fluctuations as stochasticparticle motion for sufficiently decohered trajectories (for

justification in using Sstoch in this way see Paper I).

C. Langevin integrodifferential equations of motion

Deriving a specific Langevin equation from the generalformalism of Paper I requires evaluating the cumulants

C()q , defined in Eq. (4.11) of [1]. Because the multiparti-cle case involves a straightforward generalization, we beginwith the single particle theory. The noise cumulants arethen defined as

C()q =

i

qqSIF [j]

zn (n) ...zm (m)

z=0

, (2.17)

with SIF given in (2.11). The superscript () indicates

that these are cumulants for an expansion of SIF withrespect to the stochastic variables () .

We define new variables v = Uu where vT=

v1, v2

and uT= (u, u) , and

U =1

2

1 11 1

. (2.18)

Then the influence action has the form

SIF[za] =

e2

vTGRv v + iv

TGHv v

, (2.19)

where

GRv = UGRUT (2.20)

=

GR11GR12+GR21GR22 GR11+GR12+GR21+GR22GR11GR12GR21+GR22 GR11+GR12GR21GR22

,

and

GHv = UGHUT (2.21)

=

GH11+G

H12+G

H21+G

H22 G

H11GH12+GH21GH22

GH11+GH12GH21GH22 GH11GH12GH21+GH22

The lowest order cumulant in the Langevin equation,

C()1,, is found by evaluating SIF/z

, and then set-

ting z = 0. There are two kinds of terms that arise:those where /z acts on v, and those where it acts onGR,H. For linear theories, GR,H is not a function of thedynamical variables, but is instead (at most) a function

of some predetermined kinematical variables that a pri-ori specify the trajectory. Hence, there is no contributionfrom GR,H/z. For the v/z terms, setting z = 0collapses the matrices (2.20 and 2.21) to one term each:only the (1, 2) term of GRv , and the (1, 1) term of G

Rv ,

survive. But, setting z = 0 gives v1 = 0, v2 = u, andsince the (1, 1) term of GHv is proportional to two factorsof v1, it also vanishes. When /z acts on GRv , onlythe (2, 2) terms survive (because it is the only term notproportional to a factor of v1). For similar reasons, theonly contributing element ofGH is also its (2, 2) term. To

evaluate

GRv(22)/z

|z=0, we note

z=

1

2

z

z

. (2.22)

After a little algebra, we are left with

GRv(22)/z

|z=0 =

GR(z(), z())

z(), (2.23)

where the derivative only acts on the z(), and not thez() argument, in GR. The same algebra in evaluating

the

GHv(22)/z|z=0 term shows that all the factors

cancel, and therefore GH does not contribute to the first

cumulant at all. Because the imaginary part of SIF[za

]does not contribute to the first cumulant, the equations ofmotion of the mean-trajectory are explicitly real, which isan important consequence of using an initial value formu-lation like the influence functional (or closed-time-path)method.

The first cumulant, describing radiation reaction, istherefore given by

C()1 [z, ) =

SIFz()

z=0

(2.24)

=

i

de2{ u(z)z ()

GR(z(), z())u(z())

+ u(z())

GR(z(), z())z()

u(z())}.

This expression is explicitly causal both because theproper-time integration is only over values < , andbecause of the explicit occurrence of the retarded Greensfunction. Contrary to common perception, the radiation

reaction force given by C()1 is not necessarily dissipative

in nature. For instance, we shall see that C()1 vanishes

for uniformly accelerated motion, despite the presence of

7


8/21

radiation from a uniformly accelerating charge. In other

circumstances, C()1 [z, ) may actually provide an anti-

damping force for some portions of the particle trajectory.Hence, radiation reaction is not a purely dissipative,

or damping, process. The relationship of radiation reac-tion to vacuum fluctuations is also not simply given by

a fluctuation-dissipation relation4. In actuality, only thatpart of the first cumulant describing deviations from thepurely classical radiation reaction force is balanced by fluc-tuations in the quantum field via generalized fluctuationdissipation relations. This part we term quantum dissi-pation because it is of quantum origin and different fromclassical radiation reaction. Making this distinction clearis important to dispel common misconceptions about ra-diation reaction being always balanced by vacuum fluctu-ations.

Next we evaluate the second cumulant. After similarmanipulations as above, we find that the second cumulantinvolves only GH. We note that the action of j/z on

an arbitrary function is given bydx

j(x)

z()f(x) (2.25)

= e

dx

d

z

z21/2

(x z())

f(x)

= e{ dd

z

u()

+

z

u()

d

d u()

z}f(z())

Because of the constraint z2 = N2, it follows that u = 0,and we may set u = N. Also, zz = 0. Then

dx j(x)z()

f(x) = e 1Nz

+ zz[

] f(z())

e w (z) f(z()). (2.26)This last expression defines the operator w (z) . We haveused d/d = z. The operator w satisfies the identity

z w (z) = z

1

N

z + z

z[]

=1

N

zz + z

zz[]

= 0. (2.27)

This identity ensures that neither radiation reaction nornoise-induced fluctuations in the particles trajectory move

4Fluctuation-dissipation related arguments have been fre-quently invoked for radiation reaction problems, but these in-stances generally entail linearizing (say, making the dipole ap-proximation), and/or the presence of a binding potential thatprovide a restoring force to the charge motion making it peri-odic (or quasi-periodic). Radiation reaction in atomic systemsis one example of this [20]; the assumption of some kind ofharmonic binding potential provides another example [21,38].

the particle off-shell (i.e., the stochastic equations of mo-tion preserve the constraint z2 = N2 for any constant N).

With these definitions, the noise () is defined by

() = e1/2 w(z)(z())

= e1/2 1

N

z + z

z[]

(z) (2.28)

and the second-order noise correlator by

C()2 [z; ,

) = {(), ()} (2.29)= e2w(z) w(z){ (z ()) , (z ())}= e2w(z) w(z)GH(z(), z()).

The operator w (z) acts only on the z in GH (z, z) ; like-wise, the operator w (z) acts only on z.

This scalar field result is reminiscent of electromag-netism, where the Lorentz force from the (antisymmetric)field strength tensor F is F = zF = z[A]. Theantisymmetry of F implies zF = 0. We may definea scalar analog of the antisymmetric (second rank) fieldstrength tensor by

F z[]. (2.30)This shows that the second term on the right hand side of(2.28) gives the scalar analog of (a stochastic) electromag-netic Lorentz force: F = zF. The first term on theRHS of (2.28), z (z) , does not occur in the treatment ofthe electromagnetic field. In the scalar-field theory, thisterm may be thought of as a stochastic component to the

particle mass.The stochastic equations of motion are

mz = V(z) + e2

d w(z)GR(z(), z())

+ e1/2w(z)(z()). (2.31)

The result (2.31) is formally a set of nonlinear stochasticintegrodifferential equations for the particle trajectoriesz[; ). Noise is absent in the classical limit found by theprescription 0 (this definition of classicality is for-mal in that the true semiclassical/classical limit requirescoarse-graining and decoherence, and is not just a matterof taking the limit

0).

The generalization of (2.31) to multiparticles is nowstraightforward. If we re-insert the particle number in-dices, the first cumulant is

C1(n)[z, ) =SIF[z]

zn ()

zm=0

(2.32)

=m

dme

2 wn(zn)

GR(zn(n), zm(m))u(zm(m),

8


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the noise term is given by

n() = e1/2wn(zn)(zn(n)), (2.33)

and the noise correlator is

C2(nm)[z; n, m) (2.34)

= {n(n), m(m)}= e2wn(zn) w

m(zm)G

H(zn(n), zm(m)).

The nonlinear multiparticle Langevin equations are there-fore

mzn () (2.35)

= V(zn ()) + e1/2 w(zn)(zn())+ e2

m

fi

d w(zn ())GR(zn(), zm(

)).

The n = m terms in (2.35) are particle-particle inter-action terms. Because of the appearance of the retardedGreens function, all of these interactions are causal. Then = m terms are the self-interaction (radiation-reaction)forces. The n = m noise correlator terms represent nonlo-cal particle-particle correlations: the noise that one parti-cle sees is correlated with the noise that every other parti-cle sees. The nonlocally correlated stochastic field (x) re-flects the correlated nature of the quantum vacuum. Fromthe fluctuation-dissipation relations found in Paper I (Eq.(5.23) and Eq. (5.24)), the n = m quantum noise is re-lated to the n = m quantum dissipative forces. Undersome, but not all, circumstances5 the n = m correlationterms are likewise related to the n = m propagation (in-teraction) terms through a multiparticle generalization ofthe FDR, called a propagation-correlation relation [39].

We have already noted that the first term in (2.28) hasthe form of a stochastic contribution to the particles effec-tive mass, thus allowing us to define the stochastic massas

m z =

m + e1/2(z)

z. (2.36)

Fluctuations of the stochastic mass automatically preservethe mass-shell condition since (mzz) |z=z = 0. Like-wise, the effective stochastic force F satisfies

zF = zzz

[](z) = z(z)z[](z) = 0. (2.37)

5See [39] and Paper I, Section IV, for a discussion of this point.In brief, the noise correlation between spacelike separatedcharges does not vanish owing to the nonlocality of quantumtheory, but the causal force terms involving GR does alwaysvanish between spacelike seperated points. These two kindsof terms are only connected through a propagation/correlationrelation (the multiparticle generalization of an FDR) when oneparticle is in the others casual future (or past).

These results show that the stochastic fluctuation-forcespreserve the constraint z2 = N2.

III. THE SEMICLASSICAL REGIME: THE

SCALAR-FIELD ALD EQUATION

We have emphasized in Paper I that the emergence of aLangevin equation (2.31 or 2.35) presupposes decoherence,which works to suppress large fluctuations away from themean-trajectories. In the semiclassical limit, the noise-average vanishes, and the semiclassical equations of mo-tion for the single particle are given by the mean of (2.31).The stochastic regime admits noise induced by quantumfluctuations around these decohered mean solutions. Inour second series of papers we explore the stochastic be-havior due to higher-order quantum effects that refurbishthe particles quantum nature.

On a conceptual level, we note that the full quan-

tum theory in the path integral formulation involves sum-ming over all worldlines of the particle6 joining the ini-tial and final spacetime positions, zi and zf, respectively.Furthermore, the generating functional for the worldline-coordinate expectation values (see Subsection 3.A of PaperI [1]) involve a sum over final particle positions zf. In thesepath integrals, there is no distinction between, say, run-away trajectories and any other type of trajectory7. Fur-thermore, no meaningful sense of causality is associatedwith individual (i.e. fine-grained or skeletonized) historiesin the path integral sum. Any particular path in the sumgoing through the intermediate point z () at worldlineparameter time bears no causal relation to it than goingthrough the point z() at some later parameter time .

Where questions of causality, uniqueness, and runawaysdo arise is in regard to the solutions to the equations of mo-tion for correlation function of the worldline-coordinates,particularly for the expectation value giving the mean tra-

jectory. It is here that an initial value formulation forquantum physics is crucial because only then are the equa-tions of motion guaranteed to be real and causal. In con-trast, equations of motion found from the in-out effectiveaction (a transition amplitude formulation) are generallyneither real nor casual [40,41]. Moreover, the equations of

6Actually, just what kinds of histories/worldlines are allowedin the sum is determined by both the gauge choice in the world-line quantization method, and by the boundary conditions.7In fact, trajectories in the path integral can be even stranger

than the runaways that appear in classical theory. Most pathsare non-differentiable (infinitely rough) and may even includethose that travel outside the lightcone and backward-in-time(see previous footnote). Clearly, non-uniqueness of paths isnot an issue in the path integral context.

9


10/21

motion for correlation functions must be unique and fullydetermined by the initial state if the theory is complete.

These observations lead to a reframing of the questionsthat are appropriate in addressing the semiclassical limitof quantum particle-field interactions. Namely, what arethe salient features of the decoherent, coarse-grained histo-

ries which qualify as semiclassical particle motion? Quan-tum theory permits the occurrence of events which wouldbe considered classically improbable or forbidden in theparticle motions (e.g., runaways, tunneling, or apparentlyacausal behavior). These effects are allowed because ofquantum fluctuations. Bearing in mind the requirementof classicality, we need to show that out of the infinite pos-sibilities in the quantum domain the interaction betweenparticle and field (including the self-field of the particle)is causal in the observable semiclassical limit.

So the pertinent questions in a quantum to stochastictreatment are the following: What are the equations ofmotion for the mean and higher-order correlation func-

tions? Are these equations of motion causal and well-defined? How significant are the quantum fluctuationsaround the quantum-averaged trajectory? When does de-coherence suppress the probability of observing large fluc-tuations in the motion? When do the quantum fluctua-tions assume a classical stochastic behavior? It is only byaddressing these questions that the true semiclassical mo-tion may be identified, together with the noise associatedwith quantum fluctuations which is instrumental for deco-herence. With this discussion as our guide, we proceed intwo steps. First, in this section, we find the semiclassicallimit for the equations of motion. Second, in the follow-ing section, we describe the stochastic fluctuations aroundthat limit. Since we are dealing with a nonlinear theory,the fluctuations themselves must depend on the semiclas-sical limit in a self-consistent fashion. These two stepsconstitute the full backreaction problem for nonlinear par-ticle field interactions in the semiclassical and stochasticregimes. For more general conceptual discussions on deco-herent histories and semiclassical domains, see [18,42,43].

A. Divergences, regularized QED, and the initial state

Even in classical field theory, Greens functions are usu-ally divergent, which calls for regularization. Ultravioletdivergences arise because the field contains modes of ar-

bitrarily short wavelengths to which a point particle cancouple. These facts have contributed to considerable con-fusion about the correct form the radiation reaction forcecan take on for a charged particle. There are also in-frared divergences that arise from the artificial distinctionbetween soft and virtual photon emissions. These diver-gences are an artifact of neglecting recoil on the particlemotion.

In this work, we take the effective field theory philos-ophy as our guide. One does not need to know the de-

tailed structure of the correct high-energy theory becauselow-energy processes are largely insensitive to them [22]beyond the effective renormalization of system parame-ters and suppressed high-energy corrections. Take QEDas an example, where relevant physics occurs at the physi-cal energy scale Ephys. In QED calculations, it is generally

necessary to assume a high-energy cutoffEcutoff that reg-ularizes the theory. One also assumes a high-energy scaleEeff well above Ephys where presumably new physics ispossible, thus Ephys


11/21

In the limit ,

lim

GR () = () /2, (3.3)

giving back the unregulated Greens function. Expandingthe function for small s gives

(s) = y (s) y (s) (3.4)

= s2 z4s4/12 +O s5 ,and

GR (s) =2e

4s4/2

23

, (3.5)

assuming a timelike trajectory z () .Now that we have defined a cutoff version of QED, we

briefly reconsider the type of initial state assumed in thisseries of papers. The exact influence functional is generally

found for initially uncorrelated particle(s) and field, wherethe initial density matrix is

(ti) = z (ti) (ti) . (3.6)Such a product state, constructed out of the basis |z,t | (x), is highly excited with respect to the interactingtheorys true ground state. If the theory were not ultra-violet regulated, producing such a state would require in-finite energy. This is not surprising since it is impossibleto take a fully dressed particle state and strip away allarbitrarily-high-energy correlations by some finite energyoperation. With a cutoff, (3.6) is now a finite energy stateand thus physically achievable, though depending on the

cutoff it may still require considerable energy to pre-pare. The important point, which we show below, is thatthe late-time behavior is largely insensitive to the detailsof the initial state, while the early-time behavior is fullycausal and unique.

B. Single-particle scalar-ALD equation

The semiclassical limit is given by the equations of mo-tion for the mean-trajectory. This is just the solutionto the Langevin equation (2.31) without the noise termwhere, as we indicated earlier, is the classical proper-

time such that z2 = N2. The regulated semiclassical equa-tions of motion are then

m0z () V(z) = C()1 (3.7)

= e2i

d

z () + z() z[ () ]

GR (z(), z(

))

where the acts only on the z () in GR , and not thez () term. We add the subscript to m0 because it is

the particles bare mass. This nonlocal integrodifferen-tial equation is manifestly causal by virtue of the re-tarded Greens function. Eq. (3.7) describes the fullnon-Markovian semiclassical particle dynamics, which de-pend on the past particle history for proper-times in therange [, i] . These second order integrodifferential equa-

tions have unique solutions determined by the ordinaryNewtonian (i.e. position and velocity) initial data at ti.

Therefore, the nonlocal equations of motion found from(3.7) do not have the problems associated with the classi-cal Abraham-Lorentz-Dirac equation. But, as an integralequation, (3.7) depends on the entire history of z () be-tween i and . For this reason, the transformation from(3.7) to a local differential equation of motion requiresevery derivative of z () (e.g., dnz () /dn). This is theorigin of higher-time-derivatives in the local equations ofmotion for radiation-reaction (classically as well as semi-classically). By integrating out the field variables, we haveremoved an infinite set of nonlocal (field) degrees of free-

dom in favor of nonlocal kernels whose Taylor expansionsgive higher-derivative terms.We may now transform the integral equation (3.7) to

a local differential equation by expanding the functionsy (s) around s = 0. In Figure 1, we show a semiclassicaltrajectory with respect to the lightcone at z () . Taking as fixed, we change integration variables using ds = d.Next, we need the expansions

y (s) = z () s + z () s2/2 (3.8)+...z

() s3/6 +O s4 ,y (s) = z () + z () s

+...z

() s2/2 +

O s3 ,

(s) = s z2s3/24 +O s4 , (3.9)We set z2 = 1 for convenience since we are concernedwith the semiclassical limit. Also, we use zz = 0, andz2 = z ...z to simplify as necessary. We denote bothdz/d by z and dy/ds by y. Using

(s)

z= 2y (s) (3.10)

d (s)

ds= 2yy = 2s z2s3/2 +O

s4

allows us to write the gradient operator as

=

zd

d= 2y

dds

1 d

ds(3.11)

=y

yy

d

ds.

We also define

r = i (3.12)to be the total elapsed proper-time for the particle sincethe initial time i. Recall that i is defined by z

0 (i) = ti

11


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where ti is the initial time at which the state (ti) =z (ti) (ti) is defined.

These definitions and relations let us write the RHS of(3.7) as

e2 r

0

ds(u(1)

GR

(s) + u(1)

s

2

d

dsGR

(s) (3.13)

+ u(2)s2

6

d

dsGR(s) + ...

+ u(n)sn

(n + 1)!

d

dsGR (s) + ....).

where

u(1) = z () (3.14)

u(2) =

zz2+

...z

...

u(n) = u

(n)

z, z,...,d

n+1

z/dn+1

The functions u(n) depend on higher-derivatives of the par-

ticle coordinates up through order n + 1. We next definethe time-dependent parameters

h (r) =

r0

ds GR (s) = a

1

1/4, 4r4/2

(1/4)

(3.15)

and

g(n) (r) = r

0

dssn

(n + 1)!

d

ds

GR (s) (3.16)

=32(n2)/4

3/2(n + 1)!Ln2

1 +n

4, L4r4/2

,

where (x) is the Gamma function, (x, y) is the incom-plete Gamma function, and (x, y) = (x) (x, y) . Theconstant a = 3 (5/4) /

21/43/2

.41 depends on thedetails of the high-energy cutoff, but is of order one forreasonable cutoffs.

The coefficients are bounded for all r, including the limitr . In fact,

limr

g(n) (r) = limn

g(n) (r) = 0, (3.17)

taking either limit. Therefore, we are free to exchange thesum over n and the integration over s to find a convergentseries expansion of (3.7). The resulting local equations ofmotion are

m0z () V(z) = C()1 (3.18)

= e2h (r) u(1) + e2n=1

g(n) (r) u(n) .

From (3.14) we see that the u(1) terms give time-dependentmass renormalization; we accordingly define the renormal-ized mass as

m (r) = m0 e2h (r) e2g(1) (r)m0 + m (r) . (3.19)

Similarly, the u(2) term is the usual third derivative ra-diation reaction force from the ALD equation. Thus, theequations of motion may be written as

m (r) z () V(z) = fR.R. (r) (3.20)where

fR.R. (r) = e2g(2) (r)

zz

2+...z

+ e2n=3

g(n) (r) u(n) .

(3.21)

These semiclassical equations of motion are almost ofthe Abraham-Lorentz-Dirac form, except for the time-dependent renormalization of the effective mass, the time-dependent (non-Markovian) radiation reaction force, andthe presence of higher than third-time-derivative terms.The mass renormalization is to be expected, indeed it oc-curs in the classical derivation as well. What an initial-value formulation reveals is that renormalization effectsare time-dependent owing to the nonequilibrium dynam-ics of particle-field interactions. The reason why renor-malization is viewed as a time-independent procedure isbecause one usually only deals with equilibrium conditionsor asymptotic in-out scattering problems.

IV. RENORMALIZATION, CAUSALITY, ANDRUNAWAYS

A. Time-dependent mass renormalization

The time-dependence of the effective mass m (r) (andparticularly the radiation reaction coefficients g(n) (r))play crucial roles in the semiclassical behavior. The ini-tial particle at time ti (r = 0) is assumed, by our choiceof initial state, to be fully uncorrelated with the field.This means the particles initial mass is just its bare massm0. Interactions between the particle and field then re-dress the particle state, one of the consequences being

that the particle acquires a cloud of virtual field excita-tions (quanta) that change its effective mass. For a par-ticle coupled to a scalar field there are two types of massrenormalization effects, one coming from the z interac-tion, and the other coming from the zz[] interaction.We commented in Section II.C that the latter interactionis essentially a scalar field version of the electromagneticcoupling, if one defines the scalar field analog of the fieldstrength tensor

Fscalar = z[].

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We find that this interaction contributes a mass shift atlate times of

mz[] = ae2/2. (4.1)

In Paper III it is shown that mz[] is just one half

the mass shift that is found when the particle moves inthe electromagnetic field. This is consistent with the in-terpretation of the electromagnetic field as equivalent totwo scalar fields (one for each polarization); therefore oneexpects, and finds, twice the mass renormalization in theelectromagnetic case. (For the same reason we see belowthat the radiation reaction force, and stochastic noise, areeach also reduced by half compared to the electromagneticcase.)

But the scalar field, unlike the electromagnetic field,also has a z interaction that gives a mass shift mz =2mz[]. The combined mass shift is then negative.At late-times, the renormalized mass is

m limr

m (r)

= m0 + mEM+ mscalar

= m0 + ae2/2 ae2

= m0 ae2/2. (4.2)

For a fixed m0, if the cutoff satisfies > 2m0/ae2, the

renormalized mass becomes negative and the equations ofmotion become unstable. It is well-known that environ-ments with overly large cutoffs can qualitative change thesystem dynamics. In the following we will assume that thebare mass and cutoff are such that m > 0, which in turnimplies that m0 > 0 and gives a bound on the cutoff:

< 2m0/ae2

(which is of the order of the inverse classical electron ra-dius). We emphasize that for the scalar field case the fieldinteraction reduces the effective particle mass, whereas inthe electromagnetic (vector) field case the field interactionincreases the effective particle mass (see paper III [2]). Ifwe followed the FOL [17,2729] suggestion of picking a cut-off = 2m0/ae2 we would then conclude that m = 0 asa consequence of the opposite sign in the mass renormal-ization effect. This is a kind of critical case balancedbetween stable and unstable particle motion, and while

there may be special instances where such behavior is ofinterest, it does not seem to represent the generic behaviorof typical (massive) particles interacting with scalar fielddegrees of freedom. We will defer further comparison withEM quantum Langevin equations to Paper III [2] wherethe mass renormalization has the more familiar sign.

We should further elaborate, however, on the role ofcausality in the equations of motion. In the precedingparagraph we see that for the long-term motion to be sta-ble (i.e. positive effective particle mass) there is an upperbound on the field cutoff. This is well-known, as detailed

in [17,2729], for example. There, the QLE is assumed tobe of the form

mx +

t

dt (t t) x (t) + V (x) = F(t) , (4.3)

where the time integration runs from to t. Causalityis related to the analyticity of (z) (the Fourier trans-form of (t t)) in the upper half plane of the complexvariable z. The traditional QLE of the type (4.3) with thelower integration limit set as ti = effectively makesthe assumption that the particle has been in contact withits environment far longer than the bath memory time. Inour analysis the lower limit in (3.7) is i, not , becauseit is the (nonequilibrium) dynamics at early times immedi-ately after the initial particle/field state is prepared wherewe have something new to say about how causality is pre-served, and runaways are avoided.

The proper-time dependence of the mass renormaliza-tion is shown in Figure 2. The horizontal axis marks theproper time in units of the cutoff timescale 1/. The verti-cal scale is arbitrary, depending on the particle mass. No-tice one unusual feature: the mass shift is not monotonicwith time, but instead first overshoots its final asymp-totic value. This occurs because there are two competingmass renormalizing interactions that have slightly differ-ent timescales. The time-dependent mass-shift is a rapideffect, the final dressed mass is reached within a few 1/.

B. Nonequilibrium radiation reaction

The constants gn (r) determine how quickly the particle

is able to rebuild its own self-field, which in turn controlsthe backreaction on the particle motion, ensuring that it iscausal. We have found, as is typical of effective field the-ories, that the equations of motion involve higher deriva-tive terms (i.e. dnz/dn for n > 3) beyond the usual ALDform. In fact, all higher derivative terms are in principlepresent so long as they respect the fundamental symme-tries (e.g., Lorentz and reparametrization invariance) ofour particle-field model. The timescales and relative con-tributions of these higher-derivative forces are determinedby the coefficients gn (r) . For a fixed cutoff , the late-time behavior of the gn () scale as

gn () =r

2n/2 (1 + n/2)

(2)3/22n (4.4)

Therefore, the g2 term has the late-time limit

g2 () =r

1

4, (4.5)

which is independent of . In the terminology of effectivefield theory, this makes the n = 2 term renormalizable

13


14/21

and implies that it corresponds to a marginal couplingor interaction. In the same terminology the mass renor-malization, scaling as , corresponds to a relevant (renor-malizable) coupling. In this vein, the higher-derivativecorrections proportional to gn>2, suppressed by powers ofthe high-energy scale , correspond to so-called irrelevant

couplings.If we had written the most general action for

the particle-field system consistent with Lorentz andreparametrization invariance, we should have includedfrom the beginning higher-derivative interactions. The ef-fective field theory methodology when carried out fullyshows that these high-derivative interactions are in factirrelevant at low-energies. Therefore, the ordinary ALDform for radiation reaction (e.g., the n = 2 term) resultsnot as a consequence of the details of the true high-energytheory, but because any Lorentz and reparametrizationinvariant effective theory behaves like a renormalizableparticle-field theory that has the ALD equation as the

low-energy limit. With this we come to the importantconclusion that for a wide class of theories with the ap-propriate symmetries the low-energy limit of radiation re-action for (center of mass) particle motion is given by theALD result. For theories with additional structure (suchas spin), there will of course be additional radiation reac-tion terms important at the energy scale associated withthe additional structure.

In Figure 3 we show the proper-time dependence ofg(2) (r), with the same horizontal timescale as in Fig-ure 2. It is evident that the radiation reaction force ap-proaches the ALD form quickly, essentially on the cutofftime. While the radiation reaction force is non-Markovian,the memory time of the field-environment is very short.The non-Markovian evolution is characterized by the de-pendence of the coefficients m (r) and g(2) (r) on the ini-tial time i, but become effectively independent ofi after 1/. This is an example of the well-known behav-ior for quantum Brownian motion models found in [44].The effectively transient nature of the short-time behaviorhelps justify our use of an initially uncorrelated (factoriz-able) particle and field state.

Also notice the role played by the particles elapsedproper-time, r = i. For particles with small aver-age velocities8 the particles proper-time is roughly thebackground Minkowski time coordinate t. In this case,the time-dependences of the renormalization effects are

approximately the same as one would find in the non-relativistic situation, since r = ( i) (t ti).For rapidly moving particles time-dilation can signifi-

8The small velocity approximation is relative to a choice ofreference frame. In paper I (see Appenix A) we discuss howthe choice of initially factorized state at some time ti picks outa special frame.

cantly lengthen the observed timescales with respect tothe background Minkowski time. This indicates thathighly relativistic particles will take longer (with respectto background time t) to equilibrate with the quantumfield/environment than do more slowly moving particles.Finally, we note that the radiation reaction force is exactly

half that found for the electromagnetic field.

C. Causality and early-time behavior

We have shown that at low-energies and late-times theALD radiation reaction force is generic (neglecting otherparticle structure, such as spin, which gives rise to ad-ditional low-energy corrections). But the classical ALDequation is plagued with pathologies like acausal and run-away solutions. The equations of motion in the form (3.7)are clearly unique and causal, they require only ordinaryinitial data, and do not have runaway solutions. However,

it is instructive to see how the equations of motions in theform (3.20,3.21), involving higher derivatives, preserve thecausal nature of the solutions with radiation reaction. Toaddress these questions we now examine the early-timebehavior of the equations of motion.

The presence of higher derivatives in (3.21) is an in-evitable consequence of converting from a set of nonlocalintegral equation (3.7) to a set of local differential equa-tions (3.20,3.21). As differential equations, (3.20,3.21)may seem to be unphysical at first-sight because of

the apparent need to specify initial data z(n) (i) =

dnz (i) /dn for n 2. But the coupling constants gn (r)satisfy the crucial property that

gn (i) = 0, (4.6)

for all n. We see this graphically for g2 (r) in Figure 3,and analytically for all gn (r) in (3.16). Consequently, theparticle self-force at = i identically vanishes (to allorders), and only smoothly rebuilds as the particles self-field is reconstituted. Therefore, the initial data at i isfully (and uniquely) determined by

mz (i) = V (z (i)) , (4.7)

which only requires the ordinary Newtonian initial data.The initial values for the higher derivative terms (e.g. n 2) in (3.21) are determined iteratively from

mdnz (i)

dni= d

n2

dn2iV (z (i)) . (4.8)

Given the (Newtonian) initial data, the equations of mo-tion are determined uniquely for all later times, to anyorder in n. In the classical ALD equations, one finds run-aways even in the case of vanishing external potential,V = 0. In our case, because gn (i) = 0 for all n, if V = 0,

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z(n) (i) = 0 for all n 2. With these initial conditions

(and V = 0), the equations of motion (3.20,3.21) are

z () = 0 (4.9)

with unique solutions

z () = z (i) + ( i) z (i) . (4.10)Runaways do not arise.

As in the classical derivation of the Abraham-Lorentz-Dirac equation, these semiclassical equations of motion areequivalent to a second order differential-integral equation.In the classical case the integral equations are derivedfrom the third-order ALD equation. If one chooses bound-ary conditions for the integral equation that eliminate run-aways, one inevitably introduces pre-accelerated solutions.In our case, we derive the higher-derivative equations ofmotion (3.20,3.21) from the explicitly casual non-local in-tegral equation (3.7). In doing this we avoid introducingunphysical solutions at short times as a consequence of

the initially vanishing time-dependent parameters gn (r) .This turns out to be true for any good regularization ofthe fields Green functions.

V. THE STOCHASTIC REGIME AND THE

ALD-LANGEVIN EQUATION

A. Single particle stochastic limit

The nonlinear Langevin equation in (2.31) shows a com-plex relationship between noise and radiation reaction. Anonlinear Langevin equation similar to these have been

found in stochastic gravity by Hu and Matacz [35] de-scribing the stochastic behavior of the gravitational fieldin response to quantum fluctuations of the stress-energytensor. We now apply the results in Paper I (see Sec-tion IV [1]) giving the linearized Langevin equation forfluctuations around the mean trajectory. In the Langevinequation (2.31) we define z = z + z, where z is thesemiclassical solution from the previous section, and wework to first order in the fluctuation variable z. The freekinetic term is given by

dz()

2Sz [z]

z()z()

= m (r)

d z() g d2

d2( )

= m (r)d2z ()

d2. (5.1)

where we have included the time-dependent mass renor-malization effect in the kinetic term. The external poten-tial term is

dz ()

2V(z)

z()z()

= z ()

2V(z)

z2, (5.2)

hence, the second derivative of V acts as a force linearlycoupled to z. The dissipative term for z involves the (func-tional) derivative of the first cumulant (see Eq. (3.18))with the mass renormalization (n = 1) term removed, as ithas already been included in the kinetic term (5.1) above.We therefore need the (functional) derivative with respect

to z of the radiation reaction term

fR.R. = e2n=2

g(n) (r)

n!u(n) . (5.3)

To O (z) , we have

fR.R. (z ()) =

dz(

)fR.R. (z)

z (). (5.4)

We note that fR.R (z) is a function of the derivatives z(n)

with n > 1, therefore we have

fR.R. (z ()) =n=1

z(n) ()fR.R. (z)

z(n)

(5.5)

= e2m+1n=1

z(n) ()

m=2

g(m) (r)

m!

u(m)

z(n)

,

where fR.R./z(n) are partial derivatives with respect tothe nth derivative of z () . The sum over n terminates at

m+1 since u(m) only contains derivatives up through order

z(m+1). Hence, the Langevin equations for the fluctuationsz (around z) are

mR (t)d2z ()

d2 z ()2V(z)

z2 (5.6)

= () + e2m+1n=1

m=2

g(m) (r)

m!

u(m)

z(n)

z(n) () .

These are solved with z found from (3.20). The noise () is given by

() = w (z ()) = e..

z () +.z.

z[ ]

(z ()) ,

(5.7)

where the lowest order noise is evaluated using the semi-classical solution z.

The correlator () () (see Eq. 5.9) is then foundusing the field correlator along the mean-trajectory:

{(z()), (z ())} = GH (z () , z ()) . (5.8)Equation (5.7) shows that the noise experienced by theparticle depends not just on the stochastic properties ofthe quantum field, but on the mean-solution z () . For

instance, the..z (z) term in (5.7) gives noise that is pro-

portional to the average particle acceleration. The second

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term in (5.7) depends on the antisymmetrized combina-

tion.z[ ](z). It follows immediately from = 0 that

() = 0.From (5.9), we see that the noise correlator is

() (

)

= w() w () (z ()) (z ())=

e2/c4 ..

z +.z .

z[ z]

..z

+.z .

z

[ z

]

GH(z, z)

=

..z +

.z.z[ y]

y2d

d

..z

+.z.z

[ y]

y2d

d

(5.9)

(4crc) GH().

We have defined y z () = z () and = y2. Thenoise (when the field is initially in its vacuum state) ishighly nonlocal, reflecting the highly correlated nature of

the quantum vacuum. Only in the high temperature limitdoes the noise become approximately local. Notice thatthe noise correlator inherits an implicit dependence on theinitial conditions at i through the time-dependent equa-tions of motion for z. But when t 1, the equations ofmotion become effectively independent of the initial time.The noise given by (5.9) is not stationary except for specialcases of the solution z().

Since the n > 2 terms represent high-energy correctionssuppressed by powers of , the low-energy Langevin equa-tions are well approximated by keeping only the n = 2term. The linear dissipation (n = 2) term becomes, using

u(2) = (

.z..z2

+...z),

e23n=1

z(n) ()

g(2) (r)

2!

u(2)

z(n)

(5.10)

=e2

2g(2) (r) (

.z..

z2

g.z...z

+...z

g.z.z

)

=e2

2g(2) (r)

a2h(1)

.z

+h(2)...z

,

where

h(1) g

.z...z (5.11)

h(2)

g.z.z

and a2 =..z2

. This lowest order expression for dissipationin the equations of motion for z involves time derivativesof z through third order. However, just as was the casefor the semiclassical equations of motion, the vanishingof g(2) (r) for r = 0 implies that the initial data for theequations of motion are just the ordinary kind involvingno higher than first order derivatives. The lowest-order

late-time (r >> 1) single particle Langevin equationsare thus

() = m..z () + z

V (z)

zz(5.12)

e2

8

a

2

h

(1)

.

z

+h

(2)

...

z

.

B. Multiparticle stochastic limit and stochastic Ward

identities

It is a straightforward generalization to construct multi-particle Langevin equations. The two additional featuresare particle-particle interactions, and particle-particle cor-relations. The semiclassical limit is modified by the addi-tion of the terms

e2 m=n

f

i

dm wn zn (n))G

R (zn(n), zm(m)) . (5.13)

The use of the regulated G is essential for consistencybetween the radiation that is emitted during the regimeof time-dependent renormalization and radiation-reaction;it ensures agreement between the work done by radiation-reaction and the radiant energy. A field cutoff impliesthat the radiation wave front emitted at i is smoothed ona time scale 1.

We note one significant difference between the singleparticle and multiparticle theories. For a single parti-cle, the dissipation is local (when the field is massless),and the semiclassical limit (obtaining when r 1) isessentially Markovian. The multiparticle theory is non-

Markovian even in the semiclassical limit because of mul-tiparticle interactions. Particle A may indirectly dependon its own past state of motion through the emission ofradiation which interacts with another particle B, which inturn, emits radiation that re-influences particle A at somepoint in the future. The nonlocal field degrees of freedomstores information (in the form of radiation) about theparticles past. Only for a single timelike particle in flatspace without boundaries is this information permanentlylost insofar as the particles future motion is concerned.These well-known facts make the multiparticle behaviorextremely complicated.

We may evaluate the integral in (5.13) using the identity

(zn zm)2

= ( )/2 (zn zm) zm, (5.14)

where is the time when particle m crosses particle nspast lightcone. (The regulated Greens function wouldspread this out over approximately a time 1). Defin-

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ing znm zn zm, we usedm

mG

R (zn, zm) (5.15)

=

dmdGR

ds

(znm)

(znm) zm

=

GR (znm)

(znm)

zm

+

dsGR

d

ds

(znm)

(znm)

zm

=1

2 (znm) zm

d

ds

(znm)

(znm) zm

.

The particle-particle interaction terms are then given by

e2m=n

zn + z z

[]

GR (zn, zm) (5.16)

= e2m=n

zn

[(znm) zm]

+ zn zn

2 (znm) zm

ddm

(znm)

(znm) zm

m=

,

where, as before, the wn (zn) acts only on the zn, and notthe zm () . Eq. (5.16) are just the scalar analog of theLienard-Wiechert forces. They include both the near-fieldand far-field effects.

The long range particle-particle terms in the Langevinequations are found using (2.35), so we have the additionalLangevin term for the nth particle, found from

e2

m=ndnz

nn

..z

n

(znm) .zm(5.17)

h(2)n,

2 (znm) .zm

d

dm

(znm)

(znm) .zm

m=

.

Equation (5.17) contains no third (or higher) derivativeterms. The multiparticle noise correlator is

{n (n) m (m)} = wn (n) wm (m) { (zn) (zm)} .(5.18)

In general, particle-particle correlations between spacelikeseparated points will not vanish as a consequence of thenonlocal correlations implicitly encoded by GH. For in-stance, when there are two oppositely charged particles

which never enter each others causal future, there willbe no particle-particle interactions mediated by GR; how-ever, the particles will still be correlated through the fieldvacuum via GH. This shows that it will not be possibleto find a generalized multiparticle fluctuation-dissipationrelation under all circumstances [39].

We now briefly consider general properties of thestochastic equations of motion. Notice that the noise sat-isfies the identity

zn () n () = zn () wn(zn)(zn) = 0, (5.19)

which follows as a consequence of zn wn = 0, for eachparticle number separately. This is an essential propertysince the particle fluctuations are real, not virtual. Wemay use (5.19) to prove what might be called (by analogywith QED) stochastic Ward-Takahashi identities [45]. Then-point correlation functions for the particle-noise are

{1 (1)...i (i) ...n (n)} (5.20)= w1 (z1) ...wi (zi) ...wn (zn)

{ (z1) ... (zi) ... (zn)} ,

with each wi (zi) acting on the corresponding (zi).These correlation functions may both involve differenttimes along the worldline of one particle (i.e. self-particlenoise), and correlations between different particles. From(5.20) and (5.19), follow Ward identities:

zii 1 (1)...i (i) ...n (n) = 0, for all i and n.(5.21)

The contraction of on-shell momenta (z) with thestochastic correlation functions always vanishes. Theseidentities are fundamental to the consistency of the rela-tivistic Langevin equations.

VI. EXAMPLE: FREE PARTICLES IN THE

SCALAR FIELD VACUUM

As a concrete example, we find the Langevin equationsfor V(z) = 0 and the scalar field initially in the vacuumstate. Using the stochastic equations of motion we canaddress the question of whether a free particle will expe-rience Brownian motion induced by the vacuum fluctua-tions of the scalar field. When V(z) = 0, we immedi-

ately see that for the semiclassical equations,..z

() =constant; the radiation-reaction force identically vanishes(for all orders of n). The fact that gn (0) = 0 uniquelyfixes the constant as zero due to the initial data bound-ary conditions for the higher derivative terms (4.8); andhence, there are no runaway solutions characterized by

a non-zero constant. We may write.z

() = v andz () = sv, where v is a constant spacetime velocityvector which satisfies the (mass-shell) constraint v2 = 1.A particle moving in accordance with the semiclassical so-lution neither radiates nor experiences radiation-reaction9.Using 2 = v2s2 = s2 = ( )2 , we immediately find

9In constast, the mean-solution for a free particle in a quan-tum scalar field found in [46] does not possess the Galilean in-variance of our results, and its motion is furthermore dampedproportionally to its velocity.

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from (2.31) and (5.6) (using a2 =..z2

= 0) the (linear-order)Langevin equation

md2z ()

d2+

e2

2g(2) (r)

g

.z.z ...

z

= () . (6.1)

The noise correlator is found from (5.9) to be

() () (6.2)

= e2

..z d

d

.z.z[ y]

y.y

d2

d

1

4221..z

+d

d

.z.z

[ y]

y.y

d2

d

1

= e2

d

d

v

v[v]s

2vvs2

42s2

1

d

d

v

v[v]s2vvs2

, (6.3)

where we have substituted the vacuum Hadamard func-tion for the field-noise correlator . The antisymmet-ric combination v[v] vanishes, and hence,

() () = 0, for..z

= 0. (6.4)

The origin of this at first surprising result is not hardto find. First, the

..z dependant terms vanish since

the average acceleration is zero, but why should the otherterms in the correlator vanish? From (6.3), we see that the

second term in the particle-noise correlator involves.z .

z[z]G

H(z, z). But the gradient of the vacuum Hadamard

kernel (which is a function of ) satisfies GH()

y,

and is therefore always in the direction of the spacetimevector connecting z () and z () . However, for the iner-tial particle, y v, and thus the antisymmetrized term,.z[ ]G

H() v[v]GH, always vanishes. It thereforefollows that a scalar field with the coupling that we haveassumed does not induce stochastic fluctuations in a freeparticles trajectory. This result is actually somewhat aspecial case for scalar fields, because in Paper III [2] weshow that the electromagnetic field does induce stochasticfluctuations in a free particle. In the case of the electro-magnetic field there is more freedom in FEM = [A](e.g., A is a vector field; is the gradient of a scalarfield) and so one does not find the cancellation that oc-

curs above. For the case of a scalar field, there are noise-fluctuations when the field is not in the vacuum state (e.g.a thermal quantum field), and/or when the particle is sub-

ject to external forces that make its average accelerationnon-zero (e.g. an accelerated particle).

We therefore find that the free particle fluctuations (inthis special case) do not obey a Langevin equation. Theequations of motion for z are

md2z ()

d2+

e2

2g(2) (r)

g

.z.z ...

z

= 0 (6.5)

Eq. (6.1) has the unique solution

..z () = 0 (6.6)

after recalling that g(2) (0) = 0 ...z

(0) = 0. That is, thesame initial time behavior that gives unique (runaway-

free) solutions for z (the semiclassical solution) apply to z(the stochastic fluctuations).

Let us comment on the difference of our result fromthat of [46]. First, the dissipation term in our equationsof motion is relativistically invariant for motion in the vac-uum, and vanishes for inertial motion. The equations ofmotion in [46] are not; the authors concluded that a par-ticle moving through the scalar vacuum will experience adissipation force proportional to its velocity in direct con-tradiction with experience. This result comes from an in-correct treatment of the retarded Greens function in 1+1spacetime dimensions- which is quite different in behaviorthan the Greens function in 3+1 dimensions. We avoid

this problem by working in the more physical three spatialdimensions. We furthermore find that a free particle in thescalar vacuum does not experience noise (at least at theone-loop order of our derivation- see Paper I). This resultis somewhat special owing to the particular symmetries ofthe scalar theory. In Paper III, we find that charged parti-cles experience noise in the electromagnetic field vacuum.We have not addressed in depth the degree of decoherenceof the particle motion, and how well the Langevin equa-tion approximates the quantum expectation values for theparticle motion is still an open question that will dependon the particular parameters (e.g., mass, charge, state ofthe field) of the model.

Our result also differs from that of Ford and OConnell,who propose that the field-cutoff take a special value withthe consequence that a free electron experiences fluctua-tions without dissipation [28]. We have already arguedwhy this assumption is unnecessary as a consequence ofrecognizing QED as an effective theory. We have also em-phasized the distinction between radiation reaction anddissipation. Hence, we find that a free (or for that mat-ter a uniformly accelerated) particle has vanishing aver-age radiation reaction, but when there exists a fluctuatingforce inducing deviations from the free (or uniformly ac-celerated) trajectory, there is then a balancing dissipationbackreaction force. What is a little unusual about thescalar-field-free-particle case is that the overly simple

monopole coupling between the scalar field and particledoes not lead to quantum fluctuation induced noise whenthe expectation value of the particles acceleration vanishes(i.e. for a free particle). In paper III [2], treating the elec-tromagnetic (vector) field, there is quantum field inducednoise and dissipation in the particle trajectory, for bothfree and accelerated motion.

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VII. DISCUSSION

We now summarize the main results of this paper, whatfollows from here, and point out areas for potential appli-cations of theoretical and practical values.

Perhaps the most significant difference between this and

earlier work in terms of approach is the use of a particle-centric and initial value (causal) formulation of relativisticquantum field theory, in terms of world-line quantizationand influence functional formalisms, with focus on thecoarse-grained and stochastic effective actions and theirderived stochastic equations of motion. This is a gen-eral approach whose range of applicability extends fromthe full quantum to the classical regime, and should notbe viewed as an approximation scheme valid only for thesemiclassical. There exists a great variety of physical prob-lems where a particle-centric formulation is more adeptthan a field-centric formulation. The initial value formu-lation with full backreaction ensures the self-consistency

and causal behavior of the equations of motion for thesemiclassical limit of particles in QED whose solutions arepathology free the ALD-Langevin equations.

For further development, in Paper III [2] we shall extendthese results to spinless particles moving in the quantumelectromagnetic field, where we need to deal with the is-sues of gauge invariance. We find that the electromagneticfield is a richer source of noise than the simpler scalar fieldtreated here. The ALD-Langevin equations for chargedparticle motion in the quantum electromagnetic field de-rived in Paper III is of particular importance for quan-tum beam dynamics and heavy-ion physics. Applying theALD-Langevin equations to a charged particle moving in a

constant background electric field (with the quantum fieldin the vacuum state), we show that its motion experiencesstochastic fluctuations that are identical to those experi-enced by a free particle (i.e., where there is no backgroundelectric or magnetic field) moving in a thermal quantumfield. This is an example of the Unruh effect [47]. Thesame effect exists for the scalar field example treated inthis work, but we leave the essentially identical derivationfor the more physically relevant case of QED presented in[2].

In a second series of papers [34], we use the same con-ceptual framework and methodology but go beyond thesemiclassical and stochastic regimes to incorporate the fullrange of quantum phenomena, addressing questions such

as the role of dissipation and correlation in charged parti-cle pair-creation, and other quantum relativistic processes.

In conclusion, our approach is appropriate for any situa-tion where particle motion (as

stochastic theory of relativistic particles moving in a quantum field

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