Radiation reaction from quantum electrodynamics and its implications for the Unruh effect

The Abraham-Lorentz-Dirac theory predicts vanishing radiation reaction for uniformly accelerated charges. However, since an accelerating observer should detect thermal radiation, the charge should be seen absorbing photons in the accelerated frame which, if nothing else occurs, would influence its motion. This means that either there is radiation reaction seen in an inertial frame or there should be an additional phenomenon seen in the accelerated frame countering the effect of absorption. In this paper I derive the Abraham-Lorentz-Dirac force from quantum electrodynamics, then I study the case of a uniformly accelerated charge. The field of the charge is treated quantum-mechanically. I show that in the accelerated frame, in addition to the absorption of photons due to the Unruh effect there is also stimulated emission. The net effect of these phenomena gives rise to a divergent term which is removed by mass renormalization.


Introduction
Accelerated charges produce electromagnetic radiation that carries off energy according to the Larmor formula for radiated power 1 [1] where e is the electric charge andẍ µ ≡ d 2 x µ (s)/ds 2 is the four-acceleration 2 . If there is a loss of energy, there must also be a recoil force acting on the particle, called radiation reaction, which can be described by the Abraham-Lorentz-Dirac (ALD) force [2] F µ ALD = e 2 6π ...
Since x µ (s) gives the world-line of a massive particle, the four-velocityẋ µ (s) is timelike, in particularẋ µẋ µ = 1 and it is orthogonal to the four-acceleration,ẋ µẍ µ = 0. For a review of the ALD radiation reaction, see Ref. [3].
This description of radiation reaction has some peculiar features. For hyperbolic motion, which is the relativistic generalization of uniform acceleration, the ALD force vanishes. This might suggest the absence of electromagnetic radiation for eternal hyperbolic motion but it has been shown to be non-paradoxical in Refs. [4,5]. In the case of bremsstrahlung, acceleration takes place in a finite time interval and it is the difference of the fields supported by the accelerated and the inertial charge that is released as radiation. Perhaps the easiest way to cope with this is realizing that there can be no clear distinction made between radiation and other field configurations.
In Refs. [6][7][8] it has been shown that a uniformly accelerated observer detects thermal radiation of temperature T U = a/2π, where a = −ẍ µẍ µ is the magnitude of acceleration. This phenomenon became known as the Unruh effect. In the case of an accelerated detector with multiple levels of internal energy, like the Unruh-DeWitt detector, the effect of absorbing photons from the thermal background seen in the accelerated frame can be explained in an inertial frame by the emission of photons [9]. Such an emission may cause recoil, however, for a uniformly accelerated charge the ALD theory predicts no radiation reaction. Thus, either the act of absorption observed in the accelerated frame should be countered by another phenomenon or the ALD formula gives the wrong result.
A problematic trait of the ALD radiation reaction is that it admits unphysical solutions, like preacceleration, i.e. the particle starts accelerating even before any external force is applied. This preacceleration has a characteristic time which in SI units is t 0 = µ 0 e 2 /(6πmc), where m is the mass of the particle. Such violation of causality might not be a serious problem. For an electron t 0 ∼ 10 −24 s, hence, in the case of subatomic particles this unphysical behavior appears in situations when quantum corrections should be applied [10]. Furthermore, Eq. (2) is valid for pointlike charges and it has been shown that when taking finite-size effects into account, there is no violation of causality [11]. This provides some clues as to under what circumstances does the ALD theory give reliable results.
In order to assess what is the full classical (or semiclassical) description of radiation reaction we should turn to quantum electrodynamics (QED) which is the most precise theory of the electromagnetic interaction to date. Its classical limit correctly reproduces Maxwell's equations. Although Dirac's original derivation of the ALD force appearing in Ref. [2] was based on Maxwell's equations, efforts have been made to reconstruct the classical radiation reaction from QED [12][13][14][15][16][17].
In the following, I will rederive the classical radiation reaction for a charged particle. I start in section 2 by constructing an effective action from QED by integrating out the electromagnetic field of the charge. The obtained expression will contain the photon propagator which I will evaluate in section 3. Then, in section 4 I proceed with deriving the classical equation of motion for the charge. Since in an accelerated frame the photon propagator contains a term related to excitations that follow a thermal distribution, I will use the previous results in section 5 to determine how the Unruh effect influences the charged particle. Finally, in section 6 I also point out that the obtained relations are satisfied in the quantized theory by the expectation values of the position and its derivatives.

Integrating over the gauge field
One way of quantizing field theories is using the path integral formalism. In this framework, we construct a generating functional using the action of the classical field theory. The action for the electromagnetic field coupled to a charged field or particle q is where the functional K[q] is the kinetic term and J µ is the charge current for q. If q is a fermion field described by the spinor ψ, these terms will be Of course, in this case K f is a functional of bothψ and ψ.
Let us consider the case when an external electromagnetic field is applied. Then the vector potential in Eq. (3) is replaced by A µ + A µ e , where the first term is the field of the charge and the second term is the external field. Assuming that the charge has little to no effect on the source of the external field, we can keep A µ e fixed and neglect its contribution in the kinetic term −F µν F µν /4. This way the only contribution of the external field will be in the interaction term of the charge.
To deduce how radiation affects the motion of its source we can use the path integral formalism and construct an effective action for the charge by considering the contribution of every configuration of the field generated by the charge. I define the effective action S ef f. [q] that arises from integrating over the field of the charge as where the integral on the right-hand side is a functional integral over the configurations of the gauge field A µ , the normalization constant N does not depend on any field variable and S ef f. [q] is independent of the gauge field A µ . The additional term 1 2λ (∂ µ A µ ) 2 takes care of gauge fixing. In order to obtain the effective action, the kinetic term of the electromagnetic field is cast in a more manageable form, and since the four-divergence will contribute only a surface term to the action, it can be dropped. Combining the obtained expression with the gauge-fixing term we get the equivalence Performing the functional integral of Eq. (5) over the gauge field we get The photon propagator ∆ µν (x, y) appearing in Eq. (8) is the Green's function defined by the equation where the derivatives are taken with respect to x µ .

Photon propagator
The photon propagator can be obtained from Eq. (9) by computing the Fourier- Choosing the Feynman gauge λ = 1, the expression for the photon propagator in position space then becomes The divergent behavior of this expression at x µ = y µ can be regularized using the i0 prescription Choosing ∆ + µν , i.e. the one with the −i0 prescription, we get the purely positivefrequency solution while ∆ − µν with the +i0 gives the purely negative-frequency solution. If we introduce we can write where t = x 0 − y 0 , r denotes the spatial distance between events at x µ and y µ and the two terms in the second line are for cases t > 0 and t < 0 respectively. Thus ∆ + µν takes the absorption of positive-frequency modes into account, while ∆ − µν describes the emission process. For the interaction between the charge and its own field we need to use the difference of the two solutions. According to the Sokhotski-Plemelj theorem we have In the following, I will express the propagator as

Classical equation of motion for a point charge
Considering a point charge with mass m coupled to the electromagnetic field, the kinetic term and the charge current in Eq. (3) takes the form where x µ (s) describes the world-line of the particle, parametrized by the proper time s. Here, K p [x] is a functional of the world-line while J µ p (x) is a function of space-time coordinates. We also need to take into account thatẋ µ (s)ẋ µ (s) = 1 which can be accomplished using the Lagrange multiplier method, adding to the action. The effective action then becomes By using this effective action, we omit the creation of charged particle-antiparticle pairs and the effect of charged loops which places additional limitation on the validity of this theory.
The classical equation of motion can be obtained by finding the extremum of the effective action, thus first we need to compute its variation, that is The first term will provide the equation of motion and the second term takes care of the constraint on the four-velocity. The functional derivative of the action with respect to x µ (s) is while the variation with respect to the Lagrange multiplier yields In Eq. (20), is the field strength tensor for the external electromagnetic field. For a stationary action Eq. (21) gives the constrainṫ x µ (s)ẋ µ (s) = 1 and Eq. (20) yields where I have already set λ p = 0. Note that Eq. (22) provides a classical solution in the sense that the charge and the external field are treated classically. However, it should be remembered that the field of the charge itself is treated quantum-mechanically.
Now we can substitute Eq. (16) for the propagator. Then the next step would be calculating the derivatives of the Dirac delta distribution. At this point I introduce the notation x µ ≡ x µ (s) and y µ ≡ x µ (s ′ ). Then the derivative of the Dirac delta with respect to x µ can be written as where the prime on the delta distribution denotes its derivative with respect to its argument. The equation of motion becomes Using integration by parts to move the derivative from the Dirac delta to the other parts of the integrand we get Since x µ (s) describes the world-line of a massive particle, (x − y) 2 = (x(s) − x(s ′ )) 2 can be zero only if s = s ′ . Thus the equation of motion will be (26) Clearly, the integral over the proper time s ′ is divergent, so let us make this singular behavior explicit by expanding y µ ≡ x µ (s ′ ) in s ′ about s. In order to make the following expressions easier to handle, I will denote ∆s = s ′ − s and Due to the Dirac delta in the integral, all O(∆s) terms of the integrand will vanish and only the pole and the finite part will remain. Thus, as we will see, it is sufficient to expand y µ up to third order in s ′ , Combining these expressions we geṫ Then up to O(∆s) the integrand of Eq. (26) without the Dirac delta becomes Using this result in the equation of motion we obtain The divergent contribution is associated with small time scales or equivalently, high energies. It can be removed by absorbing it into the mass of the particle. Let us define the physical mass of the point charge as This is basically the one-loop renormalized mass of the particle. With this modification, the equation of motion for the point charge takes the form ...
The last term on the right-hand side is exactly the ALD force of Eq. (2).
As I have mentioned before, by using Eq. (17) we omit the creation of charged particle-antiparticle pairs and loop effects caused by charged particles. This means that if the charge is accelerated for proper time 3 ∆τ , the average power of radiation should be much smaller than m R c 2 /∆τ . Substituting this in Eq. (1) and temporarily restoring SI units we get where a 2 is the average squared magnitude of the acceleration. This can be expressed in terms of the average squared change in four-velocity ∆v 2 as t 0 ∆τ ∆v 2 c 2 ≪ 1.

Radiation reaction and the Unruh effect
For hyperbolic motion the squared four-acceleration is constant,ẍ µẍ µ = −a 2 . The world-line of a particle undergoing such motion in a global inertial coordinate frame can be expressed as where s is the proper time and the initial conditions have been chosen such thaṫ x 1 (0) =ẋ 2 (0) =ẋ 3 (0) = 0, x 1 (0) = 1/a and x 2 (0) = x 3 (0) = 0. The coordinates are chosen such thatẍ µ is in the (x 0 , x 1 ) plane. The invariant interval between two points of the world-line is Substituting this into Eq. (12) we get the following expression for the photon propagator where ∆(x, x ′ ) is defined by the final equation and it can be also written as [18] ∆(x, which makes it explicit that an observer undergoing hyperbolic motion is immersed in a thermal bath of photons with temperature T U = a/2π as seen from the accelerated frame. The divergent behavior of this ∆(x, x ′ ), however, must be regularized. Let us introduce the notation Now we can write the propagator for the positive-frequency modes as g µν (∆ + S (x, x ′ ) + ∆ U (x, x ′ )) and for the negative-frequency modes as g µν (∆ − S (x, x ′ ) + ∆ U (x, x ′ )). Keep in mind that this split is done with respect to the frequency measured by an accelerated observer which is not the same as the one measured by an inertial observer. In fact, the modes described as of positive frequency by the accelerated observer are seen by the inertial observer as mixtures of positive and negative frequency modes.
where Γ µ αβ is the Christoffel-symbol, F µ rr is the term that describes radiation reaction and in an inertial frame it takes the form Note that although this expression is manifestly covariant and its indexed constituents behave as tensors under Lorentz-transformations, it is unsuitable for use in an arbitrary coordinate frame. The reason is thatẋ µ (s) andẋ µ (s ′ ) are the tangent vectors of the world-line at different points in space-time and one of them must be subjected to parallel transport before any sensible comparison can take place. This problem can be circumvented by replacing g µν in the expression of the propagator with the tensor P µν (x, x ′ ) for parallel transport between points x and x ′ . This would require the calculation of products of the Christoffel-symbol Γ µ αβ . Alternatively, we can use Eq. (42) to express F µ rr in terms of scalars and constant vectors before changing to a non-inertial frame, The derivative ∂ µ ∆(x, x ′ ) can be computed along the lines of Eq. (23) since ∆(x, x ′ ) is a function of (x(s) − x(s ′ )) 2 , . (44) At this point it is useful to introduce the Rindler coordinates ξ µ which are related to the original Minkowski coordinates by where ξ 1 ∈ ]0, ∞[ and ξ 0 , ξ 2 , ξ 3 ∈ ] − ∞, ∞[ . These coordinates cover only the right Rindler wedge but that is sufficient. In this frame the first two coordinates of the particle are ξ 0 (s) = s and ξ 1 (s) = 1/a and the non-zero elements of the metric are g 00 = (aξ 1 ) 2 , g 11 = g 22 = g 33 = −1.
The non-trivial relations between the different coordinate bases arê wheret µ ,x µ ,η µ andζ µ are the basis vectors adapted for the coordinates x 0 , x 1 , ξ 0 and ξ 1 respectively. The non-zero Christoffel-symbols are Now we can write the vectors in Eq. (44) aṡ x µ (s ′ ) =t µ cosh(as ′ ) +x µ sinh(as ′ ) =η µ cosh(a(s − s ′ )) −ζ µ sinh(a(s − s ′ )), (49) Finally, the term responsible for radiation reaction takes the form Since ∆(x, x ′ ) ultimately depends on the difference (s − s ′ ) we can use d∆/ds = −d∆/ds ′ and we get Following along the lines of section 3 to compute the propagator for the radiation field we can express ∆(x, x ′ ) as where the last two terms cancel but the difference is kept explicit as a reminder. We can also write this expression as In order to interpret this result let us consider a source minimally coupled to a bosonic field. That way the interaction Hamiltonian is just the linear combination of the creation and annihilation operatorsâ † k andâ k with coefficients C + (k) and C − (k), integrated over the phase space. The probability amplitude for emitting an additional particle into a background of n identical particles with wave vector k µ is For n = 0 we get that C + (k) is simply equal to the amplitude A k (0 → 1) of emitting a single particle into the vacuum. If the particles that constitute the background follow a Bose-Einstein distribution with temperature T then to obtain the probability of emitting a particle with frequency k 0 = ω > 0 we need to compute the sum of squared amplitudes weighted with p n = e −nω/T 1 − e −ω/T , which are just Boltzmann factors normalized such that n p n = 1. The probability of emission becomes ∞ n=0 p n dΩ k |A k (n → n + 1) where integration over the spatial direction of k µ has been carried out. Now it is clear that the last term in Eq. (54) is due to emission into a thermal background while the absorption of quanta is taken into account via the first term. The emission of photons by the charge seen in the accelerated frame is stimulated by the Unruh effect.
Let us return to the computation of the radiation reaction in the accelerated frame.
We can use Eq. (53) to calculate F µ rr , which is exactly the self-energy correction for the charge. Considering that in this frameẍ µ (s) = 0 andẋ µ (s) =η µ , we havë and thus the equation of motion can be reduced to where m R is the physical mass of the particle introduced in Eq. (31). This result shows that the Unruh effect also causes stimulated emission and the net influence on the charge will be a divergent term proportional to the acceleration, which can be removed by renormalizing the mass of the particle.
This means that the Unruh effect does not influence the motion of a charged particle. If the accelerated charge has multiple levels of internal energy, like the Unruh-DeWitt detector [8,9] or an atom [19], the inertial observer should see Unruh radiation, which is the spontaneous emission of quanta by the accelerated object. This claim is also supported by Refs. [20,21]. In Ref. [19] it has been shown that the contribution of radiation reaction to the change in energy for both an inertial and a uniformly accelerating atom coupled to a scalar field is the same. This implies that radiation reaction vanishes even for particles that can be excited by the absorption of photons. However, it must be stressed that from the viewpoint of a co-accelerating observer the uniformly accelerated charge undergoes stimulated emission and that is the reason why the Unruh effect does not cause recoil.

Radiation reaction through Ehrenfest's theorem
Although in the classical limit the description based on the effective action given by Eq. (18) yields the ALD formula for radiation reaction, it would be interesting to see whether measurements of a quantum system should yield results that are consistent with the ALD force. Building on the results of section 4, we can use Ehrenfest's theorem to see how radiation reaction manifests on the level of expectation values.
If the system is prepared in state φ at proper time s 1 and found by measurement to be in state χ at proper time s 2 , the transition amplitude can be written as where χ(x 2 ; s 2 ) and φ(x 1 ; s 1 ) are the wave functions for states χ and φ respectively. Now we need to compute how the system evolves between s 1 and s 2 , for which we can use the path integral formalism detailed in Ref. [22]. In this section I repeat Feynman's derivation in a language and notation consistent with the previous sections of the current paper. The evolution of the system is described by where the functional integral is over all paths that start from x 1 at proper time s 1 and end in x 2 at proper time s 2 . If we want to compute the transition amplitude for an arbitrary quantity denoted by F we need to calculate whereF is the operator representation of the quantity over the space of state vectors.
In the path integral formalism F is represented by a functional F [x] and Using this we get δG δx µ (s) If we substitute unity for G, the expression on the left-hand side will vanish and we are left with δS p δx µ (s) = 0 which is exactly Ehrenfest's theorem. This leads to 4 The limitations discussed at the end of section 4 apply here as well, meaning that pair production and the effect of charged loops are omitted and the radiation field has a negligible effect on the source of the external field. Here however, as opposed to previous sections, only the external field is treated classically.

Conclusions
Classical radiation reaction can be obtained from QED and it is exactly the ALD force. Using Ehrenfest's theorem in the form presented in Ref. [22], I also showed that the relationship between the expectation values of particle position, velocity and acceleration (and jerk) yields the same expression for radiation reaction as the ALD theory. The reliability of these results is, however, limited. Firstly, it has been assumed that the effect of the charge on the source of the external electromagnetic field is negligible. Secondly, the creation of charged particle-antiparticle pairs and the contribution of charged loops are omitted.
For eternal hyperbolic motion, it has been shown in Ref. [5] that it is possible to find a closed surface that contains the charge and the net flow of energy through that surface is zero. This brings into question whether the creation of charged particleantiparticle pairs is possible by the electromagnetic field supported by a charge that undergoes perpetual uniform acceleration. The case of bremsstrahlung is different since the charge is accelerated only for a finite time and it is the difference of the field of the accelerated charge and the Coulomb field that is responsible for the radiated energy.
When a charge undergoes eternal hyperbolic motion, in the accelerated frame it is seen to be absorbing photons from the thermal background, which in turn causes the charge to also emit photons. The net effect of absorption and emission gives a divergent term that is removed by renormalizing the mass of the charged particle. These results are consistent with the vanishing radiation reaction obtained in an inertial frame where the particle mass is renormalized in the same manner. Thus, in order to observe the Unruh effect one must use a detector that has multiple levels of internal energy. It can be said, however, that the Unruh effect does not cause recoil.
Finally, I note that the stimulated emission described in this paper should also occur due to Hawking radiation since near the horizon it resembles the Unruh-effect [23].