Accelerated-Cherenkov radiation and signatures of radiation reaction

In this manuscript we examine an accelerated charged particle moving through an optical medium, and explore the emission of accelerated-Cherenkov radiation. The particle's reaction to acceleration creates a low-frequency spectral cutoff in the Cherenkov emission that has a sharp resonance at the superluminal threshold. Moreover, the effect of recoil on the radiation is incorporated kinematically through the use of an Unruh-DeWitt detector by setting an energy gap, i.e., the change in electron energy, to the recoil momentum of the emitted photon. The simultaneous presence of recoil and acceleration conspire to produce a localized resonance peak in the emission. More generally, we discuss how non-trivial dispersion relations, either in optical media or by effective curved spacetimes, can give rise to the particle recoil by its quantized radiation emission, forming a novel type of radiation reaction phenomenon. These theoretical considerations could be used to construct high precision tests of radiation reaction using Cherenkov emission under acceleration.


I. INTRODUCTION
The Unruh-DeWitt detector [1,2] has found a variety of applications across several fields of physics. Its most famous manifestation, and original use, was in the discovery of the Unruh effect [1]. There, the transition rate of a uniformly accelerated Unruh-DeWitt detector implied that Minkowski vacuum was comprised of a thermalized bath of particles at temperature T = a 2π . In addition to its original use in exploring the particle content of general relativistic space times, it has also played a key role in the exploration of relativistic quantum information [3][4][5][6][7]. This wide ranging applicability is born out of the fact that it enables an exploration of particle systems independent of the major details, i.e. it only cares about the change in energy. This has the effect of taking an otherwise complicated process and rendering it a two level system like an atom with one excited state or a "qubit" of information. By ignoring the fine details and focusing on the particle's quantized recoil due to its radiation, the Unruh-Dewitt detector approach may be able to provide a new perspective on a long standing question: the problem of radiation reaction [8]. Put simply, radiation reaction is the effect that photon emission has on a charged particle's trajectory while it radiates. In turn, this change in the particle's trajectory will then have a back reaction on the emitted photon's energy, e.g. a quantum correction due to recoil momentum. This is particularly exciting since recent theoretical and experimental works involving high intensity lasers and channeling radiation [9][10][11][12][13] have reported the first signatures of radiation reaction which appears to have quantum aspects involved in them [14,15].
Here we develop the Unruh-DeWitt detector formalism for its use in incorporating recoil and apply it to Cherenkov radiation [16]. Moreover, we also include acceleration in the analysis to gain insight into the effects of radiation reaction beyond the conventional models that consider the recoil as a result of Larmor radiation [17]. Fundamentally, this method is based in quantum field theory in curved spacetimes capability to analyze radiation emission [18,19]. By incorporating the techniques used to explore radiation in more exotic spacetimes, we can gain considerable insight into the more simple setting of Minkowski space. To explore vector currents coupled to photons via the QED interaction, we use spacetime trajectories/world lines that are often utilized in quantum field theory in curved spacetime to investigate the emission of radiation [1,[20][21][22][23][24][25][26][27] and apply them to superluminal velocities with acceleration in an optical medium; revealing a novel accelerated-Cherenkov effect. We derive a generalized Frank-Tamm [16,28,29] formula that simultaneously takes into account the quantum recoil due to a single photon emission as well as the acceleration; the inclusion of both effects yields a novel resonant spectral cutoff at the superluminal threshold. We also demonstrate how to incorporate the recoil, e.g. radiation reaction, by using the recoil momenta of the emitted photon as the change in energy of the electron. The preliminary example of Cherenkov radiation is given to outline the functionality of the technique, how to incorporate the recoil, and is then applied to the case where an electron is accelerating in the superluminal regime. The presence of the acceleration sets an energy scale that sharply cuts off the photon emission. This energy scale is quite tunable due to the presence of a strong resonance as the electron velocity approaches the in-medium speed of light. Starting from the cutoff energy, the emitted photon spectra rapidly climbs and asymptotes to the flat Cherenkov emission spectra. The incorporation of the particle recoil due to the emission of a single photon quanta affects the higher frequencies to produce a monotonically decreasing spectra. The combination of both yields a peak in the Cherenkov emission spectrum which may be measurable experimentally [30][31][32]. To finalize, we outline our general approach of incorporating radiation reaction and its interpretation as a recoil momentum in the presence of non-trivial dispersion relations, e.g. in the presence of an index of refraction [17]. Based on the interpretation of an indexed media being analogous to a curved spacetime [33], we conjecture that there are, in general, classes of analogue spacetimes that readily yield non-trivial dispersion relations for the null photon 4-momentum [34] and will therefore cause Cherenkov radiation as well as recoil, subject to certain conditions on the electron's motion. One example is that of the classical picture of radiation reaction, i.e. recoil imparted on a radiating electron accelerated by an electric field. The recoil can be viewed as coming from photon emission with a dispersion relation due to an effective index of refraction produced by the polarization of the vacuum in strong fields [35].
The first section of this paper explores photon emission in QED [36] and its relationship to an Unruh-DeWitt detector by the with the only assumption being that the electron momentum is much bigger than the emitted photon momentum (i.e. the weak recoil approximation). By propagating the detector along an inertial, i.e. non accelerated, world line moving superluminally through an optical medium, we are then able to analyze Cherenkov emission. The overall computational technique is presented and we show how to take into account the quantum recoil of photon emission by setting the Unruh-DeWitt detector energy to the recoil momentum of the photon emission. Then, in Sec. III we retrace the computation except we move the detector along a uniformly accelerated world line. The effects of incorporating acceleration, including the particle recoil due to the interaction with the second-quantized photon field, is then explored. Strong signatures of the acceleration can be tuned by a resonance at the superluminal threshold and its combined effect with recoil produces a peak in the emission spectra. Section IV outlines the details of the overall model of implementing radiation reaction using the recoil momentum produced by non-trivial dispersion. A schematic outline as to how the Unruh-DeWitt detector can be used to explore the total change in energy between the final and initial electron states is presented. Then a kinematic analysis is presented to explain why we take the energy gap to be the recoil momentum. The generalization to photons subject to non-trivial effective space time curvature is also presented. We end the paper discussing how this process can contribute to classical radiation reaction where the recoil momentum is produced by the effective index of refraction due to vacuum polarization. Finally, a summary of conclusions are presented in Sec. V. All calculations are performed using the natural units = c = k B = 1.

II. CHERENKOV EMISSION VIA UNRUH-DEWITT DETECTORS
The emission of radiation produced by inertial currents offers the easiest setup to analyze recoil since the kinematics are purely Minkowskian. This setting provides a rather simple introduction to both radiation emission as well as the recoil. We begin by examining the emission of a photon in a refractive medium,Â µ (x), using the current interaction for QED [36],Ŝ To model the electron current semi-classically we will make use of the weak recoil limit. To see how this is accomplished we begin by recalling that for spinors u(s, p) and v(s, p) of spin s and momentum p that are created bŷ a † s,p andb † s,p , we have the following electron field operatorŝ The positive and negative frequency modes are given by φ p (x, t) and χ p (x, t) respectively. Normally these modes are plane waves, however in more general spacetimes our only requirement is that they are positive and negative frequency modes with respect to the particle's/detector's proper time. Using these fields, we will formulate the electron current, j µ =ψγ µψ . In order to enforce the weak recoil approximation we will ensure that our initial and final Fock states are labeled by their respective energies but have the same momentum, p µ . The legitimacy of this approximation is restricted to the regime where the energy of the emitted radiation is much less than the rest mass of emitting the particle, e.g. ∼ 1 MeV for an electron. We will also neglect any spin effects which are described elsewhere, see e.g. [29]. Focusing strictly on electrons, i.e. no antiparticles, our current then reduces to the followinĝ Now, in keeping with the weak recoil approximation we make use of the Gordon identity [36]. As such, we havē u(p)γ µ u(p) = u µ . We now make use of the fact that our positive and negative frequency mode solutions can be separated into their spatial and temporal components via φ(x, τ ) = g(x)e −iEτ . We have chosen to parametrize our fields via the electron's proper time, τ , to incorporate the Rindler coordinate chart when analyzing the accelerated case [18]. Our current now reduces toĵ We note that for sufficiently localized electronic wave functions, e.g. with a wavelength much smaller than the wavelength of emitted radiation, we have g * (x)g(x) = δ(x − x tr ) along the classical trajectory of the electron; which is assumed to be uniform in the weak recoil limit. This can also be seen by examining a smeared Unruh-DeWitt detector, see e.g. [37]. There, the smearing function which encodes a detector's spatial extent is given by F (x) = −iψ f (x)∇ψ i (x). Applying this formula for the inertial plane waves that are typically used to model electrons in the Cherenkov effect yields F (x) = pg * f (x)g i (x)e i∆Eτ , where p is the momentum of the electron and ∆E = E f −E i . Then, by inspection of Eq. (4), we see that our use of a delta function to model the semi-classical trajectory is equivalent to a detector with a sharply defined spatial extent. It would be interesting indeed to look for finite size effects, i.e. smearing, of an electron in the emission spectrum in processes like Cherenkov radiation. We should also comment on the nature of our current. Here we assume it to be completely uniform, but there has been considerable work detailing how the current itself will shift when taking into account radiation reaction, see e.g. [38][39][40]. We leave the exploration of that to a later work but comment that incorporating the position shift into the current may provide an additional technique to investigate recoil corrections to radiation emission. Finally, by attaching the time-dependence to the creation and annihilation operators we have Here we defined the Heisenberg evolved charge monopole moment operatorq = e iĤτq (0)e −iĤτ whereq(0) is defined asq(0) |E i = |E f with E i and E f the initial energy and final energy of the electron moving along the trajectory, x tr , of the current. The energy gap of the detector is ∆E = E f − E i and the charge q of the electron is given by q = | E f |q(0) |E i |. As such, we have transformed our fermionic current into a semi-classical charged current coupled to an Unruh-DeWitt detector [1,18,[23][24][25] This formalism allows us to examine transitions both up and down in energy. In the case of uniform motion through an indexed medium, transitions up in energy are associated with the anomalous Doppler effect [41]. This extra degree of freedom will allow us to incorporate electronic excitations to higher energy, i.e. the recoil. With the intent to examine Cherenkov radiation, i.e. photon emission from a charge moving faster than the speed of light in medium, we formulate the following amplitude; The square of the this quantity determines the transition probability per momentum of the final state photon. Note, by having a detector we are relieved of integrating over the final electron momentum states. The final momentum is essentially fixed by the detector gap and is tuned directly to the energy change experienced during the recoil. This is equivalent to a delta function density of states, dP The nature of the recoil momentum p r will be explored later on when we define the Unruh-DeWitt detector energy gap. Using the mode decomposition for a massless vector field in a dielectric medium with index of refraction n, we havê Here we have a dispersion relation which relates the frequency to momenta via nω = k [17]. This non-trivial dispersion relation is at the heart of the recoil correction to radiation reaction and we will see how it leads to a nonzero recoil momentum by strict conservation of energy arguments; see section IV below. Constructing the probability from Eq. (II) yields Evaluation of the square of the electron current matrix element yields Next, we shall evaluate the vector field inner product. For this we will need to integrate over the final state momenta, thereby developing the total emission probability. The result is the Wightman function [18] for the photon. Hence, Here we have factored out the completeness relation, dk |k k| = 1 over momentum eigenstates. The resultant two point function, with vector indices, factors into a scalar two point function weighted by the sum over indexed vector polarizations. Evaluation of this two point function, and being sure to keep polarization sum inside the integral, and without evaluating the integral over the momentum yields Combining all the pieces we can formulate the total transition probability. Hence, Note, in the last line we changed our integration variable from proper time to coordinate time via dt = u 0 dτ . Then, by defining the difference and average laboratory times, namely ξ = t − t and η = (t + t)/2 respectively, we can decouple the temporal integrations [26,27]. Note, this parameterization has a unit Jacobian, i.e. dtdt = dξdη. In the presence of trajectories that are linear in time, i.e. unaccelerated, the trajectory will obey the following relation . As a consequence we will have ∆x tr = ∆x tr (ξ). Moreover, with a trajectory that is constant in velocity we will have µ i and u µ (x) independent of time but dependent on the photon emission angle. Noting the resultant expression is therefore independent of the laboratory time η we can formulate the transition rate via dP dη = Γ. As such, We have defined the ξ-dependent Lorentz gamma γ in the last line to boost the proper time difference to the lab frame. We can then define the scalar quantity U = . Moving into momentum space spherical coordinates, with the z-axis aligned along the direction of the velocity, we have Here we used the dispersion relation ω = k/n. Now, for inertial spacetime trajectories under constant velocity with respect to the lab frame we have ∆x tr (ξ) = βξ. As such we can integrate over time along the trajectory yielding Let us examine the scalar quantity U in more detail. Since it is the 4-velocity of the charged particle contracted with the polarization vector we can evaluate it using the properties of the polarization. Since the polarization vector has no zero component and will always be orthogonal to the wave vector of the photon, we find Here we see the sin (θ) rather than cos (θ) comes out of the dot product. This is because the angle θ lies between the charged particle velocity and momentum of the photon. Since the polarization is everywhere perpendicular to the momentum we get cos (π/2 − θ) = sin (θ) from the dot product. As such, our emission rate now becomes Prior to integrating over the angle we note the presence of the delta function which enforces conservation of energy contains the celebrated Cherenkov emission angle deformed by the presence of the energy gap, this is the anomalous Doppler effect [41,42]. Hence, the root yields the Cherenkov angle, If we are examining a recoil from a photon emission we can find an explicit form for the energy gap, see section IV below for further discussion. Considering an initial "dressed" electron with total energy comprised of the electron rest mass and the emitted photons energy E i = √ m 2 + ω 2 and a final state electron energy comprised of a bare electron and recoil momentum k = nω (where n is the frequency-dependent index of refraction) produced by the photon emission, E f = m 2 + (nω) 2 , see Fig. 1 for an illustrative example. The recoil momentum, nω, is characteristic of the non-trivial dispersion relation of the photon. As such, we can determine the energy difference to be From this relation we also obtain the fully relativistic quantum corrected Cherenkov emission angle as computed by Ginzburg [41]. We also note this is consistent with mass renormalization where the electron's energy contains losses on account of it propagating in a material with dispersion [43]. These losses manifest themselves as a frequency dependent change in the electron's mass. Our Cherenkov angle, including the recoil correction, is then given by Next, to integrate over the angles we note the root of the delta function, either given as above in the case of recoil or from Eq. (19) for the more general anomalous Doppler effect. For angles which satisfy this root we define θ cr . We also pick up a factor of ωnβ in the denominator from the Jacobian. As such we obtain We note the prefactor is nothing more than the fine structure constant α = q 2 4π . Moreover, we have now derived the Frank-Tamm formula generalized to the exact quantum result that includes the reaction of the particle to its recoil by the photon emission [29,44]. Hence, To conclude this section, we have outlined the general procedure for the computation of the emission rate of Cherenkov radiation, including recoil, via the use of an Unruh-DeWitt detector. We outlined the general idea for how to incorporate recoil with the Unruh-DeWitt detector; this procedure is discussed in more detail in Section IV.
What is most important is the fact that we are able to successfully incorporate recoil into the emission process. In the following section we will follow the same procedure while placing the current along an accelerated trajectory which is also superluminal. We then explore the emission spectrum of Cherenkov radiation, under acceleration, while simultaneously taking into account recoil and how these inherently quantum mechanical processes alter the emission spectrum.

III. CHERENKOV EMISSION VIA ACCELERATED CURRENTS
Building upon the formalism of the previous section, let us now endeavor to compute the emission spectra by an accelerated electron current in the Cherenkov regime. The overall prescription will be to utilize the kinematics of Rindler space to determine trajectories, velocities, etc. and incorporate their dynamics into the emission. Recalling the photon emission probability, Eq. (13) Note that we are back to integrating over the proper time of the electron and thus reabsorbed our u 0 components. To analyze accelerated Cherenkov emission we integrate over the proper time rather than laboratory time since the accelerated trajectories are parametrized this way. We will begin by examining the polarization vectors that are contracted with our velocity tensor. Recalling that under proper acceleration a, the four-velocities at proper time τ will be given u µ = (cosh (aτ ), 0, 0, sinh (aτ )) [19]. Hence, From here we utilize the same change of variables to the difference and average proper times, ξ and η . Moreover we will make use of the hyperbolic double angle formulas to obtain sinh (aτ ) sinh (aτ ) = 1 2 2 cosh 2 (aη ) − 1 − cosh (aξ ) .
This analysis is simplified using the time-dependent formalism presented in [27]. Combining all the above pieces we can now formulate the emission rate, Γ η = dP dη . Thus Note that with respect to the proper time η , the Lorentz gamma and velocity are given by γ = cosh (aη ) and β = tanh (aη ) respectively. The trajectory is determined by ∆x tr = β∆t. The time difference is given by ∆t = 2 a sinh (aξ /2)γ for uniform acceleration and we will make use of the short time approximation such that ∆t = ξ γ. Now, to examine the momentum integrations we move to spherical coordinates with the momentum aligned along the z-axis. This will yield our emission spectrum dΓ η dω . Hence The integration over proper time can now be evaluated by the use of delta functions. Note, the first term in the square bracket is the standard Cherenkov term and the hyperbolic cosine encodes the acceleration into the emission. Conversion of the acceleration dependent term into exponents and integration yields, Converting the delta functions for the subsequent cos(θ) integration yields a Jacobian of nβωγ for each delta function as well for each of the following three roots; Evaluation of the angular integral then yields By substituting the acceleration dependent angle so that everything is written in terms of the standard Cherenkov angle we find Recalling that the emission rate is parametrized by the electron's proper time, we need to boost back into the lab frame via dΓ η dω = γ dΓ ω . We also simplify the following boost factor via 2γ 2 − 2 = 2β 2 γ 2 . Incorporating these manipulations, we have Care must be taken here since we have the proper acceleration and we need to boost back into the lab frame to have a well formulated emission rate. Recalling the relationship between the laboratory acceleration,ā, and proper acceleration is given by a =āγ 3 we have Note, we have successfully recovered the Frank-Tamm formula with an additional acceleration-dependent term. Now presented with our generalized Cherenkov emission formula, we have the following expressions for the various cases of Cherenkov emission. Hence The above expressions show the acceleration dependence in Cherenkov emission for the standard case, anomalous Doppler effect, and with the recoil correction [29,41]. These are our acceleration-dependent Cherenkov emission rates. Note they reduce to the Frank-Tamm formula when the acceleration vanishes. Here, and throughout the manuscript, we assume a constant index of refraction. Figure 2 below details the general structure of Cherenkov radiation with contributions from both acceleration and recoil. Note that both contributions are required to produce a local maximum. Without acceleration this resonance vanishes. We see that the recoil correction serves to give the Cherenkov spectrum a monotonically decreasing dependence on the frequency. Consequently, a tilted spectra is, modulo other effects such as material dispersion, a signature of recoil in Cherenkov emission. However, with acceleration present in the system we will see that there is clearly a local maximum in the spectrum which depends explicitly on the acceleration. This local maximum requires both acceleration and recoil to be present in the system and therefore may provide a new scenario to explore the phenomena of radiation reaction. To better characterize this process let us note there is a low energy cutoff, i.e. when the emission rate goes to zero, subject to the following condition Without an internal energy level, i.e. no anomalous Doppler effect, we have the following cutoff Here we defined the characteristic energy scale set by the particle's acceleration, ωā =ā γ √ 2β . Note the presence of a resonance at the Cherenkov condition. Moreover, the overall scale of the cutoff frequency from Eq. (37) is specific to having an accelerated particle, e.g. an electron, with characteristic energy ωā in the Cherenkov regime. As with the Unruh effect [1], the energy scale ωā set by acceleration is often vanishingly small. However, the resonance can potentially be used to make this acceleration scale more accessible. It is important to note that the cutoff frequency of ωā = 4.2 × 10 −4 eV used in our figures is based off an acceleration,ā ∼ 1 × 10 20 m/s 2 , that is indeed experimentally realizable in common laser systems with an amplitude of ∼ 1 GV/m. Moreover, table top plasma wakefield accelerators are capable of producing accelerations,ā ∼ 100 × 10 21 m/s 2 [45]. These systems may also provide a setting to explore this cutoff structure of Cherenkov emission. Unlike the above case with no internal energy levels, i.e. with ∆E = 0, the inclusion of an internal energy gap we will yield a quartic equation that determines the cutoff frequency in the presence of radiation reaction and acceleration. As before, the inclusion of recoil is accomplished by setting the energy gap to ∆E = ω 2 (n 2 −1) 2m in the Cherenkov angle. The fact that we will have a quartic cutoff is due to the frequency dependence in the recoil correction. As such, the cutoff frequency is determined by the solution to the following equation, Moreover, the presence of recoil also produces a high frequency cutoff even in the absence of acceleration. From Eq. (35) we can determine the high frequency cutoff due to recoil (in agreement with [29]) to be, We can also compute the peak frequency that is produced by the simultaneous presence of acceleration and recoil. A local maxima in the Cherenkov spectrum in these systems therefore implies both acceleration and recoil. The peak itself is determined by computing the maximum of our generalized Frank-Tamm formula Eq. (35). It is also quartic in nature but can be solved numerically from the following equation, In Fig. 3 we examine the structure of the cutoff frequency produced by both the recoil and the acceleration as an electrons velocity approaches Cherenkov threshold from above for an index of refraction of n = 1.5. The peak resonance that is converged to is given by the solution of Eq. (40) and gives the peak frequency emitted for any superluminal velocity under the assumption of a constant index. As such, we note that by examining the solution to the cubic resonance equation, we can tune the dominant emission frequency by driving both the velocity and acceleration of a Cherenkov system at the superluminal threshold. Moreover, we note that the index of refraction will also depend on frequency. This can also be taken into account by insertion of the frequency dependent index of refraction into the cutoff and peak frequency equation and numerically solving that equation as well. Any vertical slice of Fig. 3 will yield the maximum frequency as well as the high and low frequency cutoffs. The full cutoff curve is determined by Eq. (38). The asymptotic low frequency cutoff produced by the acceleration is determined by Eq. (37). The asymptotic high frequency cutoff produced by the recoil is determined by Eq. (39). The maximum frequency emitted is determined by Eq. (40) and it should be noted that this maximum determines the unique frequency that is emitted at the β threshold and can be used to "tune" the emission. This threshold is also shifted due to the presence of both acceleration and recoil. In addition, we can see from Fig. 3 that radiation emission is suppressed at higher energies due to the recoil. This further serves to ensure that the electron, modeled as an Unruh-DeWitt detector, remains pointlike with a wavelength much smaller than that of the emitted photons. If this were not the case then we would have to take into account smearing of the detector and its modification on the emission spectrum; see e.g. [37] for a QED-based analysis. We should also comment that additional factors such as a strong dependence of the index of refraction on frequency or higher order processes could further modify the spectrum which would undoubtedly give rise to another rich area of investigation. The structure of Cherenkov spectral cutoffs and peak frequency emitted as a function of β near the Cherenkov threshold. The shaded region determines the allowed frequencies of light emitted. The higher frequencies are bounded by the recoil correction while the lower frequencies are bounded by the acceleration energy scale ωā. These bounds converge to the peak frequency at threshold. The asymptotes for the low and high frequency cutoffs are given by Eqs. (37) and (39), respectively. The peak frequency is given by Eq. (40) and the solid curve (both blue and red) of the spectral cutoff is given by Eq. (38). Here, we use the experimentally realizable parameters of ωā = 4.2 × 10 −4 eV and index of refraction n = 1.5.
The ability to tune an electron's velocity with high precision at or near the Cherenkov threshold would imply the ability to control the peak frequency emitted. It is this peak frequency that the low and high energy cutoffs converge to at the threshold limit and will thus determine the only frequency emitted at resonance. The broadness of this peak will be determined by the control of the electron velocity; a sharp velocity distribution in an electron beam will then yield a sharp Cherenkov emission spectrum. As we drive the electrons with specific velocity and acceleration profiles we will get different peak frequencies. Figure 4 shows the various peak frequencies as a function of velocity and acceleration at resonance; We substitute an index of refraction n = 1/β everywhere. Note that for initial electron velocities and accelerations that are currently accessible in electron microscopes we would expect to see a peak frequency in, or near, the lower end of the visible spectrum. To be clear, this analysis assumes a relatively constant index of refraction as a function of frequency in the vicinity of the threshold. More specifically, given a beam velocity β we must then select a material with index of refraction that will place the beam at or near the threshold, i.e. n ∼ 1/β. Then, we must also require that material has an index that is fairly constant with respect to frequency, near the peak, in the vicinity of the threshold. With these conditions met, it stands to reason that a signal of the peak frequency can be measured. We can see from Fig. 4 that for typical electron velocities in electron microscopes β ∼ 0.6 − 0.7 and with laser field amplitudes currently in regular use ∼ 1 GV/m, we should then see a signal between .1 − 1 eV. This signal is well within the measurement capabilities of electron microscopes, e.g. an infrared detector or indirectly via electron energy loss spectroscopy [46]. FIG. 4: The structure of the peak frequency as a function of the particle velocity and local acceleration. Here, the index of refraction is tuned to n ∼ 1 β so we are always near resonance. Note that all velocity and acceleration profiles used are currently accessible experimentally and produce a peak frequency that is also well within current detection capabilities.
For the sake of completeness, we also consider the anomalous Doppler effect [41,42], i.e. an energy gap that does not depend on the emitted photons' frequency, we then have cos θ c = 1 nβ + ∆E nβωγ and a frequency cutoff given by following roots of Eqn. (36); Again, we have a resonance but we see that the cutoff frequency can shift by any internal energy level. We also note the resonances can provide an additional method to probe the anomalous Doppler effect. Note that for both excitations and decays we potentially have two cutoff frequencies. This phenomenon exists both with and without acceleration. In the limit of zero energy gap we arrive back at the Cherenkov cutoff in the presence of acceleration, Eq. (37). The fact we have an acceleration energy scale ωā is also reminiscent of the Unruh effect [1]. There is indeed a close relationship between the anomalous Doppler effect and the Unruh effect [47,48] that has been explored in detail. The results presented here further stress the similarities between these two processes and outline the details of a new effect due to acceleration that is present in Cherenkov radiation.
To conclude this section, we have revealed a new aspect in the theory of Cherenkov emission which incorporates both acceleration as well as recoil. Its high tunability, accompanied by resonance peaks, may be employed as a high precision test of radiation reaction using Cherenkov emission under acceleration since the recoil is easily incorporated. We explored the various ways in which to characterize the radiation reaction, namely the cutoff frequency and local maxima in the spectrum. In the next section we develop the conceptual theory of radiation reaction and its compatibility with the use of Unruh-DeWitt detectors.

IV. RADIATION REACTION FROM NON-TRIVIAL DISPERSION AND VACUUM POLARIZATION
There are a wide variety of systems which exhibit non trivial photon dispersion relations. In general, these systems can be characterized by an effective spacetime metric which, along with null photon normalization, affects the relationship between the photons energy and momentum. For these effective spacetimes we can then define the recoil correction produced by the non trivial dispersion. We noted that the recoil can be taken into account setting the Unruh-DeWitt detector energy gap to the recoil momentum. Let us examine the conceptual foundation of this in further detail. First, let us revisit the duality of the Unruh-DeWitt detector and solutions to wave equations, [26].
Considering a system with a time-like Killing vector such that we can define positive and negative frequency modes, f ω and f * ω , where the energy of the state is given by ω = ±ω(k) with k being a suitable index which labels the states. Consequently, second quantization of, e.g. a scalar field, in this spacetime will produce a second quantized field operator of the formψ = k â k f ω +â † k f * ω , [18]. Positive and negative frequency modes will be eigenvectors of the time-like Killing vector τ which we take to be the proper time of an observer that is instantaneously at rest in the reference frame of the electron at the time of emission. As such, for the positive frequency modes, we have the relation ∂ τ f ω = −iωf ω and the complex conjugate for the negative frequency mode. Therefore, we can write each mode as f ω = g(x)e −iωτ with g(x) being the normalized spatial variation of the mode [1]. The interaction term for a generic photon emission is given byψ fψiφ [36]. Considering theψ field as an electron and focusing on its inner product between initial and final states labeled by the indices κ i and κ f , we find Note, what we are left with is merely the phase difference in energy between the final and initial electron states after the photon emission. The assumption that the spatial wave forms are the same between the initial and final states is known as the "no recoil" condition, i.e. you allow for recoil in the phase but not the spatial variance. Moreover, considering the QED currentψγ µψ [36] which is structurally similar to the scalar interaction we explored above is equivalent to an Unruh-DeWitt detector coupled to a classical vector current [23][24][25]. To see this, we recall the same analysis from Eq. (9) above. Hence Here, we left off the charge dependence to better compare the results. As such, by inspection we see the equivalence between the two techniques. The energy gap of the Unruh-DeWitt detector is merely the difference between the initial and final electron energy eigenstates, as defined by our time-like Killing vector, τ . This general equivalence is outlined in detail in [1,26] and is but a small sample of the utility of the Unruh-DeWitt detector. Let us now consider a purely kinematic analysis of recoil using conservation of energy and momentum. Given an electron energy and momentum, E and p, and photon energy and momentum, ω and k, we have Given the dispersion relations E 2 = p 2 + m 2 and k 2 = n 2 ω 2 [17], we find the Cherenkov condition by solving the above system of equations. Note, this is precisely the conservation of energy condition that is enforced by the Dirac delta function over the detector proper time in Eqs. (16) and (29). Hence This, kinematically speaking, is the Cherenkov angle deformed by the explicit incorporation of recoil into the computation. By inspection of Eqs. (19), (20), and (21) we see that by setting the detector energy gap equal to the difference in electron energy after photon emission, while taking into account the photon energy in the initial state and recoil momentum in the final state, and incorporating non-trivial dispersion we have an energy gap of ∆E = ω 2 (n 2 −1) 2m .
We also note the above analysis also applies to more generic settings; to be explored now. There is an abundance of literature utilizing the fact that indexed media can place a photon in an effective spacetime with metric, g µν with the following line element [33,34]; For these effective spacetimes we see that there may be scenarios where we can further explore these phenomena, and perhaps model a variety of general relativistic phenomena via the clever use of electric fields. For these electromagnetic scenarios, the photon dispersion produced by null normalization, i.e. k µ k µ = 0, will take the following form [18,34], One assumption that is necessary for these specific kinematics is that the electron does not couple to this effective spacetime, i.e. the electron's dispersion relation remains Minkowskian. Examining the solution of Eq. (44), taking into account the above geometric dispersion yields the following general "spacetime Cherenkov recoil". Hence, It could be an interesting avenue of research to examine potential electromagnetic environments which can be recast into this form of geometric Cherenkov radiation. Interestingly enough, this also gives us a rather intuitive insight into the classical problem of radiation reaction. Note, in the absence of an effective index of refraction the recoil vanishes. It is well known that electric fields will not only accelerate an electron but also polarize the vacuum. We can hypothesize that this polarization will create an effective index of refraction which then leads to the recoil during emission. Explicitly considering the relationship between polarizability and index of refraction we may classically express these quantities via [17], hence n 2 = 1 + χ g tt g xx = 1 + χ.
Considering the vacuum polarization from an external electric field, E, and assuming for the sake of simplicity a uniform electric field, we have the susceptibility given by [49][50][51] Here, the critical field [35] is given by E c = m 2 e q ∼ 1.32 × 10 18 V/m, with m e and q the electron mass and charge respectively. Incorporating this effective geometry, and subsequent recoil, produced by the vacuum polarization of the driving electric field may lend insight into the classical problem of radiation reaction. It appears that the vacuum polarization produced by incredibly intense electric fields may, in fact, also be able to produce Cherenkov radiation [52]. Such scenarios may also provide a setting to investigate the effects of acceleration and recoil produced by electric fields at or near the Schwinger limit. To conclude, this section outlined the general formalism for the incorporation of radiation reaction or recoil experienced by an accelerated superluminal electron. In general, there is an energy scale associated with an electric field, or acceleration, which may give rise to non-classical electromagnetic phenomena, e.g. Schwinger pair production and the Unruh effect [53]. At magnitudes of the electric field near the Schwinger limit, strong vacuum polarization is expected to occur. It is near this regime that we expect to see signatures of radiation reaction similarly produced by the recoil momentum enhance by the effective index of refraction. It appears that at least one of the causes of radiation reaction may in fact be the electron's recoil being coupled to this vacuum polarization. Further implications on the effect of radiation reaction from charge renormalization would be interesting to investigate as well. Our work showed that the recoil correction, which is consistent with mass renormalization, manifests itself as a change in the electron's energy due to the recoil momentum imparted on the electron by the photon emission. The specific effect of charge renormalization [36] would not only change the effective charge of the electron but should, in principle, contribute a new term in the energy gap as well. This can be seen from Eqs. (4) and (5). The charge functions as the monopole moment operator in the interaction and any renormalization of the charge may have an explicit presence there. Another potential correction, which is not discussed in this work, is field renormalization. That could provide another promising avenue to investigate sources of radiation reaction. In the end, we have that the utility of the Unruh-DeWitt detector applied to the problem of radiation reaction may be of particular value.

V. CONCLUSIONS
In this manuscript we have made use of an Unruh-DeWitt detector to examine the effect of recoil and acceleration on Cherenkov radiation. The combined presence of recoil and acceleration was then used to gain insight into the paradoxical nature of radiation reaction. We found a resonant energy cutoff resulting from the acceleration and tilted spectra due to the recoil which conspire together to produce a local maximum in the emission spectra. We also introduced a view according to which the phenomenon of radiation reaction is a by product of nontrivial photon dispersion and presented its general properties as well as its geometric interpretation. In light of this analysis, the traditional picture of radiation reaction can be viewed, in part, as the electron's recoil coupling to the vacuum polarization produced by the accelerating electric field. The analysis was carried out via the use of an Unruh-DeWitt detector and we hope this work may open the way to expand its utility and also help develop a better understanding for the phenomenon of radiation reaction.