Unruh radiation produced by a uniformly accelerating charged particle in thermal random motions

# Unruh radiation produced by a uniformly accelerating charged particle in thermal random motions

Naritaka Oshita, Kazuhiro Yamamoto, and Sen Zhang Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
Department of Physical Science, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Okayama Institute for Quantum Physics, Kyoyama 1-9-1, Kita-ku, Okayama 700-0015, Japan

## I Introduction

Hawking radiation refers to the theoretically predicted black hole emission of thermal radiation with a temperature of , where is the mass of the black hole and is the gravitational constant HawkingRadiation (). The detection of Hawking radiation is extremely difficult because the Hawking temperature of a black hole with solar mass [kg] is of the order of [K], which is much smaller than the temperature of the cosmic microwave background [K]. The Unruh effect refers to the prediction that a uniformly accelerated observer sees the Minkowski vacuum as a thermal bath with a temperature of , where is the acceleration of the observer Unruh ().

The Unruh effect is related to the Hawking radiation via the principle of equivalence. Therefore, any experimental verification of the Unruh effect will be useful as an analogy of the Hawking radiation. P. Chen and T. Tajima pointed out the possibility of testing the Unruh effect using an extremely intense laser ChenTajima (). An electron accelerated by the laser will see the Minkowski vacuum as a thermal bath; then, radiation might be produced via the Unruh effect, which is referred to as Unruh radiation. Using a laser with an intensity of the order of [W/cm] ELI (), the Unruh temperature reaches [K], which is much higher than room temperature. Thus, Unruh radiation is interesting as evidence of the Unruh effect.

Possible signatures of Unruh radiation have been theoretically investigated by several authors. The authors in Refs.3; 5; 6 investigated the Unruh radiation produced by scattering between an accelerated electron and the Rindler particle of the thermal bath originating from the Unruh effect. However, the importance of quantum interference in predicting the Unruh radiation has been pointed out Raine (); Raval (); LinHu (); IYZ13 (); IYZ (); OYZ15 (); we take this interference into account in our theoretical investigation of the signature of the Unruh radiation OYZ16 ().

The paper is organized as follows. In Sec. II, we review our results in Ref.13. Adopting a model consisting of a charged particle and an electromagnetic field, we calculate the energy flux while taking the quantum interference term into account. In Sec. III, we show that the results in Sec. II are consistent with the first-order quantum correction to the Larmor radiation, which was previously obtained using a different approach in Refs.16; 17; 18; 19; 20.

## Ii Quantum radiation produced by an accelerated charged particle

We adopt a model consisting of a point particle (with a mass and a charge ) and an electromagnetic field , which are coupled to each other. The action of the model is given by with , , and , where is the field strength and is the metric of the Minkowski spacetime, for which we follow the convention . The equations of motion are derived from the action as follows:

 m¨zμ=e(∂μAν−∂νAμ)˙zν+fμ(ABG), (2.1) ∂μ∂μAν=e∫dτ˙zν(τ)δ4D(x−z(τ)), (2.2)
 with ∂μAμ=0 (gauge condition), (2.3)

where we introduce the external force that accelerates the particle with acceleration using a uniform electromagnetic field . The solution of Eq. (2.2) is given by

 Aμ(x)=ABGμ(x)+Ahμ(x)+e∫dτGR(x,z(τ))˙zμ(τ), (2.4)

where satisfies , and represents the retarded Green’s function that satisfies . Note that the background part of the electromagnetic field does not contribute to the radiation energy flux, but instead, simply uniformly accelerates the particle. We introduce the inhomogeneous solution of the electromagnetic field as follows

 Aμinh(x)≡e∫dτGR(x,z(τ))˙zμ(τ). (2.5)

Decomposing the trajectory of the particle into two parts

 zμ=¯zμ+δzμ (2.6) ¯zμ=(a−1sinh(aτ),a−1cosh(aτ),0,0), (2.7)

and substituting (2.4) into (2.1), we obtain the linearized equation of (2.1),

 mδ¨zi(τ)=e26π(δ...zi−a2δ˙zi)+e(ηiν˙¯zα−ηiα˙¯zν)∂νAhα(x)|x=z(τ). (2.8)

In the following discussion, we omit the third-order time derivative term of the radiation reaction force in (2.8), because the contribution of this term to the perturbation is of the order of ***We need to impose the condition so as to avoid the boiling vacuum that is due to the Schwinger effect. Therefore, is much smaller than unity., as mentioned in the paper by S. Zhang Zhang:2013ria (). Moreover, this term corresponds to the effect caused by the short-distance dynamics, whose scale is of the order of , which is much smaller than the Compton length .

To obtain the energy flux of the Unruh radiation, we first calculate the two-point correlation function,

 ⟨Aμ(x)Aν(y)⟩−⟨Aμh(x)Aνh(y)⟩−AμBG(x)AνBG(y) =⟨Aμh(x)Aνinh(y)⟩+⟨Aμinh(x)Aνh(y)⟩+⟨Aμinh(x)Aνinh(y)⟩, (2.9)

where represents the Wightman function for vacuum fluctuations of the field and does not contribute to the energy flux; therefore, we subtracted it from the two-point function. The inhomogeneous solution is written as

 Aμinh(x)=e˙zμ(τx−)4πρ(τx−) (2.10)

with , where satisfies the relation and the trajectory is in the R-region (see the left panel of Fig. 1). Using the expansion in (2.6), we obtain , where and are defined as and , respectively. Then, we obtain the formula for the inhomogeneous part

 Aμinh(x)=e4πρ0(x)(˙¯zμ(τx−)−Eμ(−)i(x)δ˙zi(τx−)), (2.11)

where we define as and satisfies the relation (see the left pane of Fig. 1). Using (2.11), we obtain the formula (see Ref.13 for details)

 [⟨Aμh(x)Aνinh(y)⟩+⟨Aμinh(x)Aνh(y)⟩+⟨Aμinh(x)Aνinh(y)⟩]S =(e4π)2˙¯zα(τx−)ρ0(x)˙¯zβ(τx−)ρ0(x)+⎡⎢⎣e4πρ0(x)e4πρ0(y)Eβ(−)i(y)Eαi(+)(x)2mI2(x,y) +e4πρ30(x)e4πρ0(y)Eβ(−)i(y)xiηαAaϵ A′AxA′i2m(I1(x,y)−I3(x,y))] +[(x,α)↔(y,β)], (2.12)

where the functions , and are defined as

 I1(x,y)=−i2πσ+iπlog(1+e−a|τy−−τx+|)+iπa(τy−−τx+)θ(τy−−τx+)+O(σ), (2.13) I2(x,y)=−aπ1ea(τx+)+1+O(σ), (2.14) I3(x,y)=−i2πσ+iπlog(1−e−a|τy−−τx−|)+iπa(τy−−τx−)θ(τy−−τx−)+O(σ), (2.15)

for the F-region , where the dimensionless parameter is defined as . Using the formulas of the energy-momentum tensor and the energy flux , we obtain the energy flux expressed as at a large distance (i.e., ). and are the classical part (the Larmor radiation) and the quantum part (the Unruh radiation), respectively, defined by

 fC=(e4π)2a21r21sin4θG(q),     fQ=(e4π)2a32πm1r21sin4θF(q), (2.16)

with

 G(q)=1−P2(1+q2)2, (2.17) F(q)=1(1+q2)3[6P(2P2−1){logaϵ−log(1+e−a|τ−−τ+|)−a(τ−−τ+)θ(τ−−τ+)}+2P(aϵ)2 +2(3−ea(τ+−τ−))(2−ea(τ+−τ−)(9−ea(τ+−τ−)))(1+ea(τ+−τ−))3], (2.18)

where we defined and with . Note that contains two terms that diverge in the coincidence limit . This divergence derives from the point particle approximation OYZ15 (); OYZ14 (). The approximation can be removed by taking into account the finite-size effect of the particle. Here, for simplicity, we omit the divergent terms. This procedure does not change our conclusions, as long as the cutoff value is of the order of The right panel of Fig. 1 plots and as functions of . We see that is typically negative; thus, the quantum part of the radiation flux is typically negative. However, the total radiation flux of the classical part and the quantum part is positive; therefore, the quantum radiation appears to suppress the total radiation. This property is consistent with the predictions of the model based on the massless scalar field OYZ15 ().

## Iii Quantum correction to the Larmor radiation

The quantum correction to the Larmor radiation has been investigated using quantum field theory for a time dependent backgroundHW (); NSY (); YN (); KNY (); NY (). The field theoretical approach to the quantum radiation from an accelerated charge is different from that described in the previous section. However, the results are consistent with those presented in the previous section. In the previous section, we adopted the unit ; here, we explicitly include .

According the results in Refs.16; 18, in the non-relativistic limit of the velocity of a charged particle, , and the energy of the quantum Larmor radiation is expressed as the combination of the classical Larmor radiation of the zeroth order of , and the quantum part of the first order of , where

 E(0)=e26π∫dt(˙v1(t))2, (3.19) E(1)=e2ℏ6π2m∫dt∫dt′¨v1(t)˙v1(t′)−˙v1(t)¨v1(t′)t−t′, (3.20)

where dots indicate differentiation with respect to or . We may use the relations , , and , for evaluating the radiation energy. We roughly estimate the order of the radiation rate as

 dE(0)dt≃e2a26π, (3.21) dE(1)dt∼−ℏe2a36π2m. (3.22)

Note that the first-order correction to the Larmor radiation (3.22) is negative and that its amplitude is of the order of ; these two statements are consistent with the results of (2.16). Therefore, the quantum radiation obtained in the previous section is related to the quantum correction to the Larmor radiation. The quantum correction to the Larmor radiation (3.20) is non-local in time, which might be useful for understanding the quantum radiation that is due to the Unruh effect.

## Acknowledgements

We would like to thank J. Yokoyama, T. Suyama, and K. Fukushima for helpful discussions. We also thank Prof. P. Chen for critical discussions that took place at the beginning of this study.

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