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

We adopt a model consisting of a point particle (with a mass $m$ and a charge $e$) and an electromagnetic field $A^{\mu }\left(x\right)$, which are coupled to each other. The action of the model is given by $S_{\text{tot}}=S_{\text{P}}+S_{\text{EM}}+S_{\text{int}}$ with $S_{\text{P}}=-m\int d\tau \sqrt{{\eta }_{\mu \nu }{\dot{z}}^{\mu }{\dot{z}}^{\nu }}$, $S_{\text{EM}}=-\int d^{4}x{F}^{\mu \nu }{F}_{\mu \nu }/4$, and $S_{\text{int}}(z,A^{\mu })=-e\int d\tau \int d^{4}x{\delta }_{\text{D}}^4(x-z(\tau \left)\right){\dot{z}}^{\mu }\left(\tau \right)A_{\mu }\left(x\right)$, where $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is the field strength and ${\eta }_{\mu \nu }$ is the metric of the Minkowski spacetime, for which we follow the convention $(+---)$. The equations of motion are derived from the action as follows:

where we introduce the external force $f_{\mu }$ that accelerates the particle with acceleration $a$ using a uniform electromagnetic field $A_{\text{BG}}$. The solution of Eq. (2.2) is given by

where $A_{\text{h}}^{\mu }$ satisfies ${\partial }_{\nu }{\partial }^{\nu }{A}_{\text{h}}^{\mu }=0$, and $G_{\text{R}}(x,y)$ represents the retarded Green’s function that satisfies ${\partial }_{\mu }{\partial }^{\mu }{G}_{\text{R}}(x,y)={\delta }_{\text{D}}^4(x-y)$. Note that the background part of the electromagnetic field $A_{\text{BG}}\left(x\right)$ does not contribute to the radiation energy flux, but instead, simply uniformly accelerates the particle. We introduce the inhomogeneous solution of the electromagnetic field $A_{\text{inh}}^{\mu }\left(x\right)$ as follows

Decomposing the trajectory of the particle $z_{\mu }=z_{\mu }\left(\tau \right)$ into two parts

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

In the following discussion, we omit the third-order time derivative term of the radiation reaction force $\left(e^2/6\pi \right)\delta {\stackrel{...}{z}}^i$ in (2.8), because the contribution of this term to the perturbation $\delta z^i$ is of the order of $O\left(\right(a/m{)}^2)$ ^***We need to impose the condition $a\ll m$ so as to avoid the boiling vacuum that is due to the Schwinger effect. Therefore, $a/m$ 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 $e^2/m$, which is much smaller than the Compton length $1/m$.

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

where $⟨A_{\text{h}}^{\mu }\left(x\right)A_{\text{h}}^{\nu }\left(y\right)⟩$ represents the Wightman function for vacuum fluctuations of the field and $A_{\text{BG}}^{\mu }$ does not contribute to the energy flux; therefore, we subtracted it from the two-point function. The inhomogeneous solution is written as

with $\rho \left(x_{-}^x\right)\equiv {\dot{z}}_{\mu }\left({\tau }_{-}^x\right)(x^{\mu }-z^{\mu }({\tau }_{-}^x\left)\right)$, where ${\tau }_{-}^x$ satisfies the relation $(x-z({\tau }_{-}^x){)}^2=0$ and the trajectory $z\left({\tau }_{-}^x\right)$ is in the R-region (see the left panel of Fig. 1). Using the expansion in (2.6), we obtain $\rho \left(x\right)={\rho }_0\left(x\right)+\delta \rho \left(x\right)$, where ${\rho }_0$ and $\delta \rho$ are defined as ${\rho }_0\left(x\right)\equiv {\dot{\overline{z}}}_{\mu }\left({\tau }_{-}^x\right)x^{\mu }$ and $\delta \rho \left({\tau }_{-}^x\right)\equiv \delta {\dot{z}}_{\mu }\left({\tau }_{-}^x\right)(x-{\overline{z}}^{\mu }({\tau }_{-}^x\left)\right)$, respectively. Then, we obtain the formula for the inhomogeneous part $A_{\text{inh}}^{\mu }\left(x\right)$

where we define $E_{\mp }^{\mu i}\left(x\right)$ as $E_{\mp }^{\mu i}\left(x\right)\equiv {\eta }^{\mu i}-\frac{{\dot{\overline{z}}}^{\mu }\left({\tau }_{\mp }^x\right)x^i}{{\rho }_0\left(x\right)}$ and ${\tau }_{\pm }^x$ satisfies the relation $(x-\overline{z}({\tau }_{\pm }^x){)}^2=0$ (see the left pane of Fig. 1). Using (2.11), we obtain the formula (see Ref.13 for details)

where the functions $I_1(x,y)$, $I_2(x,y)$ and $I_3(x,y)$ are defined as

for the F-region $x^0>|x^1|$, where the dimensionless parameter $\sigma$ is defined as $\sigma \equiv e^{2}a/6\pi m$. Using the formulas of the energy-momentum tensor $T_{0\mu }=-(A_{\alpha ,0}-A_{0,\alpha })(A_{,\mu }^{\alpha }-A_{\mu }^{,\alpha })$ and the energy flux $f=-{\Sigma }_{i=1}^3{T}_{0i}{n}^i$ $(n^i\equiv x^i/\sqrt{\Sigma x^i{x}^i}\equiv x^i/r)$, we obtain the energy flux expressed as $f\left(x\right)=f^C\left(x\right)+f^Q\left(x\right)$ at a large distance (i.e., $r\to \infty$). $f^C\left(x\right)$ and $f^Q\left(x\right)$ are the classical part (the Larmor radiation) and the quantum part (the Unruh radiation), respectively, defined by

with

where we defined $ϵ=|{\tau }_{-}^x-{\tau }_{-}^y|$ and $P=-\tanh a({\tau }_{+}-{\tau }_{-})/2=q/\sqrt{1+q^2}$ with $q=a(t-r-1/2a^{2}r)/\sin \theta$. Note that $F\left(q\right)$ contains two terms that diverge in the coincidence limit $y\to x$. 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 $aϵ=O\left(1\right).$ The right panel of Fig. 1 plots $G\left(q\right)$ and $F\left(q\right)$ as functions of $q$. We see that $F\left(q\right)$ 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 ().

According the results in Refs.16; 18, in the non-relativistic limit of the velocity of a charged particle, $|v|\ll 1$, and the energy of the quantum Larmor radiation is expressed as the combination of the classical Larmor radiation $E^{\left(0\right)}$ of the zeroth order of $\hslash$, and the quantum part $E^{\left(1\right)}$ of the first order of $\hslash$, $E=E^{\left(0\right)}+E^{\left(1\right)}$ where

where dots indicate differentiation with respect to $t$ or $t^{\prime }$. We may use the relations $t={\overline{z}}^0\left(\tau \right)=a^{-1}\sinh a\tau$, ${\overline{z}}^1\left(\tau \right)=a^{-1}\cosh a\tau$, and $v^1\left(t\right)=\tanh a\tau$, for evaluating the radiation energy. We roughly estimate the order of the radiation rate as

Note that the first-order correction to the Larmor radiation (3.22) is negative and that its amplitude is of the order of $\sim \hslash e^2{a}^3/m$; 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.