Stochastic analysis of an accelerated charged particle: Transverse fluctuations

An accelerated particle sees the Minkowski vacuum as thermally excited, and the particle moves stochastically due to an interaction with the thermal bath. This interaction fluctuates the particle’s transverse momenta like the Brownian motion in a heat bath. Because of this fluctuating motion, it has been discussed that the accelerated charged particle emits extra radiation (the Unruh radiation [P. Chen and T. Tajima, Phys. Rev. Lett. 83, 256 (1999).]) in addition to the classical Larmor radiation, and experiments are under planning to detect such radiation by using ultrahigh intensity lasers constructed in near future [P. G. Thirolf, D. Habs, A. Henig, D. Jung, D. Kiefer, C. Lang, J. Schreiber1, C. Maia, G. Schaller, R. Schutzhold, and T. Tajima, Eur. Phys. J. D 55, 379 (2009).][http://www.extreme-light-infrastructure.eu/.]. There are, however, counterarguments that the radiation is canceled by an interference effect between the vacuum fluctuation and the fluctuating motion. In fact, in the case of an internal detector where the Heisenberg equation of motion can be solved exactly, there is no additional radiation after the thermalization is completed [D. J. Raine, D. W. Sciama, and P. G. Grove, Proc. R. Soc. Lond. A 435, 205 (1991).][A. Raval, B. L. Hu, and J. Anglin, Phys. Rev. D 53, 7003 (1996).]. In this paper, we revisit the issue in the case of an accelerated charged particle in the scalar-field analog of QED. We prove the equipartition theorem of transverse momenta by investigating a stochastic motion of the particle, and show that the Unruh radiation is partially canceled by an interference effect.

Figure 1

Stochastic trajectories induced by quantum field fluctuations.

Figure 2

The hyperbolic line in the right wedge denotes the world line of the particle. The points $O_{\mathrm{F}}$ and $O_{\mathrm{R}}$ are observers in the future and right wedges, respectively. For an observer in the right wedge, the light-cone of the observer has two intersections with the world line, and the proper time of the intersections is given by ${\tau }_{\pm }^R$. For an observer in the future wedge, there is only one intersection on the particle’s real trajectory which corresponds to ${\tau }_{-}^F$. The other solution $T_{+}^{\mathrm{F}}={\tau }_{+}^{\mathrm{F}}+i\pi /a$ is complex. One may interpret this complex proper time as the intersection between the light-cone of the observer and the world line of a virtual particle with a real proper time ${\tau }_{+}^{\mathrm{F}}$ in the left wedge. The superscript letters $R$ or $F$ are used to distinguish two different observers, but we do not use them in the body of the paper to leave the space for the observer’s position $x$.

Figure 3

$T1$ is a trajectory of the original accelerated particle. If the particle is kicked at $P1$ to a point $P2$, it follows a different trajectory $T2$ in the constant external force. It has a different asymptote from the original one, and the original Rindler coordinate $\xi$ of the particle on $T2$ diverges at $\tau =\infty$.