Classical and quantum radiation reaction in conformally flat spacetime

We investigate the physics of a charged scalar particle moving in conformally flat spacetime with the conformal factor depending only on time in the framework of quantum electrodynamics (QED). In particular, we show that the radiation-reaction force derived from QED agrees with the classical counterpart in the limit $\hslash \to 0$ using the fact that to lowest order in $\hslash$ the charged scalar field theory with mass $m$ in conformally flat spacetime with conformal factor $\Omega (t)$, which we call Model B, is equivalent to that in flat spacetime with a time-dependent mass $m\Omega (t)$, which we call Model A, at tree level in this limit. We also consider the one-loop QED corrections to these two models in the semiclassical approximation. We find nonzero one-loop corrections to the mass and Maxwell’s equations in Model A at order ${\hslash }^{-1}$. This does not mean, however, that the corresponding one-loop corrections in Model B are nonzero because the equivalence of these models through a conformal transformation breaks down at one loop. We find that the one-loop corrections vanish in the limit $\hslash \to 0$ in Model B.

Figure 1

The Feynman diagrams at order $e^2$ contributing to the correction to the mass term.

Figure 2

The Feynman diagrams contributing to the correction due to the quadratic interaction term. The cross indicates the “interaction term” $-[(4-D)/2]Q_{\mu \nu }{A}^{\mu }{A}^{\nu }$.