Back-action from inertial and non-inertial Unruh-DeWitt detectors revisited in covariant perturbation theory

TL;DR

<3-5 sentence high-level summary> The paper presents a covariant, second-order perturbative analysis of back-action from a spatially pointlike Unruh–DeWitt detector on a quantum scalar field, focusing on the renormalised stress-energy tensor in Minkowski spacetime without conditioning on detector measurements. It shows a clean decomposition of detector back-action into deterministic and fluctuating parts, with the latter dominating for energy-eigenstate detectors; a massless field in the Minkowski vacuum is used to derive a fully local SET that remains well-defined away from the detector. For inertial motion, the energy flux from the field matches the detector’s de-excitation energy, while for uniform acceleration the fluxes observed in Minkowski, Rindler, and Milne frames reproduce the Unruh-energy exchange, including novel negative energy regions near horizons. The work underlines the causal, local character of back-action and discusses regularisation, quantum energy inequalities, and avenues for extending the framework to curved spacetimes and smeared detectors.

Abstract

We investigate the back-action from a spatially pointlike particle detector on a quantum scalar field, as characterised by the expectation value of the field's stress-energy tensor, without conditioning on a measurement of the detector. First, assuming the field to be initially in a zero-mean Gaussian Hadamard state in a globally hyperbolic spacetime, we evaluate the field's two-point function in second-order perturbation theory by techniques of covariant curved spacetime quantum field theory, which allow a full control of the time and space localisation of the interaction, and do not rely on field mode decompositions or non-local particle countings. The detector's two-point function splits into a deterministic and a fluctuating part, and we show that this split is maintained in the back-action. We then specialise to a two-level Unruh-DeWitt detector, prepared in an energy eigenstate, for which the back-action is fully fluctuating. We compute the renormalised stress-energy tensor for a massless scalar field in $(3+1)$-dimensional Minkowski spacetime for a general detector trajectory, using the manifestly causal two-point function. We present explicit analytic and numerical results for an inertial detector and a uniformly linearly accelerated detector, switched on in the asymptotic past. The energy flux into and out of the accelerated detector accounts exactly for the energy gained and lost by the detector in its transitions due to the Unruh effect. The same holds for the outward flux associated with de-excitations of the inertial detector, which has a vanishing excitation rate and no inward flux. A novelty with the accelerated detector is two regions of negative energy density when the detector is initially prepared in its ground state, one near the Rindler horizon that bounds the causal future of the trajectory, the other in the far future of the trajectory.

Figures (11)

• Figure 1: A spacetime diagram showing the detector's trajectory $\mathsf{Z}(\tau)$ alongside a spacetime point $\mathsf{x}$ that is not on the trajectory. The point $\mathsf{Z}(\tau_-)$ is the latest point on the trajectory that intersects the past null cone of $\mathsf{x}$; any earlier intersections, as may exist for example in spatially compact spacetimes, are beyond the spacetime region shown. Similarly, the point $\mathsf{Z}(\tau_+)$ is the earliest point on the trajectory that intersects the future null cone of $\mathsf{x}$; any later intersections are beyond the spacetime region shown. For some trajectories and some $\mathsf{x}$, $\mathsf{Z}(\tau_-)$ and/or $\mathsf{Z}(\tau_+)$ do not exist; in this case we understand $\tau_-$ as $-\infty$ and $\tau_+$ as $\infty$. Note that $\mathsf{Z}(\tau)$ is spacelike separated from $\mathsf{x}$ if and only if $\tau_- < \tau < \tau_+$.
• Figure 2: The energy density $\braket{T_{tt}}^{(2)}$ of the field for an inertial detector in the long-time limit \ref{['eq:inertial-T_(tt)']}, as a function of the radial distance $r$ and the energy gap $E$, in arbitrary units. Clearly seen are the different fall-off rates in $r$ for a detector initially in its ground state ($E > 0$) and a detector initially in its excited state ($E < 0$). Also seen is the expected property that larger de-excitation gaps emit more energy into the field. An $r^{-4}$ singularity, which is independent of the energy gap, is seen close to $r=0$.
• Figure 3: The energy flux $\braket{T_{tr}}^{(2)}$ of the field for an inertial detector in the long-time limit \ref{['eq:Ttr-inertial-final']}. $\braket{T_{tr}}^{(2)}$ is nonvanishing only for a detector initially in its excited state ($E < 0$), and it is continuous across $E=0$. Its magnitude is a multiple $E^2/r^2$, and its negative sign indicates an energy flux away from the detector.
• Figure 4: A spacetime diagram of the Rindler trajectory \ref{['eq:linear-acc-trajectory']}, labelled here $\mathsf{Z}(\tau)$, in the $(t, z)$ plane. The lightlike coordinates are $u = t - z$ and $v = t + z$. The left and right Rindler wedges are labelled respectively as $L$ and $R$, and the future and past wedges are labelled respectively as $F$ and $P$. The causal future of the trajectory is at $v>0$. The future branch of the Rindler horizon is at $u=0$, $v>0$.
• Figure 5: $a^{-4}\xi^{-2}\braket{T_{\eta\eta}}^{(2)}$ as a function of $a\xi$ and $E/a$, obtained from \ref{['eq:rindler-T_(eta-eta)']}. At the detector's trajectory, $a\xi \to 1$, there is divergence to $\infty$, with the asymptotics \ref{['eq:Tetaeta-norm-trajectory-as']}. At $\xi\to0$, there is divergence to $-\mathop{\mathrm{sgn}}\nolimits(E)\infty$, with the asymptotics \ref{['eq:Tetaeta-norm-horizon-as']}. The curve where $\braket{T_{\eta\eta}}^{(2)}$ passes through zero is shown in white. At $a\xi\to\infty$, there is a $\xi^{-6}$ falloff \ref{['eq:Tetaeta-norm-infty-as']}, with a positive coefficient.