Accelerated Detector Response

An Unruh–DeWitt detector turns field correlations along one worldline into a transition probability. For eternal uniform acceleration in the Minkowski vacuum, the pulled-back Wightman function is periodic in imaginary proper time and its stationary Fourier transform obeys thermal detailed balance at $T=a_0/(2\pi)$. Finite switching adds nonthermal transients, so a Planckian long-time rate and a finite experiment are not identical claims.

Required background. Special Relativity provides the uniformly accelerated trajectory and proper time.

Helpful background. Flat-Spacetime Quantum Field Theory provides Wightman functions and perturbation theory. Partial Differential Equations and Green Functions supplies distributional pullbacks. Causality and Global Structure develops the acceleration horizon. Quantum Fields in Curved Spacetime provides detector and Hadamard-state conventions.

Detector model and response function

Take a two-level detector with energy gap $E>0$, monopole operator $\mu(\tau)$, smooth switching $\chi(\tau)$, and worldline $x(\tau)$. To first order in a small coupling $\lambda$,

$$H_I(\tau)=\lambda\,\chi(\tau)\mu(\tau)\Phi(x(\tau)).$$

For a pointlike detector in a four-dimensional Hadamard state, the response function is

$$\mathcal F(E) =\int d\tau\,d\tau'\, \chi(\tau)\chi(\tau') e^{-iE(\tau-\tau')} W\!\left(x(\tau),x(\tau')\right).$$

The excitation probability is $\lambda^2|\langle e|\mu(0)|g\rangle|^2\mathcal F(E)$ at this perturbative order. The observable therefore depends on the trajectory, gap, switching, spatial smearing if used, field state, and coupling model. A detector click is not a basis-independent measurement of a pre-existing global particle number.

The pullback of a distribution to a worldline and the sharp-switching limit require care. Louko and Satz 2008, §§2–3 and §4, arXiv version, PDF give a regulator-free formulation for arbitrary four-dimensional Hadamard states and analyze which sharp-switching quantities remain finite.

Wightman function on a hyperbola

For the massless scalar Minkowski vacuum,

$$W_0(x,x') =-\frac{1}{4\pi^2} \frac{1}{(t-t'-i0)^2-|\mathbf x-\mathbf x'|^2}.$$

Pull it back to

$$t(\tau)=a_0^{-1}\sinh(a_0\tau), \qquad x(\tau)=a_0^{-1}\cosh(a_0\tau).$$

Stationarity makes the result depend only on $s=\tau-\tau'$:

$$W_0(s) =-\frac{a_0^2}{16\pi^2} \frac{1}{\sinh^2\!\left[\frac{a_0}{2}(s-i0)\right]}.$$

The poles repeat under $s\mapsto s+2\pi i/a_0$. More than formal periodicity is needed: the open strip $-2\pi/a_0<\operatorname{Im}s<0$ is pole-free, and its two distributional boundary values obey $W(s-i0)=W(-s-2\pi i/a_0+i0)$. These are the proper-time KMS properties. They follow from the hyperbolic invariant separation without introducing Rindler creation operators.

Stationary rate and detailed balance

For smooth long-duration switching, the response per unit proper time approaches

$$\dot{\mathcal F}(E) =\int_{-\infty}^{\infty}ds\,e^{-iEs}W_0(s).$$

Closing the contour in the half-plane selected by the sign of $E$ and summing the repeated poles yields, for excitation,

$$\dot{\mathcal F}(E) =\frac{E}{2\pi}\frac{1}{e^{2\pi E/a_0}-1}, \qquad E>0.$$

De-excitation includes spontaneous emission, but the ratio has the universal form

$$\frac{\dot{\mathcal F}(E)} {\dot{\mathcal F}(-E)} =e^{-2\pi E/a_0}.$$

This detailed-balance relation identifies $T=a_0/(2\pi)$. The normalization has three checks: the exponent is dimensionless; excitation vanishes as $a_0\to0$ at fixed positive gap; and the ratio approaches one in the high-temperature or small-gap limit. The standard derivation appears in Crispino, Higuchi, and Matsas 2008, arXiv version, §§III.A.1–5, pp. 19–25, PDF, after translating their metric signature.

For completeness, the residue sum explains where the Bose denominator enters. For $E>0$ close downward, with clockwise orientation. The double poles lie at $s_n=-2\pi i n/a_0+i0$, $n=1,2,\ldots$; near each pole $W(s)\sim-[4\pi^2(s-s_n)^2]^{-1}$. Therefore the residue of $e^{-iEs}W(s)$ is $iE e^{-iEs_n}/(4\pi^2)$, and

$$\dot{\mathcal F}(E) =\frac{E}{2\pi}\sum_{n=1}^{\infty}e^{-2\pi nE/a_0} =\frac{E}{2\pi(e^{2\pi E/a_0}-1)}.$$

The pole at $s=i0$ lies above the contour and is excluded from excitation. It contributes to de-excitation, accounting for the spontaneous term. This fixes the sign without an appeal to thermal intuition.

Finite-time measurements

With compact switching, time-translation invariance is broken by the apparatus even when the pulled-back state is stationary. The probability contains switching transients and generally is not exactly Planckian. In the four-dimensional model discussed here, the sharply switched total response diverges logarithmically, while a controlled derivative with respect to the detection time can remain finite. A different switching prescription or dimension needs its own limiting analysis.

Long-time thermality is itself a limit with a specified scaling of the switching plateau and waiting time. Fewster, Juárez-Aubry, and Louko 2016, §§4.1–4.3, Proposition 4.3, Theorem 4.6, and Proposition 4.7, arXiv version, PDF connect KMS correlations to detailed balance and analyze long interaction times; their §§5.1–5.3 and Theorems 5.2–5.3 show that accelerated-detector waiting-time behavior depends on how the limit is taken.

Finite-size smearing also changes ultraviolet sensitivity and may introduce orientation dependence. Higher perturbative orders include backreaction on the detector and field. Therefore an experimental prediction must state the switching, smearing, gap range, coupling strength, and duration instead of citing the eternal rate alone.

Interpretation and limitations

The detector calculation agrees with wedge KMS thermality under its stationary hypotheses, but it is a local operational statement. It does not prove that an accelerated source emits a thermal flux to inertial infinity. Nor does every click correspond to a Rindler particle: nonadiabatic switching can excite an inertial detector in vacuum.

The formula above assumes a pointlike detector, a massless scalar, four-dimensional Minkowski vacuum, uniform acceleration, and leading perturbative order. Massive fields, different dimensions, derivative couplings, boundaries, and nonvacuum states alter the spectral density even when a KMS detailed-balance ratio survives.

Exercise

Use the KMS spectral relation $\dot{\mathcal F}(-E)=e^{\beta E}\dot{\mathcal F}(E)$ to determine the late-time ratio of excited- to ground-state populations for a weakly coupled two-level detector.

Solution

In the weak-coupling rate equation, upward transitions occur at rate $\dot{\mathcal F}(E)$ and downward transitions at $\dot{\mathcal F}(-E)$. Stationarity sets $p_g\dot{\mathcal F}(E)=p_e\dot{\mathcal F}(-E)$. Hence $p_e/p_g=e^{-\beta E}$, the Gibbs ratio. For uniform acceleration $\beta=2\pi/a_0$. This conclusion uses the long-time Markovian rate regime; finite switching probabilities need not yield the same population ratio.

The ratio is more robust than the individual rates because field-dependent spectral prefactors cancel. It would fail as an equilibrium inference if the trajectory, state, or switching prevented a stationary late-time limit.

References

• Crispino, Luís C. B., Atsushi Higuchi, and George E. A. Matsas. “The Unruh Effect and Its Applications.” Reviews of Modern Physics 80 (2008): 787–838. DOI. Open PDF.
• Fewster, Christopher J., Benito A. Juárez-Aubry, and Jorma Louko. “Waiting for Unruh.” Classical and Quantum Gravity 33 (2016): 165003. DOI. Open PDF.
• Louko, Jorma, and Alejandro Satz. “Transition Rate of the Unruh–DeWitt Detector in Curved Spacetime.” Classical and Quantum Gravity 25 (2008): 055012. DOI. Open PDF.