Stationary Detector Response and Detailed Balance

A stationary detector infers temperature from a ratio, not from a single click rate. If the worldline pullback of the Wightman function is stationary and satisfies the KMS condition at inverse temperature $\beta$, its infinite-time spectral response obeys detailed balance: excitation at positive gap is suppressed relative to de-excitation by $e^{-\beta\Omega}$. This theorem has a fixed-gap long-interaction limit. A finite switching time, or a large-gap limit taken too quickly, need not be thermal.

Required background. Partial Differential Equations and Green Functions supplies distributional Fourier transforms and boundary values of analytic functions. Geodesics and Jacobi Fields provides proper time along the stationary observer’s worldline; no Jacobi-field stability estimate is required.

Helpful background. Flat-Spacetime Quantum Field Theory supplies two-level transition rates. Causality and Global Structure provides stationary Killing regions and horizons. Quantum Fields in Curved Spacetime defines KMS states on the field algebra.

Stationary pullback and spectral response

Let $x(\tau)$ be a detector orbit and let the field state be invariant under the same one-parameter time flow. With the Wightman ordering

$$W(\tau,\tau') =\omega\!\left(\Phi(x(\tau))\Phi(x(\tau'))\right),$$

stationarity means $W(\tau,\tau')=W(s)$ for $s=\tau-\tau'$. Because $W$ is of positive type, its distributional Fourier transform

$$\widehat W(\Omega)=\int_{-\infty}^{\infty}ds\, e^{-i\Omega s}W(s)$$

is a positive, polynomially bounded measure. For the pointwise rate and long-time theorems below, impose the additional hypothesis that this measure has a continuous, nonnegative, polynomially bounded density, also denoted $\widehat W(\Omega)$. Positive $\Omega$ is detector excitation; $-\Omega$ is the reverse transition. A discrete cavity spectrum has delta-function lines and does not satisfy this extra hypothesis: it requires a measure-valued or frequency-smeared formulation rather than evaluation of a rate at an arbitrary single gap.

For real smooth compact switching $\chi$, the leading response can be written spectrally as

$$\mathcal F_\chi(\Omega) =\frac{1}{2\pi}\int_{-\infty}^{\infty}d\nu\, |\widehat\chi(\nu)|^2\widehat W(\Omega+\nu),$$

Here $\widehat\chi(\nu)=\int d\tau\,e^{-i\nu\tau}\chi(\tau)$. For a general spectral measure, the same convolution is interpreted as a pairing with that measure. This formula already shows why finite-time response is not automatically thermal: switching convolves neighbouring frequencies, and the KMS exponential varies across them.

KMS implies detailed balance

Assume $\beta>0$ and that $W$ is the distributional boundary value of a function holomorphic in the strip $-\beta<\operatorname{Im}z<0$, with polynomial bounds on closed substrips. Assume its lower boundary value exists and satisfies

$$W(s-i0)=W(-s-i\beta+i0).$$

These are the stationary two-point KMS hypotheses, not merely imaginary-time periodicity of a formal expression. Under these assumptions,

$$\widehat W(-\Omega)=e^{\beta\Omega}\widehat W(\Omega),$$

or, for $\Omega>0$ where both rates are defined and the denominator is nonzero,

$$\frac{\dot{\mathcal F}(+\Omega)} {\dot{\mathcal F}(-\Omega)}=e^{-\beta\Omega}.$$

This is detailed balance at temperature $T=\beta^{-1}$. Fewster, Juárez-Aubry, and Louko prove the equivalence between the KMS boundary condition and spectral detailed balance for a stationary positive-type distribution with their stated regularity and growth assumptions Fewster, Juárez-Aubry, and Louko 2016, §4.1, Proposition 4.3, arXiv version, PDF.

The proof mechanism fixes the sign. Insert the KMS boundary relation into the Fourier integral and shift the contour from the lower edge of the strip back to the real axis. For positive gap, $e^{-i\Omega(-z-i\beta)}$ contributes $e^{-\beta\Omega}$; polynomial bounds justify the distributional contour argument. Thus excitation is Boltzmann-suppressed relative to de-excitation.

The precise long-interaction theorem

There are two useful smooth switching families. In adiabatic scaling, set

$$\chi_\lambda(\tau)=\chi(\tau/\lambda), \qquad \lambda\to\infty,$$

so the switch-on, plateau structure, and switch-off are all stretched. After division by $\lambda$,

$$\frac{\mathcal F_{\chi_\lambda}(\Omega)}{\lambda} =\frac{1}{2\pi}\int d\nu\, |\widehat\chi(\nu)|^2 \widehat W\!\left(\Omega+\frac{\nu}{\lambda}\right) \longrightarrow \frac{\|\widehat\chi\|_2^2}{2\pi}\widehat W(\Omega)$$

for each fixed $\Omega$. Continuity of the spectral density gives pointwise convergence; its polynomial bound and the rapid Fourier decay of smooth compact switching give an integrable bound. Thus all three assumptions enter. Parseval’s identity gives $\|\widehat\chi\|_2^2/(2\pi)=\int d\tau\,|\chi(\tau)|^2$: dividing by this effective interaction time as well yields the normalized spectral rate.

In plateau scaling, the switch-on and switch-off profiles keep fixed duration while the constant central plateau grows. The response divided by its growing duration again approaches a positive constant times $\widehat W(\Omega)$ for each fixed gap. Under the standing assumptions, both limiting responses satisfy detailed balance if and only if $W$ is KMS Fewster, Juárez-Aubry, and Louko 2016, §4.2, Theorem 4.6, arXiv version, PDF.

The quantifiers are essential:

$$\text{first fix }\Omega, \qquad\text{then take }\lambda\to\infty.$$

The convergence is not uniform in the gap. It does not say that one finite experiment has an exact Planck spectrum, and it does not permit sending $\Omega\to\infty$ first.

Why finite-time thermality can fail

Suppose the continuous spectral density is bounded below by a positive constant on an interval around zero. For a fixed nonzero $C_0^\infty$ switching, the response cannot fall exponentially at large positive gap: exponential Fourier decay of that strength would force the compactly supported switching function to vanish identically. This is the adverse case established in Proposition 4.7 Fewster, Juárez-Aubry, and Louko 2016, §4.3, Proposition 4.7, arXiv version, PDF. Exact thermal detailed balance at large gap requires exponential excitation suppression, so a fixed finite-duration response generally fails the test even when the underlying state is KMS.

The order and shape of the waiting limit also matter. For a uniformly accelerated detector coupled to a massless scalar in four-dimensional Minkowski vacuum, adiabatically scaled switching with sufficiently strong subexponential Fourier decay becomes asymptotically thermal with a power-law waiting time. In the notation of the source, if the decay exponent is $q\in(0,1)$, one may take $\lambda(\Omega)\propto|\Omega|^{1+p}$ with $p>q^{-1}-1$ Fewster, Juárez-Aubry, and Louko 2016, §5.2, Theorem 5.2, arXiv version, PDF. By contrast, plateau scaling with fixed-duration ramps cannot achieve the same large-gap thermality with any polynomially bounded growing plateau Fewster, Juárez-Aubry, and Louko 2016, §5.3, Theorem 5.3, arXiv version, PDF.

Thus “long time” is not a single operation. The detector’s trajectory, gap, switching family, Fourier decay, and order of limits remain part of the theorem.

Acceleration and gravitational redshift

For uniform proper acceleration $a$ in four-dimensional Minkowski vacuum, the stationary pullback is

$$W_a(s)=-\frac{a^2}{16\pi^2} \frac{1}{\sinh^2\!\left[a(s-i0)/2\right]},$$

and its spectrum is

$$\widehat W_a(\Omega) =\frac{\Omega}{2\pi\left(e^{2\pi\Omega/a}-1\right)}.$$

Consequently $\widehat W_a(\Omega)/\widehat W_a(-\Omega)=e^{-2\pi\Omega/a}$, so the long-time detector temperature is $T_U=a/(2\pi)$ Fewster, Juárez-Aubry, and Louko 2016, §5.1, Eqs. (5.2)–(5.3), arXiv version, PDF. Crispino, Higuchi, and Matsas obtain the same proper excitation/de-excitation relation in their detector analysis; their opposite signature has been translated here Crispino, Higuchi, and Matsas 2008, arXiv version, §§III.A.1–5, pp. 19–25, PDF.

More generally, let a stationary detector follow an orbit of a timelike Killing field $K^a$ with constant lapse $N=\sqrt{-K^2}$, so $d\tau=N\,dt$. If the state is KMS at inverse temperature $\beta_K$ with respect to Killing parameter $t$, the proper-time strip has width

$$\beta_{\mathrm{loc}}=N\beta_K, \qquad T_{\mathrm{loc}}=\frac{T_K}{N}.$$

This is the Tolman redshift relation. It applies where $K^a$ is timelike and the detector and state are stationary under that same flow. It does not turn a finite switched response into an exact local thermometer.

Check your understanding

Use the uniformly accelerated spectrum above to compute the excitation/de-excitation ratio for $\Omega>0$. Then explain why replacing the spectral rate by a finite switched response does not preserve the result automatically.

Solution

Substitution gives $\widehat W_a(-\Omega)=e^{2\pi\Omega/a}\widehat W_a(\Omega)$, hence the ratio is $e^{-2\pi\Omega/a}$ and the inferred temperature is $a/(2\pi)$. For finite switching, the response is instead an integral of $\widehat W_a(\Omega+\nu)$ weighted by $|\widehat\chi(\nu)|^2$. Because the Boltzmann factor varies with $\nu$, this convolution does not factor into the same exponential ratio. Detailed balance follows only after a justified long-time limit with the gap and switching family treated in the stated order.

References

• Crispino, Luis 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.