Particle Detectors versus Field Observables

A detector click is a probe outcome produced by a field–probe interaction. It is operationally meaningful, but it is not generally the value of a unique particle-number observable that existed before the interaction. The response depends on the trajectory, switching, spatial profile, energy gap, initial probe state, and readout. This page makes that dependence explicit by applying inertial and uniformly accelerated probes to the same Minkowski vacuum.

Required background. Localized detector models defines the response function and its apparatus dependence.

Helpful background. Restricted states and subregions explains why local observables are primary even when a global Fock-space factorization is unavailable.

Chapter map. From a local coupling to a field instrument follows the complete measurement chain. The claim-validity table records which hypotheses license each conclusion, while three independent validity questions separates mathematical definition, causal implementation, and empirical interpretation.

The field quantity sampled by a detector

At leading nontrivial order, a ground-state two-level detector measures the positive quadratic functional

$$\mathcal F_{\chi,F,x}(\Omega) =\int d\tau\,d\tau'\, \chi(\tau)\chi(\tau')e^{-i\Omega(\tau-\tau')} W_F\!\left(x(\tau),x(\tau')\right).$$

The excitation probability is $\lambda^2\mathcal F+O(\lambda^4)$. A concrete probe readout induces a field observable, but changing the trajectory $x$, switching $\chi$, smearing $F$, or gap $\Omega$ changes that observable. The general distinction between a measured probe observable and its induced system observable is formalized in Fewster and Verch 2020, § 3.2, Theorems 3.2–3.3, PDF.

For stationary motion and idealized long operation, the pulled-back two-point function depends only on $s=\tau-\tau'$. The transition rate per unit proper time is then

$$\dot{\mathcal F}_x(\Omega) =\int_{-\infty}^{\infty}ds\, e^{-i\Omega s}W_x(s).$$

A detector prepared in its ground state supplies only positive-gap excitation data. Recovering the full signed Fourier spectrum also requires the complementary experiment with an initially excited detector, whose de-excitation samples negative signed energy transfer. If both directions are measured for every positive gap, this particular trajectory-pulled-back correlator distribution is recovered through

$$W_x(s)=\frac1{2\pi} \int_{-\infty}^{\infty}d\Omega\, e^{i\Omega s}\dot{\mathcal F}_x(\Omega).$$

A finite set of gaps and switching windows recovers only filtered combinations. It cannot determine an arbitrary two-point distribution, still less the full field state, without a model class and a stability analysis.

Exact benchmark: one state, two trajectories

Work with a massless real scalar in the Minkowski vacuum. Use the standard pointlike stationary limit: introduce a detector-frame spatial regulator, take its zero-size limit in the regulated response, and only then quote the long-time rate. A detector with finite spatial profile generally carries an additional momentum filter and need not have the rates below. For an inertial detector at rest,

$$x_{\mathrm{in}}(\tau)=(\tau,0,0,0), \qquad W_{\mathrm{in}}(s) =-\frac1{4\pi^2(s-i0)^2}.$$

For uniform proper acceleration $a$,

$$x_a(\tau)= \left( \frac{\sinh(a\tau)}a, \frac{\cosh(a\tau)}a, 0,0 \right),$$

and the pullback is

$$W_a(s) =-\frac{a^2}{16\pi^2} \frac1{\sinh^2[a(s-i0)/2]}.$$

These are two different restrictions of the same Minkowski-vacuum two-point function. Their Fourier transforms are

$$\dot{\mathcal F}_{\mathrm{in}}(\Omega) =-\frac{\Omega}{2\pi}\Theta(-\Omega),$$ $$\dot{\mathcal F}_a(\Omega) =\frac{\Omega}{2\pi\left(e^{2\pi\Omega/a}-1\right)}.$$

The inertial and accelerated pullbacks and the Planckian rate are derived in Sriramkumar and Padmanabhan 1996, § 2, Eqs. (29)–(35), PDF. A detector-frame spatial regulator that preserves stationarity gives the same uniformly accelerated zero-size spectrum; see Schlicht 2004, § 4, Eqs. (19)–(20) and the contour evaluation on pp. 15–16, PDF.

For $\Omega>0$, excitation uses $\dot{\mathcal F}(\Omega)$ and de-excitation uses $\dot{\mathcal F}(-\Omega)$. Set

$$\Omega=1,\qquad a=2\pi.$$

Then

Probe and transition	Rate
Inertial excitation	$0$
Inertial de-excitation	$0.159154943092$
Accelerated excitation	$0.0926244696626$
Accelerated de-excitation	$0.251779412754$

The accelerated detailed-balance ratio is exactly

$$\frac{\dot{\mathcal F}_a(\Omega)} {\dot{\mathcal F}_a(-\Omega)} =e^{-2\pi\Omega/a} =e^{-1} \simeq0.367879441171.$$

The corresponding KMS temperature is $T_U=a/(2\pi)=1$ in these units. All benchmark values are analytic; the displayed decimal uncertainty is only rounding. They are stationary rates, not probabilities accumulated over a finite experimental duration.

The conclusion is precise but narrower than “the vacuum contains particles.” Along the inertial time flow, positive-gap excitation vanishes in the stationary limit. Along the boost-time flow sampled by the accelerated probe, detailed balance is thermal. The field state has not changed; the probe dynamics and the pulled-back field observable have.

Adversarial control: change trajectory or switching

First keep the Minkowski vacuum and $\Omega=1$ but change the acceleration from $a=2\pi$ to $a=\pi$. The detailed-balance ratio changes from $e^{-1}$ to $e^{-2}$. A supposed detector-independent particle count has therefore failed a trajectory-invariance test; the surviving observable statement is the pair of response functions.

Second retain inertial motion but replace eternal operation by the finite-width Gaussian

$$\chi_T(\tau)=e^{-\tau^2/(2T^2)}.$$

For a pointlike massless detector, the leading excitation coefficient is

$$\mathcal F_{\mathrm{in},T}(\Omega) =\frac{ e^{-(\Omega T)^2} -\sqrt\pi\,\Omega T\,\operatorname{erfc}(\Omega T)} {4\pi}.$$

At $\Omega T=1$,

$$\mathcal F_{\mathrm{in},T}=0.00708827223264,$$

although the stationary inertial excitation rate is zero. With $\lambda=0.05$, the leading probability is $1.77206805816\times10^{-5}+O(\lambda^4)$. This is switching-induced spectral broadening, not evidence for an invariant bath of Minkowski particles. Finite-time inertial transients and Gaussian windows are analyzed in Sriramkumar and Padmanabhan 1996, § 3(a), Eqs. (51)–(59), PDF.

Smooth switching and a controlled regulator are essential here. A naive $i\epsilon$ prescription combined with sharp switching can generate nonstationary or noncovariant artifacts. Spatial-profile regularization is developed in Louko and Satz 2006, §§ 2–4, especially Eqs. (2.1)–(2.5), PDF; the smooth-switching analysis and controlled sharp-switching comparison are given separately in Satz 2007, §§ 3–4, especially Eq. (3.8), PDF.

When a particle interpretation is justified

A detector response can approximate mode occupation in a restricted regime:

• the background and state select a stationary or asymptotically stationary positive-frequency decomposition;
• the detector operates long enough to resolve energy;
• its spatial profile and gap are matched to the target wavepacket;
• switching transients, counter-rotating terms, and regulator effects are bounded;
• the same calibration maps response to occupation over the stated bandwidth.

Even then, the statement is “this calibrated instrument estimates occupation in this mode family,” not “every local detector measures the same particles.” Accelerated, cosmological, horizon, and black-hole applications require additional geometric and state analysis and belong to the curved-spacetime treatment.

Common pitfalls

Equating detailed balance with a thermal global state. The KMS relation above concerns the vacuum restricted to the accelerated proper-time flow sampled by the probe. It does not turn the global Minkowski vacuum into a thermal density matrix for inertial modes.

Subtracting the detector model from the result. A response without its trajectory, gap, switching, smearing, and readout is not a fully specified observable.

Confusing a long-time rate with a finite-time probability. The stationary rate assumes a limit in which transients have been divided out. Multiplying it by a short duration does not reproduce an arbitrary switched experiment.

Exercises

1. Pull back the Wightman function

Starting from $W(x,x')=-[4\pi^2((\Delta t-i0)^2-|\Delta\mathbf x|^2)]^{-1}$, derive $W_a(s)$.

Solution

For the accelerated trajectory,

$$(\Delta t)^2-(\Delta x)^2 =\frac4{a^2}\sinh^2\!\left(\frac{as}{2}\right).$$

Inserting the boundary prescription $s\to s-i0$ gives

$$W_a(s) =-\frac{a^2}{16\pi^2} \frac1{\sinh^2[a(s-i0)/2]}.$$

It depends only on $s$, confirming stationarity with respect to proper-time translations.

2. Verify detailed balance

Use the accelerated rate to show $\dot{\mathcal F}_a(\Omega)/\dot{\mathcal F}_a(-\Omega)=e^{-2\pi\Omega/a}$.

Solution

For $\Omega>0$,

$$\dot{\mathcal F}_a(-\Omega) =\frac{-\Omega}{2\pi(e^{-2\pi\Omega/a}-1)} =\frac{\Omega e^{2\pi\Omega/a}} {2\pi(e^{2\pi\Omega/a}-1)}.$$

Dividing the excitation rate by this expression gives $e^{-2\pi\Omega/a}$.

3. Derive the finite Gaussian response

Set $L=0$ in the Gaussian vacuum spectral integral and evaluate it.

Solution

The response is

$$\mathcal F_{\mathrm{in},T} =\frac{T^2}{2\pi} \int_0^\infty dk\,k\,e^{-T^2(k+\Omega)^2}.$$

With $q=T(k+\Omega)$ and $z=\Omega T$,

$$\int_z^\infty dq\,(q-z)e^{-q^2} =\frac12e^{-z^2} -\frac{\sqrt\pi z}{2}\operatorname{erfc}(z).$$

Multiplication by $1/(2\pi)$ yields the stated formula.

4. What can a gap scan reconstruct?

Suppose the detector is stationary and $\dot{\mathcal F}(\Omega)$ is known for every real $\Omega$. What does it determine, and what does it not determine?

Solution

Fourier inversion determines the pulled-back, spatially smeared two-point distribution $W_F(s)$ along that trajectory. It does not determine unsmeared correlations away from the worldline, higher $n$-point functions, or the full state in a non-Gaussian theory. A finite or noisy gap scan determines still less and requires a regularized inverse problem.

References

• Fewster, C. J., and Verch, R. (2020). “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378, 851–889. DOI. Open PDF.
• Louko, J., and Satz, A. (2006). “How Often Does the Unruh–DeWitt Detector Click? Regularisation by a Spatial Profile.” Classical and Quantum Gravity 23, 6321–6344. DOI. Open PDF.
• Satz, A. (2007). “Then Again, How Often Does the Unruh–DeWitt Detector Click If We Switch It Carefully?” Classical and Quantum Gravity 24, 1719–1732. DOI. Open PDF.
• Schlicht, S. (2004). “Considerations on the Unruh Effect: Causality and Regularization.” Classical and Quantum Gravity 21, 4647–4660. DOI. Open PDF.
• Sriramkumar, L., and Padmanabhan, T. (1996). “Response of Finite-Time Particle Detectors in Non-Inertial Frames and Curved Spacetime.” Classical and Quantum Gravity 13, 2061–2079. DOI. Open PDF.