Particle Detectors versus Field Observables

A detector response is an operational observable built from a field state, a trajectory or worldtube, switching, smearing, internal dynamics, and readout. It can diagnose a particle interpretation in a stationary regime, but it is not generally the expectation value of a unique, observer-independent particle-number operator.

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.

What the detector actually samples

At leading nontrivial order, a two-level detector samples the pulled-back Wightman distribution,

$$P_{g\to e}=\lambda^2\int d\tau\,d\tau'\, \chi(\tau)\chi(\tau')e^{-i\Omega(\tau-\tau')} W_F(\tau,\tau')+O(\lambda^4).$$

The response therefore changes if one changes the worldline, spatial profile, temporal window, energy gap, or initial probe state while holding the field state fixed. The field observable induced by a concrete readout is well defined; the interpretation “number of particles already present” requires additional stationarity, asymptotic, and mode-matching hypotheses.

For inertial motion in the Minkowski vacuum and long smooth operation, the excitation rate of a ground-state detector vanishes in the ideal stationary limit. For uniform acceleration, the same state restricted along the accelerated trajectory satisfies a KMS relation with respect to boost time, producing a thermal detailed-balance ratio. This contrast changes the detector’s dynamical sampling, not the underlying Minkowski state.

The probe trajectory and switching belong to the measurement definition. A field-state claim should therefore be expressed through the induced observable before adding a particle interpretation. The diagram is schematic.

Inertial and accelerated probes in one state

For a stationary pullback, define the response rate

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

Along a uniformly accelerated trajectory with proper acceleration $a$, the Minkowski-vacuum two-point function has imaginary-proper-time periodicity. With controlled regularization, one obtains detailed balance

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

This is a thermal response at temperature $T=a/(2\pi)$ in natural units. It does not imply that an inertial Minkowski number operator has a thermal expectation. Curved-spacetime and horizon applications require further geometric analysis and are outside this chapter.

To reconstruct what is sampled, repeat the experiment over gaps $\Omega$ and switching profiles $\chi$. This probes filtered combinations of $W_F$. A finite family of filters cannot reconstruct an arbitrary two-point distribution, much less the full field state, without a model class and a stability analysis.

Invariance test for a particle claim

Hold the field state fixed and vary the trajectory or switching. If the inferred “particle number” changes, the invariant statement is the collection of detector probabilities, not a unique pre-existing count. In a scattering problem with asymptotic free regions, matched wavepackets, and long interaction times, detector transitions may approximate Fock occupation. State those limiting hypotheses explicitly.

Spatial regularization is especially important for accelerated motion: Schlicht 2004, §§ 2–4, pp. 4649–4658 shows that a naive prescription can produce unphysical time dependence, whereas a detector-frame spatial profile yields the expected stationary response. Louko and Satz 2006, §§ 3–5, pp. 6327–6339 analyze the regulator and transition rate in detail.

Detector-model dependence is not automatically an error; it becomes an error when a response conditioned on a particular apparatus is advertised as a unique field particle number. The map is schematic.

Common pitfalls

Equating thermality with a thermal global state. Detailed balance refers to the chosen detector dynamics and stationary flow. It need not identify the global density operator.

Calling a click local particle detection. A click is a probe outcome caused by an interaction with a field. Its localization and interpretation follow from the full interaction and readout, not from the word “detector.”

References

• 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.
• Schlicht, S. (2004). “Considerations on the Unruh Effect: Causality and Regularization.” Classical and Quantum Gravity 21, 4647–4660. DOI. Open PDF.