Localized Probe and Detector Models

A localized detector is a controlled probe model, not a definition of particles. It becomes a measurement model only when its Hilbert space, worldtube, switching, smearing, coupling, initial state, approximation order, and readout are all specified. The measurable response is then a functional of field correlation functions and of those apparatus choices.

Required background. Current correlators and response supplies the response-theory viewpoint. Relativistic causality supplies the support constraint. Use the protocol specification on operational locality throughout.

The smeared two-level probe

For a probe with ground state $|g\rangle$, excited state $|e\rangle$, and gap $\Omega>0$, the monopole operator along proper time $\tau$ is

$$\mu(\tau)=e^{i\Omega\tau}\sigma^++e^{-i\Omega\tau}\sigma^-.$$

An extended detector couples through

$$H_I(\tau)=\lambda\chi(\tau)\mu(\tau) \int d^d\boldsymbol\xi\,F(\boldsymbol\xi) \phi\!\left(x(\tau,\boldsymbol\xi)\right),$$

where $x(\tau,\boldsymbol\xi)$ describes the detector worldtube. To second order, the excitation probability from $|g\rangle$ is

$$P_{g\to e}=\lambda^2\mathcal F(\Omega)+O(\lambda^4),$$

with

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

Here $W_F$ is the field Wightman two-point distribution smeared over both spatial profiles. This expression makes the operational content explicit: the detector samples a frequency-filtered, trajectory-pulled-back, spatially averaged correlator. It does not read a universal number operator.

The detector response is determined upstream by the worldtube and switching and downstream by the selected probe readout. These choices must remain attached to any interpretation of a click rate. The diagram is schematic.

Vacuum and a one-particle wavepacket

Fix an inertial trajectory, a smooth switching function, and a normalized profile $F$. For the vacuum, $W_F=W_{0,F}$. For a one-particle wavepacket $|1_h\rangle=a^\dagger(h)|0\rangle$, the two-point function has the form

$$W_{1_h,F}(\tau,\tau')=W_{0,F}(\tau,\tau') +u_h^F(\tau)\overline{u_h^F(\tau')} +\overline{u_h^F(\tau)}u_h^F(\tau'),$$

where $u_h^F$ is the packet mode evaluated along and smeared across the worldtube. Thus the difference in excitation probability is finite once the same apparatus profile is used in both states, and it measures overlap with the detector’s spacetime filter. Poor mode matching can make the one-particle correction small even though the state contains one excitation in the chosen Fock representation.

For a reproducible calculation, report $d$, the field mass, the normalization of $F$, the Fourier convention, $\Omega$, $\lambda$, the support or tail tolerance of $\chi$, and the integration error. Check the $\lambda^2$ scaling and verify $0\le P_{g\to e}\le1$ within the perturbative regime.

Independent shrinking limits

Introduce a temporal scale $T$ and spatial scale $L$ through $\chi_T(\tau)=\chi(\tau/T)$ and $F_L(\boldsymbol\xi)=L^{-d}F(\boldsymbol\xi/L)$. The limits $T\to0$ and $L\to0$ probe different singular structures. Sudden switching broadens the frequency response; pointlike localization removes spatial ultraviolet suppression. Their joint limit need not exist, and the two iterated limits need not agree. The spatial-profile limit is derived in Louko and Satz 2006, §§ 3–5, pp. 6327–6339, while the smooth-switching construction is developed in Satz 2007, §§ 2–4, pp. 1722–1728.

Smooth spatial regularization can also be essential to preserving the expected causal response of an accelerated detector; Schlicht 2004, §§ 2–4, pp. 4649–4658 gives the classic analysis. The broader lesson is model-independent: convergence must be demonstrated in the family of observables actually used, not inferred from a formal delta coupling.

Pointlike and sudden limits enter the ultraviolet branch independently. A stable response under one limit does not license the other, and neither alone validates a particle interpretation. The map is schematic.

Common pitfalls

Calling the model pointlike before taking a limit. A worldline interaction is distributional. Start from a smooth spacetime profile and state the topology or observable in which a pointlike limit is claimed.

Interpreting excitation as pre-existing particles. The excitation probability depends on trajectory, switching, smearing, gap, and initial detector state. Only in controlled stationary or scattering regimes can it agree with an appropriate particle notion.

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.
• Satz, A. (2007). “Then Again, How Often Does the Unruh–DeWitt Detector Click If We Switch It Carefully?” Classical and Quantum Gravity 24, 1719–1731. DOI. Open PDF.
• Schlicht, S. (2004). “Considerations on the Unruh Effect: Causality and Regularization.” Classical and Quantum Gravity 21, 4647–4660. DOI. Open PDF.