Switching, Smearing, and Detector Regularization

Switching and smearing are part of a detector observable, not disposable regulators. They determine causal support, frequency selectivity, ultraviolet suppression, and the order in which pointlike or instantaneous idealizations may be taken. A credible response calculation therefore studies a family of smooth test functions and reports which limit, if any, is stable.

Required background. Point splitting and Wick products explains local ultraviolet singularities. Test functions and distributional support supplies the correct pairing with fields. The response model is defined on localized detectors.

Helpful background. Zero modes, boundaries, and infrared effects helps separate long-distance sensitivity from switching ultraviolet noise.

Response as a distributional pairing

For a stationary trajectory, write the pulled-back Wightman distribution as $W_F(s)$ after spatial smearing. The response is

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

In Fourier variables, schematically,

$$\mathcal F(\Omega)=\int\frac{d\omega}{2\pi}\, \left|\widehat\chi(\omega)\right|^2 \widehat W_F(\Omega+\omega).$$

A $C_0^\infty$ switching has rapidly decreasing Fourier transform, while a sharp step has a slow tail. Similarly, a smooth spatial profile suppresses large momentum, whereas a spatial delta distribution does not. These facts predict which numerical domains require resolution before any integral is evaluated.

Switching and smearing enter at the coupling stage and propagate into every later probability and state update. They cannot be changed while calling the resulting instrument the same physical measurement. The diagram is schematic.

Gaussian and compact profiles

A Gaussian family is convenient for analysis,

$$\chi_T(\tau)=e^{-\tau^2/(2T^2)}, \qquad F_L(\mathbf x)=\frac{e^{-\mathbf x^2/(2L^2)}}{(2\pi L^2)^{d/2}},$$

but has nonzero tails. A compact bump such as

$$b_T(\tau)=C_T \begin{cases} \exp\!\left[-\dfrac{1}{1-(\tau/T)^2}\right],&|\tau|<T,\\ 0,&|\tau|\ge T, \end{cases}$$

supports exact causal-separation statements. Normalize profiles according to the experimental quantity held fixed—integrated coupling, peak amplitude, or $L^2$ norm—because these normalizations scale differently as $T$ or $L$ changes.

For the Minkowski vacuum of a free scalar, compute the same $\mathcal F(\Omega)$ using both profiles. A useful convergence study varies the time grid, momentum cutoff, and integration domain independently, compares against a known stationary-rate or contour result where available, and plots the residual against each control parameter. Agreement of two quadratures using the same unresolved cutoff is not an independent check.

Noncommuting idealizations

Define $\mathcal F_{T,L}$. The questions

$$\lim_{T\to0}\lim_{L\to0}\mathcal F_{T,L}, \qquad \lim_{L\to0}\lim_{T\to0}\mathcal F_{T,L},$$

are mathematically different. Sudden switching can generate a boundary divergence even with finite spatial size; pointlike localization can expose the coincident singularity even with smooth temporal switching. If either iterated limit diverges or the two differ, report finite-$T$, finite-$L$ predictions rather than a regulator-independent number.

The careful smooth-switching analysis of Satz 2007, §§ 2–4, pp. 1722–1728 shows how otherwise problematic instantaneous transition rates can be replaced by finite, covariant quantities. Spatial-profile regularization and the pointlike limit are analyzed by Louko and Satz 2006, §§ 3–5, pp. 6327–6339. The causal regulator for stationary acceleration in Schlicht 2004, §§ 2–4, pp. 4649–4658 shows why a formally pointlike prescription can give an unphysical response.

Ultraviolet response, causal support, and inference have separate checks. A numerical value stabilized at fixed cutoff can still fail the regulator branch if the pointlike or sudden family has not converged. The map is schematic.

A minimal error record

For each reported response, give the profile formulas and normalization; support or tail tolerance; field state and $i0$ prescription; integration variables and cutoffs; estimated quadrature, truncation, and perturbative errors; and the sequence in which limits were taken. The strongest defensible claim is then tied to a finite instrument or to a demonstrated limit, not to a formal delta function.

Exercises

Show that fixing $\int d\tau\,\chi_T(\tau)$ and fixing $\int d\tau\,|\chi_T(\tau)|^2$ lead to different amplitude scalings as $T\to0$.

Solution

For $\chi_T(\tau)=A_T\chi(\tau/T)$, the first integral scales as $A_TT$, so fixed integrated coupling requires $A_T\propto T^{-1}$. The squared norm scales as $A_T^2T$, so fixed $L^2$ norm requires $A_T\propto T^{-1/2}$. Because the response is quadratic in $\chi_T$, the two sudden-switching families do not describe the same physical control.

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.