Unruh–DeWitt Detectors

An Unruh–DeWitt detector is a localized quantum system used to sample a scalar field along a prescribed timelike trajectory. At leading nontrivial order, its transition probability is controlled by the Wightman two-point distribution pulled back to the detector’s world tube or, in the pointlike limit, its worldline. The result depends jointly on the field state, trajectory, energy gap, switching, spatial profile, and perturbative order. A click is consequently neither a measurement of a basis-independent global particle number nor, by itself, evidence that radiation was emitted to infinity.

Required background. Partial Differential Equations and Green Functions supplies distributional two-point functions and their smearing. Geodesics and Jacobi Fields provides proper time and a transported local frame along the timelike worldline; the response derivation uses no geodesic-deviation theorem.

Helpful background. Flat-Spacetime Quantum Field Theory supplies interaction-picture perturbation theory. Causality and Global Structure provides the distinction between a local coupling and asymptotic radiation. Quantum Fields in Curved Spacetime defines states and Hadamard regularity on the prescribed background.

A switched two-level probe

Let the detector have energy eigenstates $|g\rangle$ and $|e\rangle$, with positive gap

$$\Omega=E_e-E_g>0.$$

It follows a smooth future-directed timelike worldline $x(\tau)$ parametrized by proper time. In the interaction picture its monopole operator is

$$\mu(\tau)=e^{iH_D\tau}\mu(0)e^{-iH_D\tau}.$$

For a real scalar field, the ideal pointlike coupling is

$$H_I(\tau)=\lambda\,\chi(\tau)\,\mu(\tau)\widehat\Phi(x(\tau)),$$

where $\lambda$ is small and $\chi\in C_0^\infty(\mathbb R)$ is real. Compact, smooth switching says exactly when and how strongly the apparatus interacts. It is part of the observable, not a disposable regulator.

For a detector of finite size, choose a transported orthonormal frame and a spacelike simultaneity prescription in a tubular neighbourhood of the worldline. With a real compact profile $F(\boldsymbol\xi)$,

$$\Phi_F(\tau)=\int d^3\xi\,\sqrt{h(\tau,\boldsymbol\xi)} F(\boldsymbol\xi)\Phi(X(\tau,\boldsymbol\xi))$$

replaces $\Phi(x(\tau))$. The profile, its physical size, and the chosen transport all affect finite-size response. A model defined by finite smearing must retain that smearing until its pointlike limit is controlled. Alternatively, a smooth timelike point detector in a Hadamard state is defined directly by the distributional pullback below; no spatial regulator is required for that leading smoothly switched response. Spatial-size, regulator, and sharp-switching limits must not be interchanged without justification.

Deriving the response function

Assume the initial detector state is $|g\rangle$ and the field is in a state $\omega$. Expanding the Dyson operator to first order in the transition amplitude, then summing the order-$\lambda^2$ probability over unobserved final field states, gives

$$P_{g\to e} =\lambda^2|\langle e|\mu(0)|g\rangle|^2 \mathcal F_{\omega,F}(\Omega) +\text{higher orders in }\lambda.$$

Declare the Wightman ordering by

$$W_\omega(x,x')=\omega\!\left(\Phi(x)\Phi(x')\right).$$

The smeared pullback $W_{\omega,F}(\tau,\tau')=\omega(\Phi_F(\tau)\Phi_F(\tau'))$ then yields

$$\mathcal F_{\omega,F}(\Omega) =\int_{-\infty}^{\infty}d\tau \int_{-\infty}^{\infty}d\tau'\, \chi(\tau)\chi(\tau')e^{-i\Omega(\tau-\tau')} W_{\omega,F}(\tau,\tau').$$

This ordering and phase convention make positive $\Omega$ an excitation and negative $\Omega$ a de-excitation. Positivity of the state makes the response nonnegative because the double integral is the positive-type distribution $W_\omega$ evaluated on the detector’s smearing test function. Louko and Satz derive this response for a four-dimensional globally hyperbolic spacetime and an arbitrary Hadamard state Louko and Satz 2008, §2, Eqs. (2.1)–(2.7), arXiv version, PDF.

The formula displays every dependency. Changing $x(\tau)$ changes the pulled-back correlations; changing $\Omega$ changes the Fourier filter; changing $\chi$ changes temporal resolution; changing $F$ changes spatial resolution; changing $\omega$ changes the correlations; and going beyond order $\lambda^2$ introduces detector–field backreaction, time ordering, and repeated transitions absent from this leading response.

Why Hadamard regularity and smooth switching matter

The Wightman function is a distribution singular at null-related and coincident points. For a Hadamard state, its wavefront orientation permits restriction to a smooth timelike curve, and pairing that pullback with the smooth compact function

$$(\tau,\tau')\longmapsto \chi(\tau)\chi(\tau')e^{-i\Omega(\tau-\tau')}$$

is well defined. This is not permission to replace the distribution by an arbitrary $i\epsilon$ expression and interchange limits.

Louko and Satz explicitly remove the regulator for smooth switching in a general four-dimensional Hadamard state Louko and Satz 2008, §3, arXiv version, PDF. In their controlled sharp-switching family, the total response has a logarithmic divergence as the switching duration tends to zero, while a derivative with respect to total detection time has a finite limit Louko and Satz 2008, §4, Eqs. (4.4)–(4.7), arXiv version, PDF. The regulator must be removed before the sharp limit. Thus “turn on instantaneously” is a singular idealization whose surviving rate requires a stated limiting prescription.

A non-Hadamard state can have additional singularities that invalidate the pullback or produce ultraviolet sensitivity not cured by smooth switching. Likewise, a point detector may be inadequate when the profile scale competes with the acceleration radius, curvature radius, or inverse gap. The finite-size model is then the physical observable.

Probability, instantaneous rate, and stationary rate

The finite-time quantity $\mathcal F(\Omega)$ is a probability factor after the coupling has ceased. Dividing it by an arbitrary duration does not automatically produce a stationary transition rate. For a general trajectory, differentiating a sharply truncated response can even yield a temporarily negative quantity; it is a comparison across ensembles with different readout times, not a negative probability.

If the state and trajectory are stationary with respect to the same time flow, the pullback depends only on $s=\tau-\tau'$. A long smooth interaction can then define a time-independent spectral rate. In four-dimensional Minkowski vacuum, an inertial detector has the limiting pointlike rate

$$\dot{\mathcal F}_{\mathrm M}(\Omega) =-\frac{\Omega}{2\pi}\,\Theta(-\Omega).$$

Hence excitation vanishes for $\Omega>0$, whereas an initially excited detector with argument $-\Omega$ can de-excite. At finite switching time, transient excitation need not vanish because the apparatus supplies temporal uncertainty. This static-vacuum limit is an essential sign and normalization check, not a claim that every finite-time response is zero.

For this same inertial detector, the mechanism is explicit. With $\widehat\chi(\nu)=\int d\tau\,e^{-i\nu\tau}\chi(\tau)$ and the massless four-dimensional vacuum,

$$\mathcal F_{\mathrm M}(\Omega) =\frac{1}{4\pi^2}\int_0^\infty d\omega\,\omega |\widehat\chi(\Omega+\omega)|^2.$$

To derive it, insert the positive-energy mode expansion $W_{\mathrm M}(s)=(4\pi^2)^{-1}\int_0^\infty d\omega\,\omega e^{-i\omega(s-i0)}$ into the response and perform the two time integrals. A finite smooth switch has a nonzero Fourier tail at $\Omega+\omega$, so it can excite the detector even though the initial inertial particle occupation is zero. Energy can be supplied by the apparatus that changes the coupling. Smoothness makes the integral ultraviolet finite; an abrupt switch does not have the same Fourier decay.

Uniform acceleration gives a different stationary spectrum in the long-interaction limit, but even there thermality requires a relation between excitation and de-excitation rates, not merely a nonzero finite-time probability. Crispino, Higuchi, and Matsas show how inertial and Rindler descriptions agree on the detector’s scalar transition rate while using different mode-particle narratives Crispino, Higuchi, and Matsas 2008, arXiv version, §§III.A.1–5, pp. 19–25, PDF. Their $+---$ convention has been translated to $-+++$. The detailed-balance conditions are developed on the next page.

What a click does and does not establish

A click establishes that this detector, with this gap, path, switching, smearing, initial state, and coupling order, made a transition. It probes a filtered two-point correlation. It does not define a global positive-frequency complex structure, so it need not equal a global number-operator measurement. It also does not by itself establish an outward energy flux: emitted radiation is a question about field energy transport, asymptotic observables, or a specified calorimetric measurement.

These distinctions resolve the apparently conflicting descriptions of an accelerated detector. An inertial account may describe excitation together with emission into Minkowski modes; a coaccelerated account may describe exchange with Rindler modes. The measurable detector probability agrees, while the global particle language depends on the chosen time evolution. Neither narrative licenses identifying every click with observer-independent emitted radiation.

Check your understanding

An inertial pointlike detector in Minkowski vacuum is coupled with a smooth compact $\chi$ for a finite time. List the data needed to predict its leading excitation probability, and decide whether a nonzero result proves that the vacuum contained particles before the interaction.

Solution

One must specify the worldline, positive gap, switching profile and duration, field state, spatial profile or pointlike limiting prescription, monopole matrix element, coupling strength, and perturbative order. Substituting these data into the pulled-back Wightman integral can give a nonzero finite-time excitation because switching broadens the detector’s frequency filter. It does not follow that the initial state had particles in a preferred global basis, and it does not prove an outgoing radiation flux. In the controlled long smooth limit, the inertial-vacuum excitation rate vanishes, which checks this interpretation.

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.
• Louko, Jorma, and Alejandro Satz, “Transition Rate of the Unruh–DeWitt Detector in Curved Spacetime,” Classical and Quantum Gravity 25 (2008), 055012, DOI, Open PDF.