Particles versus Local Observables

Most finite-region questions in curved-spacetime quantum field theory do not require a global particle number. Correlation functions, a switched detector response, and renormalized local composite fields can all be defined while no preferred positive-frequency split exists. These quantities are related through the state, but none is a universal proxy for the others: a detector click is not a global particle count, and neither one alone establishes emitted radiation or determines the renormalized stress–energy tensor.

Required background. Partial Differential Equations and Green Functions supplies distributional field correlations and local smearing. Geodesics and Jacobi Fields provides observer worldlines and proper time; no congruence or deviation theorem is used in the comparison.

Helpful background. Flat-Spacetime Quantum Field Theory supplies Fock number operators. Causality and Global Structure provides asymptotic regions and radiation boundaries. Quantum Fields in Curved Spacetime defines the algebraic state and prescribed-background scope.

Four questions, four observables

The first step in a physical calculation is to identify the question before choosing its mathematical representative.

Question	Appropriate object	Essential extra choices	What it does not determine
How are field fluctuations correlated in a finite region?	$W_\omega(f,g)=\omega(\Phi(f)\Phi(g))$ and higher $n$-point functions	State and compact test functions	A preferred global mode basis
Will a specified probe change energy level?	$\mathcal F_{\omega,x,\chi,F}(\Omega)$	Worldline, gap, switching, smearing, coupling order	A basis-independent particle count or radiative flux
What local energy and momentum source the background semiclassically?	$\langle T_{ab}(x)\rangle_{\omega,\mathrm{ren}}$	Hadamard state, subtraction prescription, finite curvature terms	A unique decomposition into particles
How many quanta occupy a selected family of modes?	$\langle N_J\rangle_\omega$	Complex structure, normalized modes, representation, domain	Local covariance or detector response

Hollands and Wald formulate the free theory through its local field algebra precisely because a general curved spacetime has no preferred particle representation, while algebraic states and local questions remain meaningful Hollands and Wald 2015, expanded arXiv version, §2.1, pp. 9–20, PDF. This is not a claim that modes are useless. It is a claim that the mode split is additional physical structure whose validity must be argued in each application.

One state, two number operators

Fix the algebraic state $\omega$ and hence its two-point distribution $W_\omega(x,x')$. Now choose a positive-frequency complex structure $J$ and a second structure $J'$ related by a Bogoliubov transformation. When both number operators are defined on the state,

$$\langle N_{J'}\rangle_\omega =\sum_{A,i}|\beta_{Ai}|^2$$

if $\omega$ is the $J$-vacuum, whereas $\langle N_J\rangle_\omega=0$. The local distribution $W_\omega$ did not change: the state was held fixed while the operator used to ask “how many?” changed. If $\beta$ is not Hilbert–Schmidt, the global $J'$ number may diverge or the representations may be unitarily inequivalent, yet the algebraic two-point distribution still defines finite smeared correlations.

A single-mode squeeze gives the finite-dimensional version. With

$$v=\cosh r\,u+\sinh r\,\bar u,$$

the same $u$-vacuum has $\langle N_u\rangle=0$ and $\langle N_v\rangle=\sinh^2r$. No local measurement changed merely because the analyst relabelled the positive-frequency subspace. Fulling’s static-coordinate construction in flat spacetime supplies the classic field-theoretic example of this nonuniqueness Fulling 1973, complete article, pp. 2850–2862.

This comparison also prevents a common mistake: two observers may use different mode narratives, but they must agree on the value of the same scalar detector probability or the same locally defined observable after translating all conventions. “Observer dependence” is not permission for contradictory predictions.

Detector response is a protocol-filtered correlation

At leading order, a two-level detector measures

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

Holding $\omega$ and its global mode occupations fixed while changing the trajectory, gap, switching, or smearing changes the response. Therefore there cannot be a universal map from $\langle N_J\rangle$ to detector clicks. Smooth switching in a four-dimensional Hadamard state makes the distributional pairing well defined, while sharp switching has its own controlled limiting behavior Louko and Satz 2008, §§2–4, arXiv version, PDF.

Two simple cases are decisive. An inertial detector coupled for finite time in Minkowski vacuum can have transient excitation even though the Minkowski number is zero. Conversely, a detector with a gap outside the spectral support selected by a long stationary experiment may not click even when some unrelated mode occupations are nonzero. Response asks whether a specified apparatus exchanges energy with the field under a specified protocol.

For uniform acceleration, the long-time excitation/de-excitation ratio is thermal. Inertial and coaccelerated descriptions can attribute the transition to different Minkowski or Rindler mode processes while agreeing on the proper transition rate Crispino, Higuchi, and Matsas 2008, arXiv version, §§III.A.1–5, pp. 19–25, PDF. Their $+---$ convention is translated here to $-+++$. The agreement concerns the operational rate, not a shared global particle ontology.

Stress–energy is local but renormalized

For a Hadamard state, the renormalized stress tensor is obtained by applying a local bidifferential operator to the difference between the two-point function and its universal Hadamard singularity, followed by the coincidence limit and permitted local curvature terms. Schematically,

$$\langle T_{ab}(x)\rangle_{\omega,\mathrm{ren}} =\lim_{x'\to x}D_{ab}(x,x') \bigl[W_\omega(x,x')-H(x,x')\bigr] +C_{ab}(x).$$

The finite tensor $C_{ab}$ is restricted by locality, covariance, conservation, dimension, and the chosen renormalization conditions. It is not fixed by a particle basis. Even in a region with a useful $J$, $\langle T_{ab}\rangle$ contains vacuum polarization and coherence terms as well as contributions that may admit a particle interpretation. Replacing it by an integral of $\omega_kN_k$ generally drops those terms and can violate conservation.

The Minkowski vacuum gives a clean separation. With the standard Minkowski subtraction, $\langle T_{ab}\rangle_{\mathrm{ren}}=0$. A uniformly accelerated detector nevertheless has a nonzero thermal stationary response. Thus nonzero detector response does not imply nonzero local stress–energy in that state, and it does not identify a bath of globally emitted Minkowski particles.

Radiation requires an energy-flux question

Radiation is diagnosed by a specified energy-transfer observable: for example, a renormalized flux through a large world tube, a component of $\langle T_{ab}\rangle$ at a declared asymptotic boundary, or a detector–source energy balance including the external agent. Such a statement needs boundary conditions and an asymptotic or finite-region measurement prescription. A click alone supplies none of these.

An accelerated detector illustrates the missing bookkeeping. The external force maintaining acceleration, the detector’s internal energy, and the field can exchange energy. In an inertial mode expansion a particular transition amplitude may be accompanied by emission of a Minkowski quantum; in a Rindler description it may be represented as absorption from or emission into Rindler modes. But observing only the internal transition does not determine the net flux, its direction, or its asymptotic persistence. Crispino, Higuchi, and Matsas explicitly separate the detector response from the more delicate question of radiation by accelerated sources Crispino, Higuchi, and Matsas 2008, §III.A, pp. 19–28, and §VI, p. 46, arXiv v1, PDF.

In cosmology, an asymptotically static in/out pair can support a particle-production statement through $\beta$. During a general expansion without an out region, local correlations and $\langle T_{ab}\rangle$ remain meaningful even when no final global particle number is selected. Metric backreaction is a further dynamical problem and lies outside the prescribed-background scope of this chapter.

A separation argument

The non-equivalence of the observables follows by varying one input at a time.

1. Hold $\omega$ fixed and change $J$: particle number changes while $W_\omega$, every detector protocol functional of it, and the fixed-scheme stress tensor remain unchanged.
2. Hold $\omega$ and $J$ fixed and change $x$, $\Omega$, $\chi$, or $F$: detector response changes while the state, number expectation, and stress tensor do not.
3. Hold the state and detector fixed and change an allowed finite renormalization condition: $\langle T_{ab}\rangle$ shifts by an allowed conserved local curvature tensor while particle number and detector response remain unchanged.

These countervariations prove that no one of the three quantities universally determines the other two. Relations can arise in special stationary, asymptotic, kinetic, or weak-coupling regimes, but every such relation inherits those hypotheses.

Check your understanding

In Minkowski vacuum, compare (i) Minkowski particle number, (ii) the long-time response of a uniformly accelerated detector, and (iii) the standard renormalized stress tensor. Does the detector response prove that radiation is flowing to null infinity?

Solution

The Minkowski vacuum has zero Minkowski-mode occupation and, with the standard subtraction, zero renormalized stress tensor. A uniformly accelerated detector nevertheless has a nonzero response with detailed-balance temperature $a/(2\pi)$ because its proper-time trajectory filters the same vacuum two-point function differently. These facts are consistent: number, detector response, and stress tensor are different observables. The click does not compute an asymptotic flux, so it cannot by itself prove emitted radiation. One would additionally need a specified stress–energy flux or complete detector–agent–field energy balance.

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.
• Fulling, Stephen A., “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time,” Physical Review D 7 (1973), 2850–2862, DOI.
• Hollands, Stefan, and Robert M. Wald, “Quantum Fields in Curved Spacetime,” Physics Reports 574 (2015), 1–35, 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.