Particles, Bogoliubov Transformations, and Detectors

Particles in curved spacetime are precise only relative to extra structure. A complex structure, stationary time translation, or asymptotic mode condition selects a positive-frequency subspace; Bogoliubov coefficients compare two such selections. A detector asks a different, operational question by filtering a two-point function along a worldline. This chapter develops both languages and marks the boundary between them.

Helpful background. Flat-Spacetime Quantum Field Theory supplies Fock modes and perturbative probes. Partial Differential Equations and Green Functions provides conserved Klein–Gordon products and distributional pullbacks. Causality and Global Structure supplies stationary and asymptotic regions. Quantum Fields in Curved Spacetime defines states, Hadamard regularity, and the prescribed-background scope.

Enter this chapter

The organizing question is not “How many particles does the field contain?” without qualification. It is:

$$\text{Which state, which positive-frequency split, which observer, and which observable?}$$

For a free scalar, the field algebra and a state can be defined without a preferred Fock representation. A compatible complex structure $J$ adds the split needed for a one-particle Hilbert space and number operator $N_J$. General curved spacetimes supply no universal analogue of the Poincaré-invariant vacuum, which is why local algebraic questions remain primary Hollands and Wald 2015, expanded arXiv version, §2.1, pp. 9–20, PDF.

A complete mode basis is useful when a timelike Killing flow, a cavity, or stationary in/out regions select it. A unitary relabelling within the same positive-frequency subspace changes mode labels but not the total particle concept. Mixing positive and negative frequency changes the complex structure. Fulling’s construction in a static coordinate patch of flat spacetime is the classic warning that geometry plus coordinates do not always privilege a unique particle interpretation Fulling 1973, complete article, pp. 2850–2862.

Mode mixing and its two levels of meaning

For two normalized bases,

$$v_A=\sum_i(\alpha_{Ai}u_i+\beta_{Ai}\bar u_i),$$

the canonical identities are

$$\alpha\alpha^\dagger-\beta\beta^\dagger=1, \qquad \alpha\beta^T-\beta\alpha^T=0.$$

The old vacuum has new occupation matrix $\beta\beta^\dagger$. In a homogeneous time-dependent geometry the mixing commonly couples opposite momenta and gives a squeezed-pair interpretation. Parker’s expanding-universe calculation makes this in/out creation statement concrete Parker 1969, complete article, pp. 1057–1068.

There are nevertheless two separate tests. The identities above hold modewise and preserve commutators. A global unitary implementation on an infinite-mode Fock space additionally requires

$$\|\beta\|_{\mathrm{HS}}^2=\sum_{A,i}|\beta_{Ai}|^2<\infty.$$

Every individual coefficient can be finite while this sum diverges. In a continuum, a finite density also need not imply a finite global norm. “Nonzero $\beta$” becomes physical particle creation only after the in and out conditions, measure, volume interpretation, and measurement question have been fixed.

During an FLRW expansion, a finite-order adiabatic basis supplies a controlled WKB particle diagnostic. Its ambiguity is suppressed in the ultraviolet adiabatic regime and disappears in an exactly stationary endpoint, but it is not a generally covariant local particle observable. Second-order adiabatic terms suffice for the mode-summed particle-number ultraviolet subtraction in Parker’s construction, while renormalizing stress–energy in four dimensions requires the distinct fourth-order subtraction problem developed by Parker and Fulling Parker 2012, §§4–8, especially Eqs. (12)–(13) and (22)–(45); Parker and Fulling 1974, complete article, pp. 341–354.

Detectors filter correlations, not global particles

For a two-level detector with positive gap $\Omega$, trajectory $x(\tau)$, smooth switching $\chi$, spatial profile $F$, and field state $\omega$, the leading response is

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

Every item in the subscript matters. Hadamard regularity and smooth compact switching make the distributional pairing well defined; a sharp-switching limit requires a declared order of limits. Louko and Satz show that in four dimensions the sharply switched total probability has a logarithmic divergence even though a controlled instantaneous rate can remain finite Louko and Satz 2008, §§2–4, arXiv version, PDF.

If the pulled-back two-point function is stationary and KMS at inverse temperature $\beta$, its infinite-time spectral response obeys

$$\frac{\dot{\mathcal F}(+\Omega)} {\dot{\mathcal F}(-\Omega)}=e^{-\beta\Omega}.$$

This detailed-balance theorem is a long-interaction statement at fixed gap. It is not inherited automatically by a finite switched probability. Fewster, Juárez-Aubry, and Louko prove both the fixed-gap long-time equivalence and the nonuniformity at large gap; adiabatic and fixed-ramp plateau switching have sharply different polynomial waiting-time behavior Fewster, Juárez-Aubry, and Louko 2016, §§4.1–4.3, Proposition 4.3, Theorem 4.6, Proposition 4.7, and §§5.1–5.3, Theorems 5.2–5.3, arXiv version, PDF.

Boundaries between the observables

Three distinctions recur throughout the chapter.

Particle number depends on modes or a complex structure. The same algebraic state and local two-point function can have zero $N_J$ expectation and nonzero $N_{J'}$ expectation. Finite $\beta$ for each mode does not prove Hilbert–Schmidt implementability.

Detector response depends on a protocol. A finite-time inertial detector can have switching transients even in Minkowski vacuum. A stationary thermal ratio requires KMS analyticity and a controlled limit. Detector clicks are therefore not a census of pre-existing global particles.

Radiation and stress–energy require their own observables. A click does not by itself compute a local renormalized stress tensor or an asymptotic energy flux. The external agent maintaining acceleration may enter the energy balance. Crispino, Higuchi, and Matsas show how inertial and Rindler observers can agree on the detector’s proper rate while describing it with different particle processes; their $+---$ formulas are translated here to $-+++$ Crispino, Higuchi, and Matsas 2008, arXiv version, §§III.A.1–5, pp. 19–25, PDF.

Birrell and Davies, Chapter 3, provides chapter-level synthesis of particle creation and detector ideas but is not used here as a theorem locator Birrell and Davies 1982, Chapter 3.

Chapter map

Read the six pages in this order:

1. Positive Frequency and Particle Number constructs the one-particle space from a symmetry or complex structure.
2. Bogoliubov Coefficients and Pair Creation derives the canonical identities, occupations, pair correlations, and Hilbert–Schmidt condition.
3. Adiabatic Particle Number builds finite-order WKB bases and identifies their controlled ambiguity.
4. Unruh–DeWitt Detectors derives the switched and smeared worldline response.
5. Stationary Detector Response and Detailed Balance states the KMS theorem and its order-of-limits boundary.
6. Particles versus Local Observables separates number, clicks, stress–energy, correlations, and radiation.

Review the chapter

Consider a smooth FLRW spacetime that is static in the remote past and future, and a finite-time detector operating during the expansion. Which objects answer (a) the late out-particle yield, (b) whether the chosen detector excites, and (c) the local energy density? What additional test is needed before claiming a unitary in/out Fock transformation, and can a finite-time click alone establish thermality or radiation?

Solution

The late out-particle yield is computed from the in/out Bogoliubov coefficients, with occupations from $\beta\beta^\dagger$. A global unitary implementation additionally requires $\beta$ to be Hilbert–Schmidt; finite coefficients mode by mode do not suffice. The detector question uses its pulled-back two-point function together with trajectory, gap, switching, smearing, and perturbative order. Local energy density comes from the renormalized stress tensor with declared subtraction conditions. Because finite switching convolves frequencies, one click does not prove KMS detailed balance, and because no flux was measured, it does not by itself prove emitted radiation.

References

• Birrell, N. D., and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1982), Chapter 3, DOI.
• 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.
• Fewster, Christopher J., Benito A. Juárez-Aubry, and Jorma Louko, “Waiting for Unruh,” Classical and Quantum Gravity 33 (2016), 165003, 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.
• Parker, Leonard, “Particle Creation and Particle Number in an Expanding Universe,” Journal of Physics A: Mathematical and Theoretical 45 (2012), 374023, DOI, Open PDF.
• Parker, Leonard, “Quantized Fields and Particle Creation in Expanding Universes. I,” Physical Review 183 (1969), 1057–1068, DOI.
• Parker, Leonard, and Stephen A. Fulling, “Adiabatic Regularization of the Energy-Momentum Tensor of a Quantized Field in Homogeneous Spaces,” Physical Review D 9 (1974), 341–354, DOI.