Particles, Detectors, and Nonadiabatic Production

Particle number, detector excitation, and energy flux are different observables in curved or accelerated settings. This chapter connects them through mode evolution and localized measurements while making their operational assumptions explicit.

Enter this chapter

Begin with Particles, Local Observables, and Detector Dependence to choose between mode occupation, detector response, and stress-energy observables.

Helpful background. Vacuum Ambiguity, Time Flow, and Observer Dependence and Green Operators, Causal Propagators, and State-Dependent Two-Point Functions develop the definitions used by the chapter’s opening pages.

Topics

1. Particles, Local Observables, and Detector Dependence — Separates number, clicks, correlators, and stress flux.
2. Mode Bases, In/Out States, and Number Operators — States when asymptotic particle counting is controlled.
3. Detector Response Along Curved and Accelerated Worldlines — Derives the response from the pulled-back two-point function.
4. Switching, Smearing, Finite-Time Response, and Transients — Controls finite duration and ultraviolet sensitivity.
5. Particle Creation in Time-Dependent Backgrounds — Computes in/out Bogoliubov production and its energy.
6. Adiabaticity, Stokes Phenomena, and Production Rates — Treats complex turning points, interference, and error domains.
7. Parametric-Oscillator and Solvable Production Benchmarks — Supplies exact quench fixtures.
8. Unruh Effect and Uniformly Accelerated Detectors — Derives acceleration-temperature detailed balance.
9. Tolman Redshift, KMS Structure, and Local Temperature — Distinguishes global KMS data from calibrated local temperature.
10. Acceleration, Gravity, and the Limits of Equivalence-Principle Arguments — Isolates local kinematics from curvature, state, and horizon data.
11. Moving Mirrors and the Dynamical Casimir Effect — Compares Bogoliubov particles, boundary work, and outgoing flux.
12. Schwinger and Gravitational Production: A Controlled Comparison — Identifies shared turning-point mathematics and different physics.

How the ideas fit together

A time-dependent background can mix positive- and negative-frequency modes, producing an in/out occupation number when the asymptotic bases are defined. A detector instead samples the field’s two-point function along its trajectory with a specified switching and spatial profile. Acceleration can therefore change detector response without changing the global Minkowski vacuum Unruh 1976, pp. 870–892.

Thermal detailed balance requires a suitable stationary or long-time limit. Finite switching introduces transients, and an instantaneous basis occupation depends on how the basis was chosen. These differences should be resolved before assigning a temperature or production rate.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Construction	State, geometry, and trajectory	Basis or interaction	Switching, asymptotics, and regulator	Reported observable	Energy accounting	Licensed claim and decisive failure
In/out number	In-state; asymptotically stationary past and future	Complete normalized in/out bases	Wavepackets or volume normalization; controlled asymptotic regions	$\langle N_f^{\mathrm{out}}\rangle$	Out energy from $\omega_fN_f$ plus vacuum and interference terms as applicable	Asymptotic occupation; fails without an out splitting
Instantaneous basis	State on one slice; chosen canonical variables	Time-local diagonalization or WKB basis	Adiabatic order and phase convention	$n_{\mathbf k}(t)$	Basis Hamiltonian, not automatically local stress energy	Diagnostic occupation; fails as an invariant under time-dependent basis changes
Localized detector	Hadamard state; declared worldline	Probe gap and field coupling	Smooth switching, spatial profile, perturbative order	Excitation probability or transition rate	Work supplied by detector control and trajectory	Response of that apparatus; fails as pre-existing density when protocol changes it
KMS or detailed balance	Stationary state and normalized flow	Stationary detector or algebra automorphism	Long-time or exact KMS domain	Spectral ratio $e^{-\beta\Omega}$	Equilibrium exchange with respect to the same generator	Thermality for that flow; fails for finite transients or a rescaled generator
Stress flux	Renormalizable state; local geometry	Local $T_{\mu\nu}$ and observer or null direction	Locally covariant subtraction and boundary data	$\langle T_{\mu\nu}\rangle_{\mathrm{ren}}u^\mu n^\nu$	Direct transported energy with source or boundary work	Local flux; fails if subtraction or conservation is uncontrolled
WKB or Stokes production	Specified in-state and analytic frequency profile	Complex turning points and connection matrices	Saddle set, Stokes graph, prefactor, uniform error	Semiclassical $\lvert\beta_{\mathbf k}\rvert^2$	Compare integrated energy with background work	Asymptotic production estimate; fails outside its saddle and asymptotic regime

Where to go next

Continue to Curved Channels, Communication, and Entanglement Harvesting to specify the local couplings and accessible algebras of a communication task.

References

• William G. Unruh, “Notes on Black-Hole Evaporation,” Physical Review D 14 (1976), 870–892, DOI.