Adiabatic and State-Selection Criteria

Curved spacetime offers several state-selection criteria, but they answer different questions. The Hadamard condition controls ultraviolet singularities; adiabatic order quantifies a finite degree of ultraviolet approximation; symmetry, averaged energy, or detector response can distinguish states only after extra geometric or operational data are chosen. A sound prescription must first be admissible for the observables under study and only then be judged as a selector within the admissible class.

Required background. Partial Differential Equations and Green Functions supplies mode equations, Sobolev regularity, and characteristic propagation. The bounded local bridge used below is the WKB expansion of a time-dependent oscillator and the fact that finite Sobolev wavefront control is weaker than exclusion of all unwanted singular directions.

Helpful background. Flat-Spacetime Quantum Field Theory provides positive-frequency modes and Bogoliubov transformations, Causality and Global Structure defines the globally hyperbolic and stationary settings in which the criteria apply, and Quantum Fields in Curved Spacetime provides the inherited state and frequency conventions.

Ultraviolet admissibility is not state selection

A Hadamard state has the full positive-frequency wavefront set required for local covariant Wick products and perturbative constructions. Any two Hadamard two-point functions differ smoothly. This makes Hadamard regularity a robust admissibility condition, but the smooth difference can contain arbitrary physical information about occupation, temperature, infrared correlations, and boundary data. Hadamard regularity therefore does not select one vacuum.

Junker and Schrohe define a quasifree state of adiabatic order $N$ by a Sobolev wavefront condition of the form

$$\operatorname{WF}'_s(\Lambda_N)\subset\mathcal C^+ \qquad\text{for every }s<N+\frac32,$$

in their four-dimensional normalization; see Junker and Schrohe 2002, Definition 3.2, p. 1122. Here the prime reverses the second covector: $\operatorname{WF}'_s(\Lambda)=\{(x,k;x',k'):(x,k;x',-k')\in\operatorname{WF}_s(\Lambda)\}$. Accordingly, $\mathcal C^+$ in this primed notation contains parallel-transported future-directed null covectors in both slots. The condition says nothing about the excluded directions at higher Sobolev order. Every Hadamard state has every finite adiabatic order, and a single state satisfying the hierarchy to all orders has full Hadamard wavefront regularity. A state obtained by stopping a WKB iteration at any fixed finite order is, in general, not Hadamard.

Finite order can nevertheless be sufficient for a specified, finite task. For example, in the same normalization, order $N>5/2$ gives local quasiequivalence to a Hadamard representation under the hypotheses of Junker and Schrohe 2002, Theorem 4.7, p. 1131; for pure states with compact Cauchy surface their theorem lowers the global unitary-equivalence threshold to $N>3/2$. These results are not permission to replace full Hadamard regularity whenever arbitrarily differentiated Wick products or a complete perturbative algebra is required.

Large-wave-number check in an expanding universe

For a spatially flat FLRW metric in conformal time,

$$ds^2=a(\eta)^2(-d\eta^2+d\boldsymbol x^2),$$

the rescaled mode $\chi_k=a\phi_k$ obeys

$$\chi_k''+\Omega_k^2\chi_k=0, \qquad \Omega_k^2=k^2+Q,$$

with

$$Q=a^2m^2+(6\xi-1)\frac{a''}{a}.$$

The WKB ansatz

$$\chi_k(\eta) =\frac{1}{\sqrt{2W_k(\eta)}} \exp\!\left[-i\int^\eta W_k(\zeta)\,d\zeta\right]$$

solves the mode equation when

$$W_k^2=\Omega_k^2 -\frac12\frac{W_k''}{W_k} +\frac34\left(\frac{W_k'}{W_k}\right)^2.$$

Expanding at fixed smooth geometry gives

$$W_k =k+\frac{Q}{2k} -\frac{Q^2+Q''}{8k^3} +O(k^{-5}).$$

The $Q''$ term shows why instantaneous diagonalization $W_k=\Omega_k$ is not the same as solving the adiabatic recursion: it matches some algebraic powers of $Q$ but misses time derivatives at the next order. Repeating the recursion improves the high-$k$ expansion to a chosen finite order. No finite truncation controls every derivative, so the calculation independently exposes the gap between finite adiabatic order and the full Hadamard condition.

Results and limitations through 2026-08-19

Evidence cutoff: 2026-08-19. The matrix separates the established ultraviolet conclusions from the extra data used to choose a state.

Criterion	Evidence and established result	Assumptions	Limitation or discriminator
Full Hadamard or infinite adiabatic order	Fulling, Narcowich, and Wald 1981, pp. 243–272 establish existence by deformation, while the microlocal hierarchy identifies full short-distance regularity.	Free Klein–Gordon theory on a smooth globally hyperbolic spacetime, with the state and commutator hypotheses fixed.	Establishes ultraviolet admissibility and nonemptiness, but does not select a unique state.
Finite adiabatic order	Junker and Schrohe 2002, Definition 3.2 and Theorem 4.7, pp. 1122 and 1131 give Sobolev control and representation thresholds.	Four-dimensional free scalar, stated order normalization, quasifree states, and additional compactness or purity assumptions for the stronger result.	A finite order may support a finite observable class but is generally not Hadamard and can retain weaker unwanted singularities.
Stationary ground or KMS condition	Hollands and Wald 2015, arXiv v2, §2.1, pp. 17–20, PDF formulate ground spectrum and KMS strip conditions; regular free-field examples are Hadamard under further hypotheses.	A complete stationary automorphism group, declared generator sign, inverse temperature, and sufficient regularity.	Strong selector when available, but generic expanding geometries have no such global flow; ultraviolet regularity must still be verified.
State of low energy	Olbermann 2007, §§2.1 and 3.1, Theorem 4.9, PDF proves the Hadamard property for states obtained by minimizing a time-smeared energy density.	The minimally coupled free scalar on a smooth Robertson–Walker spacetime, homogeneous isotropic pure quasifree states, and a nonzero smooth compactly supported time-smearing function; the mode minimization must be well defined, including its infrared sector.	Selects relative to the smearing and observer family; changing that data can change the state, especially in the infrared.
Instantaneous or detector-based prescription	Junker and Schrohe 2002, §7, pp. 1164–1167 show that high-gap detector response distinguishes finite orders by power-law decay, whereas Hadamard response falls faster than every inverse power in their setting.	A declared worldline, compact switching, detector coupling, gap limit, and a quasifree adiabatic state.	Operationally meaningful but apparatus-dependent; a low response for one detector does not prove a global wavefront condition or unique vacuum.

Competing positions and their domains

Full-admissibility position. Require a Hadamard state before defining the unrestricted local observable algebra. This is the conservative choice when the calculation may involve arbitrarily high derivatives, perturbative products, or a change of observable not known in advance. Its strength is stability under local covariant operations; its cost is that it leaves a large smooth family of states.

Task-relative position. Accept sufficiently high but finite adiabatic order when only a fixed finite family of observables, detector moments, or representation properties is needed. Junker and Schrohe provide real support for this position, but only with order thresholds tied to the operation. The position becomes unsound if “sufficient for this finite calculation” is silently promoted to “Hadamard” or “sufficient for every local product.”

Geometry-sensitive selection position. First impose Hadamard admissibility, then use a complete symmetry, an averaged-energy functional, initial data, or a specified observer family to select within it. Stationary ground and KMS states and Robertson–Walker states of low energy fit here. The selected state is distinguished relative to that extra structure, not naturally preferred on all spacetimes.

Operational position. Rank states by detector response or another measurement protocol. This directly addresses an experiment but depends on worldline, switching, smearing, coupling, and gap regime. It can discriminate states with identical ultraviolet class, yet it cannot replace a global admissibility theorem.

These positions are compatible once their questions are separated. The unresolved issue is not whether finite order equals Hadamard—it does not—but which weakest regularity and which extra selection data suffice for a declared observable class in a declared geometry.

Adverse case: identical ultraviolet order, different infrared state

Let $u_k$ define a homogeneous quasifree state of some adiabatic order, and choose a smooth Bogoliubov coefficient $\beta_k$ supported in a finite band $0\lt k_{\min}\lt k\lt K$. Take $\alpha_k=\sqrt{1+\lvert\beta_k\rvert^2}$, so it equals one outside the band, and set

$$\widetilde u_k=\alpha_k u_k+\beta_k\overline{u_k}.$$

Because $\beta_k$ vanishes at large $k$, the original and modified modes have the same high-wave-number expansion and therefore the same adiabatic order. Smoothness of the modes on the finite band permits differentiation under the compact momentum integral to every order, so their two-point difference is smooth. Keeping the band away from $k=0$ also avoids assuming that an infrared-divergent zero-mode sector has been repaired by an ultraviolet argument. Yet the modification changes long-wavelength occupation, equal-time correlations, and the response of detectors sensitive to this band. Thus neither finite order nor even the Hadamard condition ranks the two states.

An instantaneous prescription can make the problem worse: choosing modes by minimizing a Hamiltonian on one time slice changes under a new time coordinate or canonical variable and may reproduce only a finite WKB order. The adverse case defeats a universal claim of uniqueness, while the derivative mismatch in the WKB check defeats a universal claim of full ultraviolet regularity.

What could change these conclusions

The conclusions could change if a new theorem changes the spacetime, state class, algebra, or observable class for which a finite adiabatic threshold is sufficient; if an independent calculation or replication changes the large-wave-number or detector-response limit; if a counterexample shows that a symmetry, low-energy, or instantaneous prescription fails Hadamard regularity under its published assumptions; or if a boundary condition, regulator, curvature coupling, or field-content extension invalidates one of the regimes in the matrix. A result that merely proposes another preferred state without proving its ultraviolet class would expand the list of selectors but would not change the admissibility conclusions.

Check your understanding

Two homogeneous states have modes related by the smooth finite-band Bogoliubov transformation above. Explain why comparing their WKB coefficients cannot select between them, and name data that can.

Solution

Above the support cutoff the Bogoliubov coefficient is exactly zero, so the two mode families agree at every sufficiently large wave number. Their asymptotic WKB series therefore has identical coefficients to all orders, and any criterion based only on ultraviolet order assigns them the same class. Smooth compact support also makes the difference of their two-point functions smooth under the stated regularity assumptions, so full Hadamard admissibility does not separate them either. They can still differ in infrared occupation and correlations. A low-frequency detector with declared worldline and switching, an averaged-energy functional with declared smearing, boundary or initial conditions, or an available symmetry criterion can compare them. Each answer is relative to that additional physical data, which is precisely why ultraviolet admissibility and state selection must be kept distinct.

References

• Fulling, Stephen A., Francis J. Narcowich, and Robert M. Wald. “Singularity Structure of the Two-Point Function in Quantum Field Theory in Curved Spacetime, II.” Annals of Physics 136, no. 2 (1981): 243–272. doi:10.1016/0003-4916(81)90098-1.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. doi:10.1016/j.physrep.2015.02.001. Open PDF.
• Junker, Wolfgang, and Elmar Schrohe. “Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties.” Annales Henri Poincaré 3, no. 6 (2002): 1113–1181. doi:10.1007/s000230200001. Open PDF.
• Olbermann, Heiner. “States of Low Energy on Robertson–Walker Spacetimes.” Classical and Quantum Gravity 24, no. 20 (2007): 5011–5030. doi:10.1088/0264-9381/24/20/007. Open PDF.