Adiabatic Vacua and Ultraviolet Regularity

An adiabatic state is built by matching each high-momentum mode to a finite-order WKB solution. The construction is valuable because the same expansion identifies the state-independent ultraviolet terms that must be subtracted from local observables. Its scope must be stated carefully: a finite adiabatic order controls a corresponding Sobolev degree of singularity, but it is not automatically the full Hadamard condition.

Required background. Compactly, the ultraviolet condition concerns the distributional singularities of the two-point function, not merely pointwise decay of individual modes; a parametrix subtracts the locally fixed singular part before coincidence. Partial Differential Equations and Green Functions supplies distributions, parametrices, and Sobolev regularity. Helpful background. Flat-Spacetime Quantum Field Theory provides WKB oscillators and composite-operator divergences; Causality and Global Structure establishes the Cauchy evolution of the chosen initial data; Quantum Fields in Curved Spacetime provides Hadamard states and local renormalization.

The WKB frequency equation

On the spatially flat FLRW background of the preceding page, a rescaled scalar mode obeys

$$\chi_k''+\omega_k^2(\eta)\chi_k=0, \qquad \omega_k^2=k^2+a^2m^2+(6\xi-1)\frac{a''}{a}.$$

Insert the normalized positive-frequency ansatz

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

Its Wronskian is exactly $i$ for every real positive $W_k$. Substitution into the mode equation gives

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

This nonlinear relation is solved asymptotically, not generally exactly. In a common counting convention, $a$ is adiabatic order zero, each derivative of $a$ raises the order by one, and

$$\Omega_k=\sqrt{k^2+a^2m^2}$$

is the zeroth-order frequency. The curvature term is second order. Starting from $W_k^{(0)}=\Omega_k$, substitute the previous iterate into the derivative terms and retain contributions only through the chosen order. Odd orders vanish for this scalar frequency convention, although the state data contain derivatives of the even-order result.

The first nontrivial iteration is explicit:

$$W_k^{(2)}=\Omega_k +\frac{(6\xi-1)a''}{2a\Omega_k} -\frac{\Omega_k''}{4\Omega_k^2} +\frac{3\Omega_k'^2}{8\Omega_k^3}.$$

To obtain it, write $W_k=\Omega_k+w_k^{(2)}+\cdots$, keep $2\Omega_kw_k^{(2)}$ on the left, and use $W_k=\Omega_k$ in derivative terms that already have order two. Retaining an unexpanded square root instead defines a different approximation at higher orders; the order-counting prescription must accompany any numerical implementation.

From an approximation to a state

Choose a Cauchy time $\eta_0$. An adiabatic state of order $N$ is specified modewise by

$$\chi_k(\eta_0)=\frac{1}{\sqrt{2W_k^{(N)}(\eta_0)}},$$ $$\chi_k'(\eta_0) =\left[-iW_k^{(N)} -\frac{W_k^{(N)\prime}}{2W_k^{(N)}}\right]_{\eta_0} \chi_k(\eta_0).$$

These data satisfy the Wronskian and are then evolved with the exact mode equation, not with a truncated equation. Positivity of $W_k^{(N)}$ is needed where the prescription is applied; at low $k$, a smooth completion may be required. Such infrared completion does not change the prescribed ultraviolet order, but it can change long-distance observables.

Adiabatic order measures large-$k$ accuracy. Schematically, increasing $N$ makes the Bogoliubov coefficient relative to the formal high-frequency solution fall by more powers of $k$. The resulting two-point distribution approaches the Hadamard wavefront structure in a Sobolev hierarchy. Junker and Schrohe formulate this hierarchy in Junker and Schrohe 2002, Definition 3.2 and Lemma 3.3; §6, Theorem 6.3, PDF. Their microlocal order and the number of WKB iterations are different labels; one must translate them before importing a regularity threshold. A finite-order adiabatic state can be sufficiently regular for a specified finite set of observables without being fully Hadamard. An actual state whose initial data have the full asymptotic expansion, with suitable remainder estimates and global completion, can satisfy the full microlocal condition; a formal series alone is not a state.

Observable-dependent subtraction order

The question “which order is enough?” has no answer until the observable is named. In four spacetime dimensions:

• the coincidence limit defining $\langle\phi^2\rangle_{\mathrm{ren}}$ requires subtraction through second adiabatic order;
• the stress tensor contains two derivatives and requires subtraction through fourth adiabatic order; and
• higher-derivative composite operators require correspondingly deeper expansions.

For a homogeneous state, write a bare observable as a momentum integral of its mode integrand. Adiabatic regularization subtracts the large-$k$ WKB terms through the degree needed to remove the quartic, quadratic, and logarithmic divergences:

$$\langle T_{\mu\nu}\rangle_{\mathrm{ren}} =\int\frac{d^3k}{(2\pi)^3} \left[t_{\mu\nu,k}^{\mathrm{exact}} -t_{\mu\nu,k}^{(0\text{--}4)}\right] +C_{\mu\nu}.$$

Here the measure is comoving, while the factors of $a$ inside $t_{\mu\nu,k}$ convert to physical densities. The local counterterm tensor $C_{\mu\nu}$ contains the allowed finite conserved curvature terms, with coefficients fixed by the chosen renormalization conditions. Mode subtraction is therefore not permission to discard those covariant finite ambiguities. Parker and Fulling construct the stress-tensor subtraction for homogeneous geometries in Parker and Fulling 1974, complete article, pp. 341–354.

Ultraviolet admissibility is not state selection

Many different states share the same sufficiently high-order ultraviolet expansion. They can differ at finite $k$, in the infrared, or by smooth contributions to the two-point function. Adiabatic regularity therefore does not select a unique cosmological vacuum. The initial time, the low-momentum completion, the background history, and any symmetry or energy-minimization criterion remain additional choices.

There is also no guarantee that a truncated $W_k^{(N)}$ stays real for every mode and time. A turning point with $\omega_k^2=0$ invalidates the elementary oscillatory WKB basis; uniform approximations or exact evolution across the turning region are then needed. This failure is physical information about the approximation, not evidence that the quantum state ceases to exist.

Conformal and flat-space checks

For $m=0$ and $\xi=1/6$, the exact rescaled frequency is $k$. Then $W_k=k$ solves the nonlinear frequency equation at every order, and the conformally transported Minkowski state terminates the mode iteration. This checks the curvature-coupling sign and absence of mode mixing. It does not justify setting the renormalized stress tensor to zero: local curvature terms and the trace anomaly can survive without particle creation. In a massless adiabatic subtraction, taking $m=0$ before the regulated momentum integrals can also discard finite anomaly terms or create spurious infrared divergences. A massive regulator followed by its controlled massless limit, or a covariant subtraction with a specified reference scale, keeps that issue distinct from the exact plane-wave solution.

If $a$ is constant, derivatives vanish and $W_k=\sqrt{k^2+a^2m^2}$ is exact. For slowly varying backgrounds, the relevant high-momentum ratios include $\mathcal H/k=H/p$ and their higher-derivative analogues. A calculation that instead uses $H/k$ mixes a cosmic-time rate with a comoving momentum: although it is dimensionless when $a$ is dimensionless, it changes under a relabeling of comoving coordinates and is not the physical adiabatic parameter.

Diagnostic: choose the order for the observable

Suppose fourth-order WKB data are evolved exactly on a smooth FLRW spacetime. May the state be called Hadamard solely from that fact? Which subtraction orders are needed for $\langle\phi^2\rangle$ and $\langle T_{\mu\nu}\rangle$, and what must be checked if $W_k^{(4)2}$ becomes negative for small $k$?

Solution

Fourth adiabatic order alone does not establish the full Hadamard wavefront condition; it gives finite Sobolev control appropriate to a corresponding class of observables. In four dimensions, subtract through second order for the field fluctuation and through fourth order for the stress tensor. A negative low-$k$ truncated frequency signals failure of that WKB prescription outside its asymptotic ultraviolet regime. One must choose a positive smooth infrared completion or use exact modes, preserve the Wronskian, and verify that the change affects only smooth or infrared state data rather than the required large-$k$ singular structure.

The distinction matters because renormalizability of one composite observable is weaker than a state-independent microlocal admissibility statement for every local product. Repeating the construction at a higher finite order improves the ultraviolet falloff but never turns the word “finite” into an all-orders proof.

References

• Junker, Wolfgang, and Elmar Schrohe. “Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties.” Annales Henri Poincaré 3 (2002): 1113–1181. DOI. Open PDF, arXiv:math-ph/0109010.
• 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.