13.09.05 · gr-cosmology / microlocal-qft-curved-spacetimes

Hadamard states by pseudo-differential calculus (Gérard-Wrochna)

shipped3 tiersLean: none

Anchor (Master): Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 7-8; Gérard & Wrochna, *Comm. Math. Phys.* 325 (2014) 713 (direct pseudo-differential construction); Gérard & Wrochna, *Ann. Henri Poincaré* 18 (2017) 2715 (in/out-state Hadamard property on asymptotically static spacetimes); Junker & Schrohe, *Ann. Henri Poincaré* 3 (2002) 1113 (adiabatic-vacuum precursor); Gérard & Stoskopf, *Lett. Math. Phys.* 111 (2021) 95 (curved-spacetime extensions); Seeley, *Proc. Symp. Pure Math.* 10 (1967) 288 (complex powers of an elliptic operator); Hörmander, *The Analysis of Linear Partial Differential Operators*, Vol. III (Springer, 1985), §18 (pseudo-differential calculus on a manifold) and §26 (propagation of singularities)

Intuition Beginner

The previous unit 13.09.04 established that every globally hyperbolic spacetime admits at least one Hadamard quasi-free state, via the Fulling-Narcowich-Wald 1981 deformation argument. The proof was conceptually clean — deform the metric to an ultrastatic reference, pull the ground state back along the deformation, verify preservation of the wave-front-set condition by propagation of singularities. It was also, by construction, non-constructive in any practically useful sense. The deformation path and the resulting two-point function are existence statements, not formulas, and no concrete Hadamard state on, say, the FLRW cosmology or the Schwarzschild exterior comes out of the FNW argument directly.

The question that drove the next twenty years of work was: can one construct a Hadamard state explicitly on a given globally hyperbolic spacetime, without invoking an auxiliary ultrastatic reference and without invoking a smooth one-parameter family of metrics? The first systematic answer came from Wolfgang Junker in 1996, who introduced the adiabatic-vacuum-approximation programme: pick a Cauchy hypersurface $Σ$ , decompose the Klein-Gordon Cauchy problem mode-by-mode using a finite-order smooth approximation of $D_{h}$ on $Σ$ , and produce a state whose Cauchy-data covariance approximates the would-be exact Hadamard form to a controlled order. Taking the order of approximation to infinity gives a genuine Hadamard state, supplied by an explicit construction rather than an existence theorem.

Christian Gérard and Michał Wrochna in 2014 completed this programme. The construction takes one Cauchy hypersurface $Σ \subset (M, g)$ and one input: a pseudo-differential operator $ϵ$ on $Σ$ that serves as a parametrix for the positive square root $D_{h}$ of the spatial Klein-Gordon operator $D_{h} := - Δ_{h} + m^{2} + V$ . The existence of such an $ϵ \in Ψ^{1} (Σ)$ , with the right principal symbol and with the error $ϵ^{2} - D_{h}$ smoothing, is the Seeley 1967 complex-powers construction applied to the elliptic positive self-adjoint operator $D_{h}$ .

From the pair $(ϵ, σ_{h})$ — operator and Cauchy-data symplectic form — one writes down a quasi-free state on the CCR algebra with an explicit covariance formula, and Gérard-Wrochna prove that this state is Hadamard in the Radzikowski wave-front-set sense.

The mechanism is microlocal. The principal symbol of $ϵ$ , by the Seeley construction, equals $p_{h} (x, ξ)$ — the positive square root of the principal symbol of the spatial Klein-Gordon operator. This encodes the future-pointing positive-frequency selection on $T^{*} Σ$ : it picks out the upper half of the spatial mass shell at each point of $Σ$ , rather than the lower (antiparticle) half.

The Cauchy evolution of these data forward in time, via the Klein-Gordon equation on $(M, g)$ , is governed by Hörmander's propagation-of-singularities theorem: the wave-front set of the two-point function on $T^{*} M$ propagates along the bicharacteristic flow of the full spacetime operator $P = □_{g} + m^{2} + V$ , with the positive-frequency boundary condition on $Σ$ propagating to the future-pointing half of the bicharacteristic relation on $M$ . This is exactly the Radzikowski criterion.

The construction is explicit and computable: pick any pseudo-differential parametrix $ϵ$ of $D_{h}$ with the right principal symbol — there are many of them, all differing by lower-order pseudo-differential operators — and a single covariance formula, built from the Cauchy-data fields paired with $ϵ$ and $ϵ^{- 1}$ and averaged over the slice $Σ$ against the induced volume element, writes down a Hadamard-state covariance directly. Different choices of $ϵ$ produce different Hadamard states, all differing by smooth corrections (the uniqueness-up-to-smooth-corrections principle of 13.09.03). For practical cosmological calculations (Bunch-Davies vacuum on de Sitter, adiabatic vacua on FLRW) and black-hole calculations (Hartle-Hawking state on Schwarzschild-Kruskal), this construction replaces the FNW existence-only result by an effective construction with controlled symbol-class error bounds.

The conceptual upshot: the existence-of-Hadamard-states programme is closed by two complementary results. FNW 1981 gives the conceptual existence theorem on every globally hyperbolic spacetime — there exists at least one Hadamard state, by deformation from an ultrastatic reference. Gérard-Wrochna 2014 gives the explicit construction whenever the Cauchy hypersurface $Σ$ supports the requisite pseudo-differential calculus on $L^{2} (Σ)$ . Together they make the algebraic-QFT-on-curved-spacetimes programme not just well-defined but computationally accessible, with the symbol calculus of $Σ$ replacing the deformation argument as the working construction.

Visual Beginner

The picture to hold is a single globally hyperbolic spacetime $(M, g)$ with a horizontal Cauchy slice $Σ$ drawn across the middle, on which a pseudo-differential operator $ϵ \approx D_{h}$ is constructed by Seeley calculus; the two-point function of the resulting state is built directly from $(ϵ, σ_{h})$ , with no auxiliary ultrastatic reference and no deformation.

Three pieces drive the picture. The Cauchy hypersurface $Σ$ supplies the working slice on which everything lives — by the Bernal-Sánchez splitting 13.09.01, every globally hyperbolic spacetime admits such a smooth spacelike Cauchy slice with induced Riemannian metric $h$ . The pseudo-differential operator $ϵ \in Ψ^{1} (Σ)$ is the Seeley square root of the spatial Klein-Gordon operator $D_{h} := - Δ_{h} + m^{2} + V$ , with principal symbol $p_{h} (x, ξ)$ and smoothing error $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ .

The covariance formula — half the slice-average over $Σ$ of the position-data paired through $ϵ$ plus the momentum-data paired through $ϵ^{- 1}$ , against the induced volume element — is the explicit Hadamard-form initial data; the two-point function on $M \times M$ obtained by Cauchy-evolving this data forward by the Klein-Gordon equation satisfies the Radzikowski wave-front-set condition.

The contrast with the FNW picture of 13.09.04 is sharp. There, the picture had two spacetimes — an ultrastatic reference $(M, g_{0})$ on the left and a target $(M, g_{1})$ on the right, connected by a deformation arrow. Here, there is only one spacetime, and the construction happens at a single Cauchy slice $Σ$ inside it.

The Seeley pseudo-differential machinery replaces the deformation argument as the technical bridge: instead of "deform the metric, pull back the state, preserve the wave-front-set condition", the workflow is "approximate $D_{h}$ pseudo-differentially on $Σ$ , write down the covariance formula, verify the wave-front-set condition by propagation of singularities". The trade-off — pseudo-differential calculus on $Σ$ versus a smooth family of globally hyperbolic metrics — is the trade-off between an analytic machine (Seeley calculus on a Riemannian manifold) and a geometric one (smoothly-varying Lorentzian metrics on a smooth manifold).

Worked example Beginner

Build the Gérard-Wrochna construction on the simplest case — Minkowski spacetime with the standard time-zero Cauchy slice — and check that it reproduces the standard Minkowski vacuum.

Step 1. Take the spacetime as four-dimensional Minkowski: the real time line crossed with three-dimensional Euclidean space, with the standard flat metric. The natural Cauchy hypersurface is the time-zero slice, three-dimensional Euclidean space itself, with the standard Euclidean metric. The spatial Klein-Gordon operator on this slice is minus the Laplacian plus the mass squared, acting on square-integrable functions on three-dimensional Euclidean space.

Step 2. Compute the Seeley square root. The spatial Klein-Gordon operator has continuous spectrum from the mass-squared upward, with plane-wave eigenfunctions. Acting on a plane wave at spatial wavenumber, the operator multiplies by the wavenumber-squared plus mass-squared. The positive self-adjoint square root acts on the same plane wave by multiplication by the corresponding angular frequency — the square root of wavenumber-squared plus mass-squared. This is a Fourier-multiplier operator with multiplier the angular frequency; in pseudo-differential terms it has principal symbol exactly the positive square root of the principal symbol of the spatial Klein-Gordon operator, with smoothing error zero (the construction is exact in this constant-coefficient case). The Seeley calculus in this special setting reduces to ordinary Fourier-multiplier calculus.

Step 3. Write down the Cauchy-data covariance. With $ϵ$ acting as multiplication by the angular frequency on the Fourier side, the covariance formula reduces to a Fourier-side expression: $η (Φ_{1}, Φ_{2}) = \frac{1}{2}$ times the Fourier-side integral of $(u_{0}^{(1) *} ω_{k} u_{0}^{(2)} + u_{1}^{(1) *} ω_{k}^{- 1} u_{1}^{(2)})$ over three-dimensional momentum space, where $ω_{k} = ∣ k ∣^{2} + m^{2}$ . The symplectic form is the standard Cauchy-data symplectic form on a free field. The two-point function obtained by Cauchy-evolving these data is the standard Minkowski vacuum two-point function — the positive-frequency Wightman function familiar from flat-space QFT.

Step 4. Verify the Hadamard wave-front-set condition. By the Fourier-mode-decomposition computation from 13.09.04, the wave-front set of the Minkowski vacuum two-point function is on pairs $(x, k; y, - k)$ with $k$ on the upper mass shell — $k^{2} + m^{2} = 0$ , $k^{0} > 0$ , by the mostly-plus signature convention. This is exactly the future-pointing half of the bicharacteristic relation of the Minkowski metric. The Radzikowski criterion holds, the Minkowski vacuum is Hadamard. The Gérard-Wrochna construction recovers this standard result in the constant-coefficient case, with the pseudo-differential machinery reducing to ordinary Fourier-multiplier calculus.

Step 5. Now move to a curved background. Take a small smooth perturbation of Minkowski — say a metric of the form $- d t^{2} + (1 + ϵ_{χ} χ (x))^{2} δ_{ij} d x^{i} d x^{j}$ for a small parameter $ϵ_{χ}$ and a smooth compactly-supported bump function $χ$ on three-dimensional space. The time-zero Cauchy slice is the same three-dimensional space, but now with the perturbed Riemannian metric $h = (1 + ϵ_{χ} χ)^{2} δ_{ij} d x^{i} d x^{j}$ . The spatial Klein-Gordon operator $D_{h}$ on this slice is a small perturbation of the flat operator — still positive self-adjoint and elliptic, with positive principal symbol.

Step 6. Apply the Seeley construction. By Seeley 1967, the operator $D_{h}$ admits complex powers as classical pseudo-differential operators on the Riemannian manifold. In particular the positive self-adjoint square root $D_{h}$ has a $Ψ^{1}$ -pseudo-differential representative $ϵ \in Ψ^{1} (Σ)$ with principal symbol $p_{h} (x, ξ) = h^{ij} (x) ξ_{i} ξ_{j} + m^{2}$ and smoothing error $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ . The covariance formula now uses the curved-space inner product — averaging over $Σ$ against the induced volume element of $h$ — on three-dimensional space with the perturbed Riemannian metric. The resulting two-point function is the Gérard-Wrochna Hadamard state on this curved background, computable as a smooth perturbation of the Minkowski vacuum.

What this tells us: the Gérard-Wrochna construction is genuinely explicit. On Minkowski it reduces to the standard vacuum, and the Seeley pseudo-differential calculus reduces to ordinary Fourier-multiplier calculus. On a curved background the same recipe applies — pick a Cauchy slice, compute the spatial Klein-Gordon operator there, take the Seeley pseudo-differential square root, write down the covariance formula. The result is a Hadamard state given by an explicit formula, not just an existence statement. For cosmological FLRW spacetimes and for Schwarzschild-like black-hole backgrounds this is what makes concrete computations of cosmological perturbation power spectra and Hawking radiation possible.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Gérard-Wrochna 2014 construction of Hadamard states differs from the FNW 1981 deformation argument primarily in that

A. It uses a smooth one-parameter family of globally hyperbolic metrics rather than a single metric B. It constructs the state directly on the target spacetime via pseudo-differential calculus on a Cauchy hypersurface, with no auxiliary ultrastatic reference and no deformation C. It produces a state that is not Hadamard in general D. It requires an explicit Killing vector field on the spacetime

Hint

The whole point of the Gérard-Wrochna construction is to bypass the FNW deformation argument and give a direct, computable construction on the target spacetime itself.

Answer

B. It constructs the state directly on the target spacetime via pseudo-differential calculus on a Cauchy hypersurface, with no auxiliary ultrastatic reference and no deformation.

The Gérard-Wrochna 2014 Comm. Math. Phys. 325 713 construction takes a globally hyperbolic spacetime $(M, g)$ , picks a Cauchy hypersurface $Σ$ , constructs a pseudo-differential parametrix $ϵ \in Ψ^{1} (Σ)$ for the positive square root of the spatial Klein-Gordon operator $D_{h} = - Δ_{h} + m^{2} + V$ , and writes down the covariance — half the slice-average over $Σ$ of the position-data paired through $ϵ$ plus the momentum-data paired through $ϵ^{- 1}$ , against the induced volume element — directly. No deformation is performed; no auxiliary ultrastatic spacetime is invoked. The trade-off is that the construction requires the full pseudo-differential calculus on $Σ$ as input, while FNW uses only the explicit ultrastatic-ground-state spectral construction.

Option A is the FNW deformation, the opposite. Option C is wrong: the central theorem of Gérard-Wrochna 2014 is that the constructed state is Hadamard. Option D is wrong: the construction works on any globally hyperbolic spacetime, not only those with extra Killing-symmetry structure.

Exercise (easy, true-false).

True or false: the pseudo-differential parametrix $ϵ \in Ψ^{1} (Σ)$ of $D_{h}$ used in the Gérard-Wrochna construction is uniquely determined by the requirement that $ϵ^{2} - D_{h}$ be smoothing.

Hint

A parametrix of an elliptic operator is unique only up to smoothing corrections. Adding any operator in $Ψ^{- \infty} (Σ)$ to $ϵ$ gives another parametrix with the same principal symbol and the same smoothing error class.

Answer

False.

A pseudo-differential parametrix $ϵ$ of $D_{h}$ is unique only up to smoothing corrections in $Ψ^{- \infty} (Σ)$ . The principal symbol $σ_{1} (ϵ) = p_{h} (x, ξ)$ is uniquely determined by the principal symbol of $D_{h}$ , but the lower-order pseudo-differential corrections (the subprincipal symbol and below) are not — different choices give different parametrices, all of which satisfy $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ . The Seeley 1967 construction picks out one canonical choice (the complex-power construction via the resolvent contour integral), but any other choice with the right principal symbol works equally well for the Gérard-Wrochna construction. Different choices of $ϵ$ produce different Hadamard states on $(M, g)$ , all differing by smooth functions on $M \times M$ (the uniqueness-up-to-smooth-corrections principle of 13.09.03).

Exercise (medium, multiple choice).

The principal symbol of the pseudo-differential operator $ϵ \in Ψ^{1} (Σ)$ used in the Gérard-Wrochna construction is

A. $p_{h} (x, ξ) = h^{ij} (x) ξ_{i} ξ_{j} + m^{2}$ B. $p_{h} (x, ξ)$ C. $1/ p_{h} (x, ξ)$ D. $p_{h} (x, ξ)^{- 1}$

Hint

$ϵ$ is a pseudo-differential parametrix for $D_{h}$ , where $D_{h} = - Δ_{h} + m^{2} + V$ has principal symbol $p_{h}$ . The principal symbol of a square root is the positive square root of the principal symbol.

Answer

B. $p_{h} (x, ξ)$ .

The operator $ϵ \in Ψ^{1} (Σ)$ is required to be a parametrix for the positive self-adjoint square root $D_{h}$ , where $D_{h} = - Δ_{h} + m^{2} + V$ has principal symbol $p_{h} (x, ξ) = h^{ij} (x) ξ_{i} ξ_{j} + m^{2}$ (which is the contribution from the Laplacian and the mass; the potential $V$ enters at lower order). The pseudo-differential composition law gives the principal symbol of $ϵ^{2}$ as $σ_{1} (ϵ)^{2}$ , which must equal $σ_{2} (D_{h}) = p_{h}$ modulo lower-order terms. So $σ_{1} (ϵ) = p_{h}$ , the positive square root.

This is exactly the Seeley 1967 Proc. Symp. Pure Math. 10 calculation: for an elliptic positive self-adjoint operator $A \in Ψ^{m}$ , the complex power $A^{z}$ is a classical pseudo-differential operator of order $m z$ with principal symbol $σ_{m} (A)^{z}$ — the order-1/2 case ( $z = 1/2$ ) gives the square root.

Option A is the principal symbol of $D_{h}$ itself, not its square root. Option C is the principal symbol of $ϵ^{- 1} \approx (D_{h})^{- 1}$ (the inverse square root, which has principal symbol $1/ p_{h}$ ). Option D is the principal symbol of $D_{h}^{- 1}$ . The choice of positive square root (versus the negative root $- p_{h}$ ) is what encodes the future-pointing positive-frequency selection: the negative-root choice would give an "antiparticle vacuum" with wave-front set on the past-pointing half of the bicharacteristic relation, the wrong sign for a Hadamard state.

Exercise (medium, numeric).

Consider $D_{h} = - Δ_{h} + m^{2} + V$ on $L^{2} (Σ, d vol_{h})$ as a pseudo-differential operator on a smooth Riemannian manifold $Σ$ . The Seeley complex power $D_{h}^{z}$ is a classical pseudo-differential operator of order $2 z$ for $z \in C$ . What is the order of $ϵ = D_{h}^{1/2}$ ?

Hint

The complex power $D_{h}^{z}$ has order $m z$ where $m$ is the order of $D_{h}$ . The spatial Klein-Gordon operator $D_{h}$ is a second-order differential operator (Laplacian-type), so its order is 2. Set $z = 1/2$ and compute $m z$ .

Answer

$1$ .

The spatial Klein-Gordon operator $D_{h} = - Δ_{h} + m^{2} + V$ is a second-order differential operator on the Riemannian manifold $Σ$ — order 2 as a pseudo-differential operator, since it is a polynomial in derivatives of degree 2. By Seeley 1967 Proc. Symp. Pure Math. 10, for an elliptic positive self-adjoint pseudo-differential operator $A$ of order $m$ , the complex power $A^{z}$ is a classical pseudo-differential operator of order $m z$ . Applied to $D_{h}$ with order $m = 2$ , the complex power $D_{h}^{z}$ has order $2 z$ .

The square root corresponds to $z = 1/2$ , so $ϵ = D_{h}^{1/2}$ has order $2 \cdot (1/2) = 1$ . This is why the Gérard-Wrochna construction takes $ϵ \in Ψ^{1} (Σ)$ — a first-order pseudo-differential operator on the Cauchy hypersurface. The matching is dimensional: the spatial Klein-Gordon operator $D_{h}$ is second-order, so its square root is first-order, and the covariance formula — half the slice-average over $Σ$ of the position-data paired through $ϵ$ plus the momentum-data paired through $ϵ^{- 1}$ , against the induced volume element — pairs first-order $ϵ$ on the position-data $u_{0}$ with order- $(- 1)$ $ϵ^{- 1}$ on the momentum-data $u_{1}$ to give a well-defined symmetric Cauchy-data covariance.

Formal definition Intermediate+

Throughout this unit $(M, g)$ denotes a smooth four-dimensional connected time-oriented globally hyperbolic Lorentzian manifold in mostly-plus signature $(-, +, +, +)$ . By the Bernal-Sánchez theorem 13.09.01 we identify $M$ with $R \times Σ$ via a smooth diffeomorphism in which the metric takes the form $g = - β^{2} d t^{2} + h_{t}$ , with $β : M \to R_{> 0}$ a smooth lapse and $h_{t}$ a smoothly $t$ -varying Riemannian metric on $Σ$ . Fix the time-zero Cauchy slice $Σ := {0} \times Σ$ with induced Riemannian metric $h := h_{0}$ . The Klein-Gordon operator $P = □_{g} + m^{2} + V$ with $V \in C^{\infty} (M)$ , $m \geq 0$ , acts on smooth scalar fields as in 13.09.02, and the spatial Klein-Gordon operator on $Σ$ is

D_{h} := - Δ_{h} + m^{2} + V ∣_{Σ},

acting on $L^{2} (Σ, d vol_{h})$ . Assume $D_{h} \geq c > 0$ on $L^{2} (Σ)$ for some constant $c$ — i.e. the spatial Klein-Gordon operator is strictly positive (the gap condition that excludes zero-modes and ensures the square root is well-defined as a positive self-adjoint operator with bounded inverse).

The pseudo-differential calculus on $Σ$ , as developed in 02.14.02, supplies the symbol classes $S_{1, 0}^{m} (T^{*} Σ)$ , the operator classes $Ψ^{m} (Σ)$ , the principal-symbol map $σ_{m} : Ψ^{m} (Σ) / Ψ^{m - 1} (Σ) \to S^{m} / S^{m - 1}$ , the composition law $Ψ^{m} \cdot Ψ^{m^{'}} \subset Ψ^{m + m^{'}}$ with $σ_{m + m^{'}} (ab) = σ_{m} (a) σ_{m^{'}} (b)$ , and the parametrix construction for elliptic operators.

Definition (Seeley complex power and the parametrix $ϵ$ of $D_{h}$ ). Let $D_{h}$ be a strictly positive self-adjoint second-order elliptic pseudo-differential operator on a smooth closed Riemannian manifold $Σ$ (or on a complete Riemannian manifold with appropriate decay assumptions on the symbol). The Seeley complex power of $D_{h}$ at $z \in C$ is the operator

D_{h}^{z} := \frac{i}{2 π} \int_{γ} λ^{z} (λ - D_{h})^{- 1} d λ,

where $γ$ is a contour in $C$ encircling $spec (D_{h}) \subset [c, \infty)$ counterclockwise and avoiding the cut along the negative real axis chosen for $λ^{z}$ . By Seeley 1967 Proc. Symp. Pure Math. 10, $D_{h}^{z} \in Ψ^{2 z} (Σ)$ is a classical pseudo-differential operator of order $2 z$ with principal symbol $σ_{2 z} (D_{h}^{z}) (x, ξ) = p_{h} (x, ξ)^{z}$ , where $p_{h} (x, ξ) = h^{ij} (x) ξ_{i} ξ_{j} + m^{2}$ is the principal symbol of $D_{h}$ .

*The Seeley square root $ϵ := D_{h}^{1/2} \in Ψ^{1} (Σ)$ is the case $z = 1/2$ : a first-order classical pseudo-differential operator with principal symbol $σ_{1} (ϵ) (x, ξ) = p_{h} (x, ξ)$ (positive square root) and with $ϵ^{2} = D_{h}$ as operators on $L^{2} (Σ)$ (exactly, not modulo smoothing, by the spectral theorem). Any other $Ψ^{1}$ -pseudo-differential operator $\tilde{ϵ} \in Ψ^{1} (Σ)$ with principal symbol $p_{h}$ satisfies $\tilde{ϵ}^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ (smoothing) and is called a pseudo-differential parametrix of $D_{h}$ .*

The Seeley square root is the canonical choice; any other parametrix with the right principal symbol differs by a smoothing operator and produces a Hadamard state differing by a smooth correction.

Definition (Cauchy-data phase space and symplectic form). The symplectic phase space of Cauchy data of the Klein-Gordon field on $(M, g)$ is

(S, σ_{h}) := (C_{c}^{\infty} (Σ) \oplus C_{c}^{\infty} (Σ), σ_{h}), σ_{h} (Φ_{1}, Φ_{2}) := \int_{Σ} (u_{0}^{(1)} u_{1}^{(2)} - u_{0}^{(2)} u_{1}^{(1)}) d vol_{h},

for $Φ_{j} = (u_{0}^{(j)}, u_{1}^{(j)}) \in S$ . The symplectic form is the standard Cauchy-data symplectic pairing on a free Klein-Gordon field.

The central definition:

Definition (Gérard-Wrochna pseudo-differential covariance). Let $ϵ \in Ψ^{1} (Σ)$ be a pseudo-differential parametrix of $D_{h}$ on $Σ$ with principal symbol $p_{h}$ . The Gérard-Wrochna covariance is the real bilinear form on $S$ defined by

η_{ϵ} (Φ_{1}, Φ_{2}) := \frac{1}{2} \int_{Σ} (u_{0}^{(1)} ϵ u_{0}^{(2)} + u_{1}^{(1)} ϵ^{- 1} u_{1}^{(2)}) d vol_{h},

for $Φ_{j} = (u_{0}^{(j)}, u_{1}^{(j)}) \in S$ , where $ϵ^{- 1}$ is the bounded inverse of $ϵ$ on $L^{2} (Σ)$ (which exists because $ϵ$ is invertible by the gap condition $D_{h} \geq c > 0$ ).

The Gérard-Wrochna two-point function is $ω_{2}^{ϵ} = η_{ϵ} + \frac{i}{2} σ_{h}$ on $S \otimes S$ , extended to a quasi-free state on the CCR algebra $A (M, g)$ by the standard symplectic-purity construction.

The covariance $η_{ϵ}$ is symmetric and positive — symmetry by direct computation, positivity by the spectral-theorem inequality $η_{ϵ} \geq \frac{1}{4} σ_{h} \otimes σ_{h}^{*}$ (which follows from the spectral identity $ϵ \cdot ϵ^{- 1} = 1$ as in the Worked example of 13.09.04 Exercise 5). The state is pure (the equality $η_{ϵ} = \frac{1}{4} σ_{h} \otimes σ_{h}^{*}$ holds, by the Araki-Woods 1963 characterisation, since the operator $ϵ^{- 1} \cdot ϵ = 1$ exactly).

Counterexamples to common slips

The pseudo-differential parametrix $ϵ$ is not unique: it is determined only up to smoothing corrections by the requirement $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ . The Seeley complex-power construction picks out one canonical choice (via the contour-integral representation of $D_{h}^{1/2}$ ), but any other $Ψ^{1}$ -operator with principal symbol $p_{h}$ and smoothing error in $Ψ^{- \infty} (Σ)$ produces an equally valid Hadamard state, differing by a smooth correction on $M \times M$ . The Gérard-Wrochna construction is intrinsically a class of Hadamard states parametrised by pseudo-differential corrections, not a single state.
The choice of positive square root $p_{h}$ versus negative $- p_{h}$ as the principal symbol of $ϵ$ is what selects the future-pointing positive-frequency component. Taking the negative-root branch gives an "antiparticle vacuum" with wave-front set on the past-pointing half of the bicharacteristic relation, which is not Hadamard in the Radzikowski sense (the criterion requires the future-pointing half). The sign convention is load-bearing.
The covariance formula $η_{ϵ} = \frac{1}{2} \int_{Σ} (u_{0} ϵ u_{0} + u_{1} ϵ^{- 1} u_{1}) d vol_{h}$ is specific to the order-1 + order- $(- 1)$ pairing of the Cauchy data $(u_{0}, u_{1})$ . The position-data $u_{0}$ is paired with the order-1 operator $ϵ$ , the momentum-data $u_{1}$ with the order- $(- 1)$ operator $ϵ^{- 1}$ . Mixing the orders — e.g. $\frac{1}{2} \int_{Σ} (u_{0} ϵ^{- 1} u_{0} + u_{1} ϵ u_{1}) d vol_{h}$ — gives a different bilinear form, not the Gérard-Wrochna covariance, and produces a state that is not in general Hadamard.
The gap condition $D_{h} \geq c > 0$ for some constant $c$ is essential. If $D_{h}$ has zero in its spectrum (e.g. for $m = 0$ and $V \geq 0$ with a non-zero null space, or for $V < 0$ producing instabilities), the Seeley square root $D_{h}$ is not bounded below away from zero, the operator $ϵ^{- 1}$ does not exist as a bounded operator on $L^{2} (Σ)$ , and the covariance formula breaks down. The Gérard-Wrochna construction requires the positive-mass + positive-potential condition or some refinement that handles the spectral gap.
The construction requires the pseudo-differential calculus on $Σ$ , which in turn requires a smooth Riemannian structure on $Σ$ — supplied by the Bernal-Sánchez splitting and the induced Cauchy-slice metric. For non-smooth Cauchy hypersurfaces (which can arise in certain non-globally-hyperbolic generalisations) the construction breaks; the smooth-splitting theorem of 13.09.01 is the geometric prerequisite.

Key derivation Intermediate+

Theorem (Gérard-Wrochna 2014). Let $(M, g)$ be a globally hyperbolic Lorentzian manifold with Bernal-Sánchez Cauchy hypersurface $Σ$ and induced Riemannian metric $h$ on $Σ$ . Let $D_{h} := - Δ_{h} + m^{2} + V ∣_{Σ}$ on $L^{2} (Σ, d vol_{h})$ with $V \in C^{\infty} (M)$ and $m \geq 0$ . Assume $D_{h} \geq c > 0$ for some constant $c$ . Let $ϵ \in Ψ^{1} (Σ)$ be a pseudo-differential parametrix of $D_{h}$ with principal symbol $σ_{1} (ϵ) (x, ξ) = p_{h} (x, ξ)$ where $p_{h}$ is the principal symbol of $D_{h}$ . Then the Gérard-Wrochna quasi-free state $ω^{ϵ}$ on the CCR algebra $A (M, g)$ with two-point function $ω_{2}^{ϵ} = η_{ϵ} + \frac{i}{2} σ_{h}$ is Hadamard in the sense of the Radzikowski wave-front-set criterion 13.09.03:

WF (ω_{2}^{ϵ}) = {(x_{1}, k_{1}; x_{2}, - k_{2}) \in T^{*} (M \times M) ∖ 0 : (x_{1}, k_{1}) \sim_{bichar}^{g} (x_{2}, k_{2}), k_{1} \in \overline{V}_{x_{1}}^{+} (g)} .

The proof is a microlocal computation that propagates the Cauchy-surface wave-front-set support of the data forward in time via the Klein-Gordon equation. The originating reference is Gérard-Wrochna 2014 Comm. Math. Phys. 325 713 ^{[Gérard-Wrochna 2014]}; the modern textbook formulation appears in Gérard 2019 ^{[Gérard 2019 Ch. 7]}.

Proof.

Step 1: existence and properties of the Seeley square root. By Seeley 1967 Proc. Symp. Pure Math. 10 ^{[Seeley 1967]}, applied to the positive self-adjoint elliptic operator $D_{h}$ on the Riemannian manifold $Σ$ , the complex power $D_{h}^{z}$ is a classical pseudo-differential operator of order $2 z$ for each $z \in C$ , with principal symbol $σ_{2 z} (D_{h}^{z}) (x, ξ) = p_{h} (x, ξ)^{z}$ where the branch of the complex power is chosen consistently with the spectral cut. In particular $D_{h}^{1/2} \in Ψ^{1} (Σ)$ has principal symbol $p_{h} (x, ξ)$ and satisfies $D_{h}^{1/2} \cdot D_{h}^{1/2} = D_{h}$ as operators on $L^{2} (Σ)$ (the spectral-theorem identity, exact rather than modulo smoothing). The chosen parametrix $ϵ$ may differ from $D_{h}^{1/2}$ by a smoothing operator $R \in Ψ^{- \infty} (Σ)$ , in which case $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ but $ϵ^{2} - D_{h}$ need not vanish exactly. By the gap condition $D_{h} \geq c > 0$ , the operators $ϵ$ and $ϵ^{- 1}$ are bounded on $L^{2} (Σ)$ (and on Sobolev spaces $H^{s} (Σ)$ by pseudo-differential mapping properties).

Step 2: the covariance is a positive symmetric bilinear form. Compute directly:

η_{ϵ} (Φ_{1}, Φ_{2}) = \frac{1}{2} \int_{Σ} (u_{0}^{(1)} ϵ u_{0}^{(2)} + u_{1}^{(1)} ϵ^{- 1} u_{1}^{(2)}) d vol_{h} .

Symmetry under $Φ_{1} \leftrightarrow Φ_{2}$ follows from self-adjointness of $ϵ$ and $ϵ^{- 1}$ on $L^{2} (Σ)$ (which holds because $ϵ$ is a positive self-adjoint pseudo-differential operator, by the Seeley spectral construction). Positivity: for $Φ = (u_{0}, u_{1}) \in S$ ,

η_{ϵ} (Φ, Φ) = \frac{1}{2} \int_{Σ} (u_{0} ϵ u_{0} + u_{1} ϵ^{- 1} u_{1}) d vol_{h} = \frac{1}{2} (⟨ ϵ^{1/2} u_{0}, ϵ^{1/2} u_{0} ⟩ + ⟨ ϵ^{- 1/2} u_{1}, ϵ^{- 1/2} u_{1} ⟩) \geq 0,

with equality only if $u_{0} = u_{1} = 0$ . The state $ω^{ϵ}$ defined by $ω_{2}^{ϵ} = η_{ϵ} + \frac{i}{2} σ_{h}$ on $S \otimes S$ extends to a positive normalised quasi-free state on the CCR algebra by the Araki-Woods 1963 construction 12.14.01.

Step 3: the state is pure. Compute the purity-saturation condition $η_{ϵ} = \frac{1}{4} σ_{h} σ_{h}^{*}$ in the appropriate weak sense. Pick a test data $Φ = (u_{0}, u_{1})$ and the conjugate test data $Φ^{'} = (ϵ^{- 1} u_{1}, - ϵ u_{0})$ . Compute the symplectic pairing:

σ_{h} (Φ, Φ^{'}) = \int_{Σ} (u_{0} \cdot (- ϵ u_{0}) - ϵ^{- 1} u_{1} \cdot u_{1}) d vol_{h} = - \int_{Σ} (u_{0} ϵ u_{0} + u_{1} ϵ^{- 1} u_{1}) d vol_{h} = - 2 η_{ϵ} (Φ, Φ) .

So $σ_{h} (Φ, Φ^{'})^{2} = 4 η_{ϵ} (Φ, Φ)^{2}$ . The covariance evaluated on the conjugate data is $η_{ϵ} (Φ^{'}, Φ^{'}) = \frac{1}{2} \int_{Σ} (ϵ^{- 1} u_{1} \cdot ϵ \cdot ϵ^{- 1} u_{1} + ϵ u_{0} \cdot ϵ^{- 1} \cdot ϵ u_{0}) d vol_{h} = \frac{1}{2} \int_{Σ} (ϵ^{- 1} u_{1} \cdot u_{1} + ϵ u_{0} \cdot u_{0}) d vol_{h} = η_{ϵ} (Φ, Φ)$ . The purity-saturation condition $4 η_{ϵ} (Φ, Φ) η_{ϵ} (Φ^{'}, Φ^{'}) = σ_{h} (Φ, Φ^{'})^{2}$ becomes $4 η_{ϵ} (Φ, Φ)^{2} = 4 η_{ϵ} (Φ, Φ)^{2}$ , which holds. The state is pure: GNS representation gives a bosonic Fock-space realisation with cyclic vacuum vector.

Step 4: wave-front-set computation on $Σ \times Σ$ . The Schwartz kernel of $ϵ$ on $Σ \times Σ$ is a distribution; its wave-front set, by pseudolocality of $Ψ^{1} (Σ)$ and the principal-symbol identity $σ_{1} (ϵ) = p_{h}$ , is

WF (ϵ -kernel) = {(x, ξ; x, - ξ) \in T^{*} Σ \times T^{*} Σ ∖ 0 : ξ \neq = 0}

(the diagonal in $T^{*} Σ$ , with the standard sign convention for the wave-front-set of the kernel of a pseudo-differential operator). Similarly $WF (ϵ^{- 1}$ -kernel $)$ is the same diagonal. The Cauchy-data covariance $η_{ϵ}$ on $Σ \times Σ$ thus has wave-front set on the diagonal ${(x, ξ; x, - ξ)}$ with $ξ \neq = 0$ . The future-pointing positive-frequency selection encoded by $σ_{1} (ϵ) = + p_{h}$ (positive square root, not negative) is what picks out the future-pointing half of the bicharacteristic relation when this data is evolved forward in time by the Klein-Gordon equation.

Step 5: extension to $M \times M$ via propagation of singularities. The two-point function $ω_{2}^{ϵ}$ on $M \times M$ is obtained from the Cauchy-data covariance $η_{ϵ} + \frac{i}{2} σ_{h}$ by solving the Cauchy problem for the Klein-Gordon equation $P ϕ = 0$ in each argument with the prescribed Cauchy data. By the well-posedness theorem of 13.09.02, this evolution is well-defined and gives a bi-distribution $ω_{2}^{ϵ} \in D^{'} (M \times M)$ .

The key microlocal step: $ω_{2}^{ϵ}$ satisfies $P ω_{2}^{ϵ} = 0$ in each argument, by construction (the Cauchy evolution is defined via the Klein-Gordon equation). By Hörmander's propagation-of-singularities theorem 02.14.03 applied to the real-principal-type operator $P = □_{g} + m^{2} + V$ in each argument, the wave-front set of $ω_{2}^{ϵ}$ on $M \times M$ is contained in the bicharacteristic-invariant set of $g$ and is invariant under the bicharacteristic flow of $g$ in each argument. The Cauchy-surface initial wave-front-set on $Σ \times Σ$ propagates forward in time along the bicharacteristic flow of $P$ in each argument.

Step 6: the positive-frequency boundary condition propagates. On the Cauchy hypersurface $Σ$ , the principal symbol $σ_{1} (ϵ) = p_{h}$ selects the future-pointing positive-frequency half of the spatial mass shell on $T^{*} Σ$ : at each point $x \in Σ$ and each spatial cotangent direction $ξ \in T_{x}^{*} Σ$ , the operator $ϵ$ contributes the eigenvalue $p_{h} (x, ξ) > 0$ (positive root). The Cauchy evolution of this data forward in time by the Klein-Gordon equation transports the positive-frequency selection along the bicharacteristic flow to give the future-pointing component of the bicharacteristic relation on $T^{*} M$ , i.e. cotangent vectors $k = (k_{0}, k) \in T_{x}^{*} M$ with $g^{μν} (x) k_{μ} k_{ν} = 0$ and $k$ in the closed future light cone $\overline{V}_{x}^{+} (g)$ . This is exactly the Radzikowski wave-front-set support condition.

Step 7: smoothing-error correction. The smoothing error $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ contributes only smooth corrections to the two-point function: any $Ψ^{- \infty} (Σ)$ operator has smooth integral kernel, so its contribution to $ω_{2}^{ϵ}$ is a smooth function on $M \times M$ (after Cauchy evolution, which preserves smoothness by 13.09.02). Smooth functions have empty wave-front set, so they do not contribute to $WF (ω_{2}^{ϵ})$ . The wave-front-set support of $ω_{2}^{ϵ}$ is therefore exactly the future-pointing bicharacteristic relation, modulo the smoothing-error contributions which do not affect the wave-front set.

Step 8: lower bound from the CCR commutator. By the CCR-commutator-condition lower bound on the wave-front set of any quasi-free state's two-point function (proved in the Full proof set of 13.09.03), $WF (ω_{2}^{ϵ})$ contains the full future-pointing half of the bicharacteristic relation. Combined with the upper bound from Step 6, $WF (ω_{2}^{ϵ})$ equals the future-pointing half of the bicharacteristic relation. The Radzikowski wave-front-set condition holds. $□$

Bridge. The Gérard-Wrochna construction supplies an explicit, computable Hadamard state on every globally hyperbolic spacetime where the spatial Klein-Gordon operator $D_{h}$ on the Cauchy hypersurface satisfies the gap condition $D_{h} \geq c > 0$ — which is the generic case for massive Klein-Gordon fields with non-negative potential. The construction extends to other free field theories (Dirac, Maxwell, Yang-Mills) by replacing the scalar spatial Klein-Gordon operator $D_{h}$ with the appropriate covariant Klein-Gordon-type operator on the relevant vector bundle over $Σ$ (Gérard-Stoskopf 2021). Downstream, 13.09.06 uses the Gérard-Wrochna construction as the explicit Hadamard reference for the construction of Wick polynomials and time-ordered products on a curved background (Brunetti-Fredenhagen-Köhler 1996; Hollands-Wald 2001). On de Sitter, the maximal-symmetric-slice Bunch-Davies state emerges as the Gérard-Wrochna construction with the de-Sitter-invariant pseudo-differential parametrix; the explicit hypergeometric form of the Bunch-Davies two-point function can be read off from the Seeley-square-root construction applied to $D_{h}$ on the spherical Cauchy slice. The pattern — explicit pseudo-differential construction of Hadamard initial data on a Cauchy slice — recurs in every cosmological / black-hole application of the curved-spacetime QFT programme.

Exercises Intermediate+

Exercise 1 (easy, symbolic).

Compute the Seeley square root $ϵ = D_{h}^{1/2}$ for the flat Euclidean Cauchy slice $Σ = R^{3}$ with $D_{h} = - Δ + m^{2}$ , $m > 0$ . Express $ϵ$ as a Fourier-multiplier operator and identify its principal symbol.

Hint

On flat $R^{3}$ , $- Δ$ acts as multiplication by $∣ k ∣^{2}$ on the Fourier side. The operator $D_{h} = - Δ + m^{2}$ acts as multiplication by $∣ k ∣^{2} + m^{2}$ . The square root acts as multiplication by $∣ k ∣^{2} + m^{2}$ .

Answer

On $Σ = R^{3}$ with the flat Euclidean metric, the operator $D_{h} = - Δ + m^{2}$ is a constant-coefficient differential operator and the Seeley square root reduces to ordinary Fourier-multiplier calculus. Acting on a Schwartz function $u \in S (R^{3})$ via the Fourier transform $\overset{u}{^} (k) = \int_{R^{3}} e^{- i k \cdot x} u (x) d^{3} x$ ,

(ϵ u) (x) = (2 π)^{- 3} \int_{R^{3}} e^{i k \cdot x} ∣ k ∣^{2} + m^{2} \overset{u}{^} (k) d^{3} k .

In pseudo-differential notation $ϵ = a (D)$ where $a (k) = ∣ k ∣^{2} + m^{2}$ is the full symbol (constant coefficient: depends only on $k$ , not on $x$ ). The principal symbol is $σ_{1} (ϵ) (x, k) = ∣ k ∣$ — the leading-order behaviour as $∣ k ∣ \to \infty$ , which is the positive square root of the principal symbol of $D_{h}$ (which is $∣ k ∣^{2}$ ). The lower-order subprincipal corrections come from the mass term: $∣ k ∣^{2} + m^{2} = ∣ k ∣ (1 + m^{2} / (2∣ k ∣^{2}) + O (∣ k ∣^{- 4}))$ , contributing $O (∣ k ∣^{- 1})$ corrections that are in $S_{1, 0}^{- 1}$ . The composition $ϵ^{2}$ equals $D_{h}$ exactly in this constant-coefficient case (no smoothing error), because the spectral theorem applied to a Fourier-multiplier operator gives the exact square root via multiplication. This is the flat-space special case where the Seeley contour integral reduces to ordinary spectral calculus and the smoothing error is zero rather than just smoothing.

Exercise 2 (easy, multiple choice).

The Seeley 1967 complex-power construction $D^{z} = \frac{i}{2 π} \int_{γ} λ^{z} (λ - D)^{- 1} d λ$ for an elliptic positive self-adjoint pseudo-differential operator $D$ of order $m$ produces, for $z = 1/2$ , an operator of order

A. $0$ B. $1$ C. $m /2$ D. $m$

Hint

Seeley 1967 Theorem 1: $D^{z} \in Ψ^{m z}$ where $m$ is the order of $D$ . Compute $m z$ for $z = 1/2$ .

Answer

C. $m /2$ .

The Seeley complex-power theorem says: for an elliptic positive self-adjoint pseudo-differential operator $D \in Ψ^{m} (M)$ on a closed Riemannian manifold $M$ , the complex power $D^{z}$ is a classical pseudo-differential operator of order $m z$ , i.e. $D^{z} \in Ψ^{m z} (M)$ . For $z = 1/2$ the order is $m /2$ . Applied to the spatial Klein-Gordon operator $D_{h} = - Δ_{h} + m^{2} + V$ which has order $m = 2$ (Laplacian-type, second-order), the square root has order $2 \cdot (1/2) = 1$ — confirming that $ϵ = D_{h}^{1/2} \in Ψ^{1} (Σ)$ is a first-order pseudo-differential operator on the Cauchy hypersurface. Option A would correspond to $z = 0$ (the identity); option B is the special case $m = 2$ , $z = 1/2$ ; option D is $z = 1$ (the operator itself).

Exercise 3 (medium, symbolic).

Show that two pseudo-differential parametrices $ϵ_{1}, ϵ_{2} \in Ψ^{1} (Σ)$ of $D_{h}$ , with the same principal symbol $p_{h}$ but possibly different subprincipal corrections, produce Gérard-Wrochna two-point functions $ω_{2}^{ϵ_{1}}, ω_{2}^{ϵ_{2}}$ that differ by a smooth function on $M \times M$ .

Hint

The difference $ϵ_{1} - ϵ_{2} \in Ψ^{0} (Σ)$ (at most), so $ϵ_{1}^{2} - ϵ_{2}^{2} \in Ψ^{1} (Σ)$ — but since both are parametrices of $D_{h}$ , $ϵ_{1}^{2} - ϵ_{2}^{2} = (ϵ_{1}^{2} - D_{h}) - (ϵ_{2}^{2} - D_{h}) \in Ψ^{- \infty} (Σ)$ is smoothing. Combined with Cauchy evolution preserving smoothness, the contributions of the difference to the two-point function are smooth.

Answer

Let $ϵ_{1}, ϵ_{2} \in Ψ^{1} (Σ)$ both have principal symbol $p_{h}$ and satisfy $ϵ_{j}^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ for $j = 1, 2$ . The difference $ϵ_{1} - ϵ_{2} \in Ψ^{0} (Σ)$ has zero principal symbol, so is at most of order $0$ — but a sharper argument is needed. Subtract the two parametrix conditions: $ϵ_{1}^{2} - ϵ_{2}^{2} = (ϵ_{1}^{2} - D_{h}) - (ϵ_{2}^{2} - D_{h}) \in Ψ^{- \infty} (Σ)$ is smoothing. Factor $ϵ_{1}^{2} - ϵ_{2}^{2} = (ϵ_{1} + ϵ_{2}) (ϵ_{1} - ϵ_{2}) + [ϵ_{2}, ϵ_{1} - ϵ_{2}]$ ; the commutator $[ϵ_{2}, ϵ_{1} - ϵ_{2}] \in Ψ^{0} (Σ)$ (composition of $Ψ^{1}$ and $Ψ^{0}$ minus the same is at most $Ψ^{0}$ by symbol calculus). The factor $ϵ_{1} + ϵ_{2} \in Ψ^{1} (Σ)$ is invertible (its principal symbol is $2 p_{h} > 0$ by the gap condition); its inverse is in $Ψ^{- 1} (Σ)$ . Multiplying:

ϵ_{1} - ϵ_{2} = (ϵ_{1} + ϵ_{2})^{- 1} (ϵ_{1}^{2} - ϵ_{2}^{2}) - (ϵ_{1} + ϵ_{2})^{- 1} [ϵ_{2}, ϵ_{1} - ϵ_{2}] \in Ψ^{- \infty} (Σ)

after iterating the argument (the commutator term is one order lower, so the iteration converges to a smoothing operator).

So $ϵ_{1} - ϵ_{2} \in Ψ^{- \infty} (Σ)$ is smoothing. Similarly $ϵ_{1}^{- 1} - ϵ_{2}^{- 1} = ϵ_{2}^{- 1} (ϵ_{2} - ϵ_{1}) ϵ_{1}^{- 1} \in Ψ^{- \infty} (Σ)$ (the composition of three $Ψ^{*}$ operators with the middle one smoothing produces a smoothing operator).

The Gérard-Wrochna covariance difference is

η_{ϵ_{1}} (Φ_{1}, Φ_{2}) - η_{ϵ_{2}} (Φ_{1}, Φ_{2}) = \frac{1}{2} \int_{Σ} (u_{0}^{(1)} (ϵ_{1} - ϵ_{2}) u_{0}^{(2)} + u_{1}^{(1)} (ϵ_{1}^{- 1} - ϵ_{2}^{- 1}) u_{1}^{(2)}) d vol_{h} .

Both operators in the integrand are in $Ψ^{- \infty} (Σ)$ , so their Schwartz kernels are smooth functions on $Σ \times Σ$ . The Cauchy-evolved two-point functions inherit this smoothness via the well-posedness of the Klein-Gordon Cauchy problem 13.09.02: smooth initial data propagate to smooth solutions on $M$ . So the difference $ω_{2}^{ϵ_{1}} - ω_{2}^{ϵ_{2}}$ is a smooth function on $M \times M$ , with empty wave-front set. This is the uniqueness-up-to-smooth-corrections principle of 13.09.03 applied to the Gérard-Wrochna construction: different parametrices give different Hadamard states, all in the same equivalence class modulo smooth functions.

Exercise 4 (medium, symbolic).

Verify that the Gérard-Wrochna construction on Minkowski spacetime, with the time-zero Cauchy slice $Σ = R^{3}$ and the standard Euclidean metric, reproduces the standard Minkowski vacuum two-point function for the massive Klein-Gordon field.

Hint

From Exercise 1, $ϵ$ on Minkowski's time-zero slice is the Fourier-multiplier operator with multiplier $ω_{k} = ∣ k ∣^{2} + m^{2}$ . Plug into the Gérard-Wrochna covariance formula and Cauchy-evolve forward in time.

Answer

On Minkowski $R^{1, 3} = R_{t} \times R_{x}^{3}$ with metric $g = - d t^{2} + δ_{ij} d x^{i} d x^{j}$ and Cauchy slice $Σ = {0} \times R^{3}$ with induced flat metric $h = δ_{ij} d x^{i} d x^{j}$ , the spatial Klein-Gordon operator is $D_{h} = - Δ + m^{2}$ , a constant-coefficient differential operator. The Seeley square root $ϵ = D_{h}^{1/2}$ is the Fourier-multiplier operator with multiplier $ω_{k} = ∣ k ∣^{2} + m^{2}$ (Exercise 1). Its inverse $ϵ^{- 1}$ has multiplier $ω_{k}^{- 1}$ .

The Gérard-Wrochna Cauchy-data covariance is

η_{ϵ} (Φ_{1}, Φ_{2}) = \frac{1}{2} \int_{R^{3}} (u_{0}^{(1)} ϵ u_{0}^{(2)} + u_{1}^{(1)} ϵ^{- 1} u_{1}^{(2)}) d^{3} x .

Pass to the Fourier side: using Plancherel and the Fourier-multiplier action of $ϵ$ ,

η_{ϵ} (Φ_{1}, Φ_{2}) = \frac{1}{2} (2 π)^{- 3} \int_{R^{3}} (\overset{u}{^}_{0}^{(1) *} ω_{k} \overset{u}{^}_{0}^{(2)} + \overset{u}{^}_{1}^{(1) *} ω_{k}^{- 1} \overset{u}{^}_{1}^{(2)}) d^{3} k .

The corresponding two-point function on $M \times M$ , obtained by Cauchy-evolving the data forward in time via the Klein-Gordon equation $(\partial_{t}^{2} + D_{h}) ϕ = 0$ with $ϕ ∣_{Σ} = u_{0}$ , $\partial_{t} ϕ ∣_{Σ} = u_{1}$ , has the explicit form on Fourier modes: each plane-wave mode at spatial wavenumber $k$ evolves as $ϕ (t, k) = cos (ω_{k} t) \overset{u}{^}_{0} (k) + ω_{k}^{- 1} sin (ω_{k} t) \overset{u}{^}_{1} (k)$ . The two-point function $ω_{2}^{ϵ} (x, y) = ω^{ϵ} (ϕ (x) ϕ (y))$ on smeared test functions is

ω_{2}^{ϵ} (f_{1}, f_{2}) = \int_{R^{3}} \frac{d ^{3} k}{( 2 π ) ^{3} \cdot 2 ω _{k}} \tilde{f}_{1} (- ω_{k}, - k)^{*} \tilde{f}_{2} (ω_{k}, k),

where $\tilde{f}_{j} (k^{0}, k) = \int f_{j} (x) e^{ik \cdot x} d^{4} x$ is the four-dimensional Fourier transform. This is exactly the standard Minkowski vacuum two-point function for the massive Klein-Gordon field: integration over the upper mass shell $k^{0} = + ω_{k}$ in four-momentum space with the standard Lorentz-invariant measure $(2 π)^{- 3} (2 ω_{k})^{- 1}$ . The Gérard-Wrochna construction on Minkowski recovers the Poincaré-vacuum two-point function exactly — the simplest case of the construction, where the pseudo-differential machinery reduces to ordinary Fourier-multiplier calculus, and the standard vacuum emerges as the canonical instance.

Exercise 5 (medium, short-answer).

Why does the Gérard-Wrochna construction select the positive square root $p_{h}$ as the principal symbol of $ϵ$ , rather than the negative root $- p_{h}$ ? What state would the negative-root choice produce?

Hint

The positive-frequency / future-pointing condition in the Radzikowski wave-front-set criterion requires the cotangent vectors at the first argument to be in the closed future light cone. The principal symbol of $ϵ$ on $T^{*} Σ$ corresponds, via the Cauchy evolution, to the positive-frequency selection on $T^{*} M$ .

Answer

The Radzikowski wave-front-set criterion 13.09.03 requires that for a Hadamard quasi-free state $ω$ , the wave-front set of the two-point function on $M \times M$ satisfies

WF (ω_{2}) = {(x_{1}, k_{1}; x_{2}, - k_{2}) : (x_{1}, k_{1}) \sim_{bichar}^{g} (x_{2}, k_{2}), k_{1} \in \overline{V}_{x_{1}}^{+} (g)},

where $\overline{V}_{x_{1}}^{+} (g)$ is the closed future light cone in $T_{x_{1}}^{*} M$ . The future-pointing condition is what distinguishes a Hadamard state from an "antiparticle vacuum" with wave-front set on the past-pointing half.

The principal symbol of $ϵ$ on $T^{*} Σ$ controls which half of the spatial mass shell at each Cauchy-slice point is selected. Choosing $σ_{1} (ϵ) = + p_{h}$ — the positive square root — corresponds, via the Cauchy evolution of the data by the Klein-Gordon equation, to the future-pointing half of the bicharacteristic relation on $T^{*} M$ . The reason is the time-evolution structure: in the Cauchy evolution $ϕ (t, k) = cos (ω_{k} t) \overset{u}{^}_{0} (k) + ω_{k}^{- 1} sin (ω_{k} t) \overset{u}{^}_{1} (k)$ on Minkowski (with $ω_{k} > 0$ by the positive-root choice), the positive-frequency phase $e^{- i ω_{k} t}$ appears with positive $ω_{k}$ , putting the wave-front-set support on $k^{0} = + ω_{k} > 0$ — the future-pointing upper mass shell.

The negative-root choice $σ_{1} (ϵ) = - p_{h}$ would reverse the sign in the Cauchy evolution: the positive-frequency phase $e^{+ i ω_{k} t}$ appears with $ω_{k} > 0$ , putting the wave-front-set support on $k^{0} = - ω_{k} < 0$ — the past-pointing lower mass shell. The resulting state would have wave-front set on the past-pointing half of the bicharacteristic relation, which violates the Radzikowski criterion and is therefore not Hadamard. Physically this state is the "antiparticle vacuum" — the time-reversed version of the Hadamard state, with all particle / antiparticle roles swapped. It is a perfectly valid quasi-free state on the CCR algebra, but it does not have the right short-distance singularity structure for the curved-spacetime renormalisation programme (its wave-front set is the wrong sign for Hadamard subtraction in the renormalised stress-energy tensor).

The positive-root selection is therefore load-bearing: it is the analytic encoding of the future-pointing positive-frequency condition that distinguishes a Hadamard state from its time-reverse.

Exercise 6 (medium, short-answer).

How does the Gérard-Wrochna construction relate to the Junker-Schrohe 2002 adiabatic-vacuum-of-order- $N$ construction? In what sense is the Gérard-Wrochna state the $N \to \infty$ limit of the Junker-Schrohe adiabatic vacua?

Hint

Junker-Schrohe 2002 use a finite-order pseudo-differential approximation of $D_{h}$ — a parametrix $ϵ_{N} \in Ψ^{1} (Σ)$ with $ϵ_{N}^{2} - D_{h} \in Ψ^{- N} (Σ)$ rather than $Ψ^{- \infty} (Σ)$ . The resulting state is not exactly Hadamard but has wave-front set differing from the Hadamard wave-front set by a $Ψ^{- N}$ -correction.

Answer

The Junker-Schrohe 2002 Ann. Henri Poincaré 3 1113 ^{[Junker-Schrohe 2002]} construction of an adiabatic vacuum of order $N$ uses a finite-order pseudo-differential approximation of $D_{h}$ . Specifically, pick a parametrix $ϵ_{N} \in Ψ^{1} (Σ)$ with $ϵ_{N}^{2} - D_{h} \in Ψ^{- N} (Σ)$ — i.e. the smoothing error is of order $- N$ , finite rather than $- \infty$ . The principal symbol is still $p_{h}$ , but the subprincipal expansion is truncated at order $- N$ . Applying the Gérard-Wrochna formula $η_{ϵ_{N}} = \frac{1}{2} \int_{Σ} (u_{0} ϵ_{N} u_{0} + u_{1} ϵ_{N}^{- 1} u_{1}) d vol_{h}$ gives a quasi-free state $ω^{(N)}$ on the CCR algebra. This state is not exactly Hadamard for finite $N$ : the wave-front set of its two-point function differs from the Hadamard wave-front set by a correction in $Ψ^{- N}$ (the smoothing-error contribution from $ϵ_{N}^{2} - D_{h} \in / Ψ^{- \infty}$ ), so the two-point function fails the Radzikowski criterion by a finite-order correction. Junker-Schrohe show that this finite-order failure is uniform in a precise sense (the "adiabatic" condition controls the asymptotic behaviour) and that the resulting state is physically admissible for renormalisation calculations: it has the right number of derivatives smooth across the diagonal to support a renormalised stress-energy tensor to a finite order, even though it is not exactly Hadamard.

The Gérard-Wrochna 2014 construction corresponds to taking $N = \infty$ in the Junker-Schrohe framework: $ϵ \in Ψ^{1} (Σ)$ with $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ smoothing exactly (in the limit sense, via the Seeley complex-power construction or an equivalent infinite-order parametrix construction). The smoothing-error contribution is now exactly smooth, the wave-front-set correction is zero, and the state is exactly Hadamard.

The relationship: Junker-Schrohe gives a family of approximate Hadamard states parametrised by the approximation order $N$ , with explicit error control; Gérard-Wrochna gives the exact Hadamard state as the $N \to \infty$ limit, supplied by the Seeley complex-power machinery. For practical computations the finite- $N$ Junker-Schrohe states are often more useful (the symbol calculations are explicit and the approximation order can be tuned to the renormalisation question at hand); the Gérard-Wrochna construction is the conceptual closure that proves the family converges to a genuine Hadamard state.

The historical sequence is: Junker 1996 Rev. Math. Phys. 8 1091 introduced the adiabatic-vacuum / pseudo-differential idea; Junker-Schrohe 2002 systematised the finite-order construction with explicit error control; Gérard-Wrochna 2014 closed the construction at infinite order with the rigorous Hadamard property. The three papers together form the pseudo-differential construction-of-Hadamard-states programme.

Exercise 7 (hard, symbolic).

Sketch the propagation-of-singularities argument that lifts the Cauchy-surface wave-front-set of the Gérard-Wrochna data on $Σ \times Σ$ to the future-pointing bicharacteristic relation on $M \times M$ . Identify the role of (i) the principal symbol $σ_{1} (ϵ) = p_{h}$ on $T^{*} Σ$ ; (ii) Hörmander's theorem applied to $P = □_{g} + m^{2} + V$ ; (iii) the future-orientation of the Cauchy hypersurface.

Hint

The wave-front set on $Σ \times Σ$ is on the diagonal ${(x, ξ; x, - ξ)}$ with $ξ \in T_{x}^{*} Σ$ non-zero. The Cauchy evolution lifts $ξ$ to a four-cotangent $k = (\pm ω_{k}, ξ) \in T_{x}^{*} M$ on the spacetime mass shell; the sign of $k^{0}$ is determined by the sign convention of $ϵ$ .

Answer

Setup. Let $ω_{2}^{ϵ}$ be the Gérard-Wrochna two-point function on $M \times M$ obtained by Cauchy-evolving the covariance $η_{ϵ} + \frac{i}{2} σ_{h}$ from $Σ \times Σ$ . The Cauchy evolution is well-posed by 13.09.02: smooth initial data on $Σ$ propagate uniquely to smooth solutions of $P u = 0$ on $M$ , and the propagation is continuous. The two-point function on $M \times M$ satisfies $P ω_{2}^{ϵ} = 0$ in each argument by construction.

Step (i): Cauchy-surface wave-front-set support. The wave-front set of $η_{ϵ}$ on $Σ \times Σ$ , as computed in Step 4 of the Key derivation, is on the diagonal ${(x, ξ; x, - ξ)}$ in $T^{*} Σ \times T^{*} Σ ∖ 0$ — by pseudolocality of $ϵ \in Ψ^{1} (Σ)$ and $ϵ^{- 1} \in Ψ^{- 1} (Σ)$ , plus the principal-symbol identity $σ_{1} (ϵ) = + p_{h}$ (positive root). The future-pointing positive-frequency selection encoded by the positive-root choice is captured in the spectral support of the operator $ϵ$ on $L^{2} (Σ)$ : the operator's spectrum is on $[c, \infty)$ (positive), and the eigenfunction with eigenvalue $p_{h} (x, ξ)$ corresponds to the positive-frequency mode at spatial wavenumber $ξ$ .

Step (ii): Hörmander's theorem for $P$ on $M$ . For $ω_{2}^{ϵ}$ regarded as a bi-distribution on $M \times M$ satisfying $P ω_{2}^{ϵ} = 0$ in each argument, Hörmander's propagation-of-singularities theorem 02.14.03 gives: the wave-front set $WF (ω_{2}^{ϵ})$ is contained in the characteristic variety $p^{- 1} (0) \times p^{- 1} (0)$ of $P \otimes P$ on $T^{*} (M \times M) ∖ 0$ , where $p (x, k) = g^{μν} (x) k_{μ} k_{ν} + m^{2}$ is the principal symbol of $P$ . The wave-front set is invariant under the bicharacteristic flow of $g$ in each cotangent argument: the Hamiltonian flow of $p$ on $T^{*} M ∖ 0$ acts on each argument, and the wave-front-set support is preserved under this flow.

Step (iii): Cauchy-surface boundary condition + future orientation. The Cauchy-surface initial wave-front-set support on $Σ \times Σ$ — the diagonal ${(x, ξ; x, - ξ)}$ in $T^{*} Σ \times T^{*} Σ$ — lifts under the Cauchy evolution to a wave-front-set support on $M \times M$ via the bicharacteristic flow of $g$ . For each $(x, ξ) \in T^{*} Σ ∖ 0$ with $ξ$ spacelike (since $Σ$ is spacelike), the bicharacteristic curve through $(x, k_{\pm})$ with $k_{\pm} = (\pm ω_{k}, ξ) \in T_{x}^{*} M$ on the mass shell ( $p (x, k_{\pm}) = - ω_{k}^{2} + ∣ ξ ∣_{h}^{2} + m^{2} = 0$ for $ω_{k} = ∣ ξ ∣_{h}^{2} + m^{2}$ ) carries the wave-front-set support to spacetime points $x^{'} = π (Φ_{t} (x, k_{\pm}))$ along the null/timelike geodesic with cotangent $k_{\pm}$ parallel-transported. The two choices $k_{+} = (+ ω_{k}, ξ)$ and $k_{-} = (- ω_{k}, ξ)$ correspond to the future-pointing and past-pointing halves of the bicharacteristic relation.

The positive-root choice $σ_{1} (ϵ) = + p_{h}$ on $Σ$ selects $k_{+}$ (the future-pointing lift) over $k_{-}$ . The mechanism: in the Cauchy evolution, the positive-frequency phase $e^{- i t D_{h}}$ acting on a spatial-Fourier-mode at wavenumber $ξ$ produces a four-spacetime-Fourier component at $(+ ω_{k}, ξ)$ — the future-pointing four-cotangent on the mass shell. The negative-root choice would have selected $(- ω_{k}, ξ)$ , the past-pointing four-cotangent. The future-orientation of the Cauchy hypersurface plus the positive-square-root choice of $σ_{1} (ϵ)$ together give the future-pointing selection on $T^{*} M$ .

Conclusion. The Cauchy-surface initial wave-front-set ${(x, ξ; x, - ξ)}$ on $Σ \times Σ$ with $ξ \neq = 0$ , lifted by the bicharacteristic flow of $g$ in each argument with the future-pointing $k_{+} = (+ ω_{k}, ξ)$ selection from the positive-root principal symbol, gives the wave-front-set support of $ω_{2}^{ϵ}$ on $M \times M$ as exactly the future-pointing half of the bicharacteristic relation:

WF (ω_{2}^{ϵ}) = {(x_{1}, k_{1}; x_{2}, - k_{2}) : (x_{1}, k_{1}) \sim_{bichar}^{g} (x_{2}, k_{2}), k_{1} \in \overline{V}_{x_{1}}^{+} (g)} .

The Radzikowski wave-front-set condition holds. The lower bound — that the wave-front-set support is all of the future-pointing bicharacteristic relation, not a subset — comes from the CCR-commutator-condition lower bound proved in the Full proof set of 13.09.03. Combined, $ω^{ϵ}$ is Hadamard. $□$

Exercise 8 (hard, short-answer).

Compare the explicit Hadamard-state constructions on three benchmark spacetimes — Minkowski, de Sitter, and Schwarzschild exterior — in terms of how the Gérard-Wrochna construction simplifies in each case. What features of the Cauchy slice $Σ$ and the spatial Klein-Gordon operator $D_{h}$ make the construction tractable, and what additional structure (Killing symmetry, conformal flatness) makes the resulting Hadamard state physically distinguished?

Hint

Minkowski: constant-coefficient $D_{h}$ , $ϵ$ is a Fourier multiplier. de Sitter: maximally symmetric, $D_{h}$ on the spherical Cauchy slice has explicit spherical-harmonic decomposition; the Gérard-Wrochna state is the Bunch-Davies vacuum. Schwarzschild: $D_{h}$ on the Killing-horizon-respecting Cauchy slice; the Gérard-Wrochna state related to the Hartle-Hawking state.

Answer

Minkowski. Take $Σ = R^{3}$ with the flat Euclidean metric, $D_{h} = - Δ + m^{2}$ a constant-coefficient differential operator. The Seeley square root $ϵ$ is the Fourier-multiplier operator with multiplier $ω_{k} = ∣ k ∣^{2} + m^{2}$ (Exercises 1, 4). The Gérard-Wrochna construction reduces to ordinary Fourier-multiplier calculus and the smoothing error $ϵ^{2} - D_{h}$ vanishes exactly (not just modulo smoothing). The resulting state is the standard Poincaré-invariant Minkowski vacuum — the unique state distinguished by Poincaré invariance. The construction is simplest here because (i) the Cauchy slice is $R^{3}$ (flat, non-compact, with translation symmetry); (ii) $D_{h}$ is constant-coefficient; (iii) Lorentz/Poincaré symmetry picks out a unique Hadamard state.

De Sitter. Take the spatial slice $Σ = S^{3}$ (three-sphere) at conformal time $τ$ , with the round metric scaled by the FLRW conformal factor $a (τ) = - 1/ (H τ)$ for $τ \in (- \infty, 0)$ in the de Sitter expansion. The spatial Klein-Gordon operator $D_{h}$ on $S^{3}$ with the time-dependent conformal factor has explicit spherical-harmonic decomposition: eigenfunctions of $- Δ_{S^{3}}$ are spherical harmonics $Y_{ℓ m}$ with eigenvalues $ℓ (ℓ + 2)$ , $ℓ = 0, 1, 2, \dots$ , and the Seeley square root $ϵ$ acts on each spherical-harmonic mode by multiplication by $ℓ (ℓ + 2) / a^{2} + m^{2}$ . The Gérard-Wrochna state with the de-Sitter-invariant choice of $ϵ$ (using the maximal-symmetric pseudo-differential parametrix) is the Bunch-Davies vacuum — the unique de-Sitter-invariant Hadamard state (Allen 1985). The construction is tractable because (i) the spatial Klein-Gordon operator has explicit spectral decomposition via spherical harmonics; (ii) the de Sitter symmetry group $S O (1, 4)$ picks out a unique invariant pseudo-differential parametrix. The explicit two-point function of the Bunch-Davies state, in terms of hypergeometric functions, can be computed mode-by-mode from the spherical-harmonic decomposition.

Schwarzschild exterior. Take the spatial slice as a constant- $t$ Cauchy slice $Σ_{t} = {r > 2 M}$ in Schwarzschild coordinates with induced spatial metric $h = (1 - 2 M / r)^{- 1} d r^{2} + r^{2} d Ω^{2}$ . The spatial Klein-Gordon operator $D_{h} = - Δ_{h} + m^{2}$ on this slice has no explicit closed-form spectral decomposition (unlike the spherical-harmonic case of de Sitter), but it is positive self-adjoint elliptic, and the Seeley complex-power construction applies. The Gérard-Wrochna state with a parametrix $ϵ$ that respects the bifurcate-Killing-horizon structure (the static Killing vector $\partial_{t}$ that generates the horizon) is the Hartle-Hawking state — the unique Hadamard state on the Kruskal extension that is invariant under the Killing-horizon flow (Kay-Wald 1991). The construction is more involved than the Minkowski or de Sitter cases because (i) the Cauchy slice is not flat and not spherically symmetric in the same way (Schwarzschild has spherical symmetry but no full $S O (4)$ acting on the spatial slice); (ii) the spectrum of $D_{h}$ is continuous from $m^{2}$ upward without explicit eigenfunctions; (iii) the Killing-horizon structure picks out the Hartle-Hawking choice of parametrix. The thermal spectrum of outgoing modes at the Hawking temperature $T = 1/ (8 π M)$ is read off from the analytic structure of the resulting two-point function under analytic continuation through the bifurcate horizon.

Common features. In all three cases the Cauchy slice $Σ$ is geodesically complete and the spatial Klein-Gordon operator $D_{h}$ is positive self-adjoint elliptic with positive principal symbol — the prerequisites for the Seeley complex-power construction. Additional symmetry (Poincaré for Minkowski, de Sitter for de Sitter, bifurcate Killing horizon for Schwarzschild) picks out a unique preferred pseudo-differential parametrix, and the Gérard-Wrochna construction with that parametrix reproduces the physically distinguished Hadamard state of each spacetime. On a generic globally hyperbolic spacetime without extra symmetry, the construction gives a non-empty class of Hadamard states parametrised by the choice of parametrix (differing by smooth corrections); on a spacetime with sufficient symmetry, the symmetry singles out one element of the class as the physically natural one.

Lean formalization Intermediate+

Mathlib has no Lorentzian-metric infrastructure, no d'Alembertian on a pseudo-Riemannian manifold, no wave-front-set machinery on a manifold, no pseudo-differential calculus on a Riemannian manifold, no Seeley complex-power construction, no CCR-algebra layer, no Hadamard-parametrix construction, and no theory of curved-spacetime QFT as of 2026-05. The closest layers are the smooth-manifold + Riemannian-metric apparatus (Geometry.Manifold.SmoothManifoldWithCorners, Geometry.Manifold.MetricSpace), the spectral theorem for unbounded self-adjoint operators (Mathlib.Analysis.NormedSpace.Spectrum), partial $C^{*}$ -algebra abstractions (Mathlib.Analysis.CStarAlgebra.Basic), and partial distribution theory (Mathlib.Analysis.Distribution).

The full chain of formalisation gaps identified in 13.09.01, 13.09.02, 13.09.03, 13.09.04, 02.14.01, 02.14.02, 02.14.03, and 12.14.01 must be filled before the Gérard-Wrochna pseudo-differential construction of Hadamard states can be stated in Lean. Above those layers, the present unit additionally requires (i) the Seeley 1967 complex-power construction $A^{z} \in Ψ^{m z} (M)$ for an elliptic positive self-adjoint operator $A \in Ψ^{m} (M)$ via the contour-integral representation, with the principal-symbol identity $σ_{m z} (A^{z}) = σ_{m} (A)^{z}$ and the verification that the resulting operator is a classical pseudo-differential operator; (ii) the parametrix theory for the square root $D_{h} \in Ψ^{1} (Σ)$ on a Riemannian manifold, with the smoothing-error class $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ ; (iii) the Cauchy-evolution formalism that maps Cauchy data on $Σ$ to a bi-distribution on $M \times M$ satisfying the Klein-Gordon equation in each argument; (iv) the propagation-of-singularities lifting of the Cauchy-surface wave-front-set to the future-pointing bicharacteristic relation on $T^{*} M$ .

Each of these is a substantial Mathlib contribution in its own right. The Seeley complex-power construction in particular is a foundational result of microlocal analysis with no Mathlib representation; the closest current Lean 4 work is in the SpaceTime project (M. Larson and collaborators, outside Mathlib) with a partial Lorentzian-metric layer, but nothing in the pseudo-differential-calculus-on-a-Riemannian-manifold direction has been formalised.

lean_status: none reflects this. No Lean module ships with this unit. the Mathlib gap analysis names the specific layered infrastructure that must be built. Tyler's review attests Intermediate-tier correctness of the Seeley pseudo-differential square root, the Gérard-Wrochna covariance formula, the purity-saturation calculation, and the propagation-of-singularities lifting argument. The Master-tier comparison with Junker-Schrohe 2002 adiabatic vacua and the Gérard-Wrochna 2017 in/out-state refinement are flagged for external review by a microlocal-QFT specialist.

Advanced results Master

Four structural developments extend the Gérard-Wrochna pseudo-differential construction to the depth required by the modern curved-spacetime QFT programme.

The Junker-Schrohe 2002 adiabatic-vacuum hierarchy as the precursor. Junker-Schrohe 2002 Ann. Henri Poincaré 3 1113 ^{[Junker-Schrohe 2002]} developed the adiabatic vacuum of order $N$ framework. Given a globally hyperbolic spacetime $(M, g)$ and a Cauchy hypersurface $Σ$ , pick a parametrix $ϵ_{N} \in Ψ^{1} (Σ)$ with the property that $ϵ_{N}^{2} - D_{h} \in Ψ^{- N} (Σ)$ — i.e. the smoothing error is of finite order $- N$ rather than $- \infty$ . The principal symbol is still $σ_{1} (ϵ_{N}) = p_{h}$ , but the subprincipal terms are truncated at order $- N$ . The resulting Gérard-Wrochna-type covariance $η_{ϵ_{N}}$ gives a state $ω^{(N)}$ that is not exactly Hadamard but has wave-front-set differing from the Hadamard wave-front-set by a $Ψ^{- N}$ correction. Junker-Schrohe show that $ω^{(N)}$ is physically admissible for renormalisation calculations to order $N$ in the Wick-polynomial / time-ordered-product programme — for large enough $N$ (depending on the perturbative-QFT order needed), the adiabatic vacuum gives the same renormalised stress-energy tensor as a true Hadamard state. The Gérard-Wrochna 2014 construction is the $N \to \infty$ limit: $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ , exact Hadamard property. The adiabatic-vacuum hierarchy is the practical computational tool; the Gérard-Wrochna closure is the conceptual existence-and-construction theorem.

The Gérard-Wrochna 2017 in/out-state refinement on asymptotically static spacetimes. Gérard-Wrochna 2017 Ann. Henri Poincaré 18 2715 ^{[Gérard-Wrochna 2017]} extends the pseudo-differential construction to asymptotically static spacetimes — spacetimes that approach ultrastatic spacetimes in the asymptotic past ( $t \to - \infty$ ) and future ( $t \to + \infty$ ), with explicit decay rates on the metric perturbation. The in-state $ω^{in}$ is the Gérard-Wrochna state on the asymptotic-past ultrastatic spacetime, evolved forward to the present; the out-state $ω^{out}$ is the analogous construction from the asymptotic-future ultrastatic spacetime, evolved backward. Both are Hadamard, by the Gérard-Wrochna 2014 construction on the asymptotic ultrastatic spacetimes plus propagation of singularities through the asymptotically-static region. The S-matrix of the curved-spacetime QFT relates $ω^{in}$ and $ω^{out}$ via the Bogoliubov transformation of Cauchy-data symplectic frames between past and future asymptotic infinity; particle creation (curved-spacetime analogue of the Hawking effect for asymptotically static rather than horizon-respecting spacetimes) emerges from the non-zero Bogoliubov coefficients. The 2017 refinement was extended to Dirac fields by Gérard-Stoskopf 2021 Lett. Math. Phys. 111 95 ^{[Gérard-Stoskopf 2021]}, showing the universality of the pseudo-differential approach across covariant Klein-Gordon-type operators on vector bundles.

The Brunetti-Fredenhagen-Köhler 1996 extension to all $n$ -point functions and locally covariant Wick polynomials. As in 13.09.04, a quasi-free state is determined by its two-point function, but the full hierarchy of $n$ -point functions $ω_{n}^{(N)}$ must satisfy the microlocal-spectrum condition $μ SC$ for the state to support renormalised Wick polynomials and time-ordered products. The Gérard-Wrochna construction supplies a two-point function satisfying the Radzikowski criterion; by the Wick-rule reconstruction of $n$ -point functions for quasi-free states, the full hierarchy automatically satisfies $μ SC$ — the BFK 1996 extension goes through. The construction therefore plugs directly into the Hollands-Wald 2001/2002 programme for renormalised Wick polynomials and time-ordered products on a curved background: pick a Gérard-Wrochna Hadamard state, use its two-point function as the Hadamard parametrix subtraction reference, and the renormalisation procedure gives finite Wick polynomials and time-ordered products with explicit symbol-class error control. This is the practical advance over the FNW deformation construction, where the existence-only Hadamard state could not be used effectively for renormalisation computations.

Reduction to the FNW deformation argument and the BFV locally covariant framework. The Gérard-Wrochna construction subsumes the FNW deformation existence theorem in the following sense: by Gérard-Wrochna 2014, every globally hyperbolic spacetime with $D_{h} \geq c > 0$ admits an explicit pseudo-differential Hadamard state, so the FNW existence theorem follows as a corollary (with the Gérard-Wrochna state as the explicit witness). The gap condition $D_{h} \geq c > 0$ excludes some pathological cases (e.g. $m = 0$ with $V \equiv 0$ on a compact Cauchy slice produces a zero-mode that the gap condition rules out), but for the generic case the Gérard-Wrochna construction is at least as strong as FNW. In the Brunetti-Fredenhagen-Verch 2003 Comm. Math. Phys. 237 31 locally covariant QFT functor framework, the Gérard-Wrochna construction gives a natural assignment $(M, g) \mapsto ω^{ϵ (M, g)}$ — pick the canonical Seeley parametrix on each Cauchy hypersurface, get a canonical Hadamard state. The locally covariant version of this assignment (compatibility with isometric embeddings) is the Sahlmann-Verch 2001 Rev. Math. Phys. 13 1203 theorem, which the Gérard-Wrochna construction makes explicit at the level of pseudo-differential symbols.

Synthesis. The Gérard-Wrochna pseudo-differential construction is the computational closure of the existence-of-Hadamard-states programme. It replaces the FNW deformation existence theorem by an explicit construction with computable error bounds, plugs directly into the Hollands-Wald renormalisation framework via the Wick-polynomial / time-ordered-product subtraction procedure, gives the canonical Hadamard state on the standard cosmological and black-hole backgrounds (Bunch-Davies on de Sitter, Hartle-Hawking on Schwarzschild-Kruskal), and extends to vector-valued fields via Gérard-Stoskopf 2021. The pattern — pseudo-differential calculus on the Cauchy hypersurface produces explicit Hadamard initial data — is now the standard working construction in the curved-spacetime QFT literature.

Full proof set Master

Proposition (Seeley square root of an elliptic positive self-adjoint operator). Let $A$ be a positive self-adjoint elliptic classical pseudo-differential operator of order $m > 0$ on a smooth closed Riemannian manifold $M$ , with $A \geq c > 0$ for some constant $c$ . The Seeley complex power

A^{z} := \frac{i}{2 π} \int_{γ} λ^{z} (λ - A)^{- 1} d λ,

with $γ$ a contour in $C$ encircling $spec (A) \subset [c, \infty)$ counterclockwise and avoiding the branch cut chosen for $λ^{z}$ , is a classical pseudo-differential operator of order $m z$ with principal symbol $σ_{m z} (A^{z}) (x, ξ) = σ_{m} (A) (x, ξ)^{z}$ . In particular $A^{1/2} \in Ψ^{m /2} (M)$ with principal symbol $σ_{m} (A)$ and $A^{1/2} \cdot A^{1/2} = A$ as operators on $L^{2} (M)$ .

Proof. By Seeley 1967 Proc. Symp. Pure Math. 10 ^{[Seeley 1967]}. The resolvent $(λ - A)^{- 1}$ for $λ \in C ∖ spec (A)$ is a classical pseudo-differential operator of order $- m$ (parametrix construction for the elliptic operator $λ - A$ ). The contour integral $\int_{γ} λ^{z} (λ - A)^{- 1} d λ$ converges in the operator-norm topology on $L^{2} (M)$ and, by careful estimates on the symbol asymptotic expansion of $(λ - A)^{- 1}$ uniformly in $λ$ on $γ$ , defines a classical pseudo-differential operator of order $m z$ with full symbol given by the contour-integral asymptotic of the resolvent symbol. The principal-symbol identity $σ_{m z} (A^{z}) = σ_{m} (A)^{z}$ follows from the leading-order behaviour of the resolvent symbol $(λ - σ_{m} (A))^{- 1}$ contour-integrated against $λ^{z}$ . The composition $A^{1/2} \cdot A^{1/2} = A$ follows from the spectral-theorem identity $f (A) g (A) = (f g) (A)$ for the functional calculus, applied with $f (λ) = g (λ) = λ$ . $□$

Proposition (Gérard-Wrochna covariance is a pure quasi-free state on the CCR algebra). Let $(M, g)$ be globally hyperbolic with Cauchy hypersurface $Σ$ and induced Riemannian metric $h$ . Let $D_{h} := - Δ_{h} + m^{2} + V ∣_{Σ} \geq c > 0$ on $L^{2} (Σ, d vol_{h})$ for some constant $c$ . Let $ϵ \in Ψ^{1} (Σ)$ be a parametrix of $D_{h}$ with principal symbol $p_{h}$ . The bilinear form $η_{ϵ} (Φ_{1}, Φ_{2}) := \frac{1}{2} \int_{Σ} (u_{0}^{(1)} ϵ u_{0}^{(2)} + u_{1}^{(1)} ϵ^{- 1} u_{1}^{(2)}) d vol_{h}$ on Cauchy data $Φ_{j} = (u_{0}^{(j)}, u_{1}^{(j)}) \in S = C_{c}^{\infty} (Σ) \oplus C_{c}^{\infty} (Σ)$ is a positive symmetric bilinear form satisfying the purity-saturation condition $\eta_\epsilon = \tfrac{1}{4}\sigma_h \sigma_h^ $, h e n ce d e f in es a p u r e q u a s i - f r ees t a t eo n t h e C C R a l g e b r a$ \mathcal{A}(M, g) $w i t h tw o - p o in t f u n c t i o n$ \omega_2^\epsilon = \eta_\epsilon + \tfrac{i}{2}\sigma_h$.*

Proof. Steps 2 and 3 of the Key derivation. Symmetry follows from self-adjointness of $ϵ$ and $ϵ^{- 1}$ on $L^{2} (Σ)$ ; positivity from the spectral-theorem identity $η_{ϵ} (Φ, Φ) = \frac{1}{2} (⟨ ϵ^{1/2} u_{0}, ϵ^{1/2} u_{0} ⟩ + ⟨ ϵ^{- 1/2} u_{1}, ϵ^{- 1/2} u_{1} ⟩) \geq 0$ with equality only for $Φ = 0$ . Purity from the explicit conjugate-data calculation $Φ^{'} = (ϵ^{- 1} u_{1}, - ϵ u_{0})$ giving $σ_{h} (Φ, Φ^{'})^{2} = 4 η_{ϵ} (Φ, Φ) η_{ϵ} (Φ^{'}, Φ^{'})$ . The Araki-Woods 1963 construction 12.14.01 gives the bosonic Fock-space realisation. $□$

Proposition (Gérard-Wrochna 2014, wave-front-set computation on the Cauchy hypersurface). Let $ϵ \in Ψ^{1} (Σ)$ be a parametrix of $D_{h}$ with principal symbol $σ_{1} (ϵ) (x, ξ) = + p_{h} (x, ξ)$ (positive square root). The Schwartz kernel of $ϵ$ on $Σ \times Σ$ is a distribution with wave-front set on the diagonal ${(x, \xi; x, -\xi) : x \in \Sigma, \xi \in T^_x\Sigma \setminus 0} $, an d s imi l a r l y f or$ \epsilon^{-1} \in \Psi^{-1}(\Sigma)$.*

Proof. Standard pseudolocality of pseudo-differential operators: for $A \in Ψ^{m} (M)$ , the Schwartz kernel $K_{A} \in D^{'} (M \times M)$ has wave-front set $WF (K_{A}) \subseteq {(x, ξ; y, η) : x = y, η = - ξ}$ — i.e. on the diagonal in $T^{*} M \times T^{*} M$ with reversed cotangent on the second factor. This holds for any $A \in Ψ^{m} (M)$ regardless of $m$ , by the symbol-asymptotic-expansion definition of $Ψ^{m}$ . The principal symbol of $A$ determines the strength of the singularity on the diagonal (the order of the conormal distribution) but not its support. Applied to $ϵ \in Ψ^{1} (Σ)$ and $ϵ^{- 1} \in Ψ^{- 1} (Σ)$ , the Schwartz kernels have wave-front set on the diagonal in $T^{*} Σ \times T^{*} Σ$ as claimed. The positive-square-root choice $σ_{1} (ϵ) = + p_{h}$ does not affect the wave-front-set support (which is purely the diagonal); it affects the time-evolution behaviour of the data, as in the next proposition. $□$

Proposition (Cauchy-evolution lift of wave-front-set support). Let $ω_{2}^{ϵ}$ be the bi-distribution on $M \times M$ obtained by Cauchy-evolving the Gérard-Wrochna data $η_{ϵ} + \frac{i}{2} σ_{h}$ from $Σ \times Σ$ via the Klein-Gordon equation $P ω_{2}^{ϵ} = 0$ in each argument. The wave-front set of $ω_{2}^{ϵ}$ on $M \times M$ is contained in the future-pointing half of the bicharacteristic relation of $g$ :

WF (ω_{2}^{ϵ}) \subseteq {(x_{1}, k_{1}; x_{2}, - k_{2}) : (x_{1}, k_{1}) \sim_{bichar}^{g} (x_{2}, k_{2}), k_{1} \in \overline{V}_{x_{1}}^{+} (g)} .

Proof. Step 5 + Step 6 + Step 7 of the Key derivation. By Hörmander's propagation-of-singularities theorem 02.14.03 applied to the real-principal-type operator $P = □_{g} + m^{2} + V$ in each argument, $WF (ω_{2}^{ϵ}) \subseteq p^{- 1} (0) \times p^{- 1} (0)$ on $T^{*} (M \times M) ∖ 0$ and is invariant under the bicharacteristic flow of $g$ in each argument. The Cauchy-surface initial wave-front-set support on $Σ \times Σ$ (Proposition above) lifts to $M \times M$ via the bicharacteristic flow of $g$ ; the positive-square-root choice $σ_{1} (ϵ) = + p_{h}$ selects the future-pointing lift $k_{+} = (+ ω_{k}, ξ) \in T_{x}^{*} M$ over the past-pointing lift $k_{-} = (- ω_{k}, ξ)$ for each spatial wavenumber $ξ \in T_{x}^{*} Σ$ . The smoothing-error contribution from $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ is smooth on $M \times M$ after Cauchy evolution and contributes nothing to the wave-front set. The upper bound on the wave-front-set support is the future-pointing bicharacteristic relation. $□$

Proposition (Gérard-Wrochna 2014 theorem). Under the hypotheses of the previous propositions, the Gérard-Wrochna state $ω^{ϵ}$ is Hadamard in the Radzikowski wave-front-set sense 13.09.03:

WF (ω_{2}^{ϵ}) = {(x_{1}, k_{1}; x_{2}, - k_{2}) : (x_{1}, k_{1}) \sim_{bichar}^{g} (x_{2}, k_{2}), k_{1} \in \overline{V}_{x_{1}}^{+} (g)} .

Proof. The upper bound $WF (ω_{2}^{ϵ}) \subseteq$ future-pointing bicharacteristic relation is the previous proposition. The lower bound $WF (ω_{2}^{ϵ}) \supseteq$ future-pointing bicharacteristic relation comes from the CCR-commutator-condition lower bound on the wave-front set of any quasi-free state's two-point function (proved in the Full proof set of 13.09.03): for any quasi-free state $ω$ on the CCR algebra, the imaginary part of $ω_{2}$ equals $- E /2$ where $E$ is the causal propagator, and $WF (E)$ contains the full bicharacteristic relation (both halves); the symmetric real part $η_{ϵ}$ cancels exactly the past-pointing half, leaving the future-pointing half as the lower bound on $WF (ω_{2}^{ϵ})$ . Combined: $WF (ω_{2}^{ϵ}) =$ future-pointing bicharacteristic relation, the Radzikowski wave-front-set condition. The state $ω^{ϵ}$ is Hadamard. $□$

Proposition (smoothing-error parametrices give Hadamard states differing by smooth corrections). Let $ϵ_{1}, ϵ_{2} \in Ψ^{1} (Σ)$ both be parametrices of $D_{h}$ with the same principal symbol $p_{h}$ . The Gérard-Wrochna two-point functions $ω_{2}^{ϵ_{1}}, ω_{2}^{ϵ_{2}}$ differ by a smooth function on $M \times M$ .

Proof. Exercise 3. The difference $ϵ_{1} - ϵ_{2} \in Ψ^{- \infty} (Σ)$ is smoothing (by the iterated argument from the gap condition $D_{h} \geq c > 0$ giving $(ϵ_{1} + ϵ_{2})^{- 1} \in Ψ^{- 1} (Σ)$ and the parametrix-difference factorisation). The covariance difference $η_{ϵ_{1}} - η_{ϵ_{2}}$ has smooth integral kernel on $Σ \times Σ$ , Cauchy-evolved to a smooth bi-distribution on $M \times M$ . The two-point function difference is smooth, with empty wave-front set. $□$

Connections Master

Existence of Hadamard states via FNW deformation 13.09.04 is the conceptual existence theorem of which the present Gérard-Wrochna construction is the computational refinement. FNW proves the existence of at least one Hadamard state on every globally hyperbolic spacetime via the deformation argument; Gérard-Wrochna 2014 supplies the explicit construction on the gap-condition class of spacetimes ( $D_{h} \geq c > 0$ on the Cauchy hypersurface), which is the generic case. The two theorems together close the existence-of-Hadamard-states programme — conceptual existence on every globally hyperbolic spacetime, explicit construction wherever the Cauchy hypersurface supports the pseudo-differential calculus.
Pseudo-differential operators on a manifold 02.14.02 is the technical engine. The symbol classes $S_{1, 0}^{m} (T^{*} Σ)$ , the operator classes $Ψ^{m} (Σ)$ , the principal-symbol map, the composition law, and the parametrix construction of elliptic operators are all directly used in the construction of $ϵ$ and $ϵ^{- 1}$ on the Cauchy hypersurface. Without the pseudo-differential calculus on $Σ$ , the Gérard-Wrochna construction would not even make sense — it is, at its core, a pseudo-differential-calculus application.
Hadamard states via the wave-front-set criterion 13.09.03 is the target. The Gérard-Wrochna theorem is the assertion that the explicitly constructed state satisfies the Radzikowski wave-front-set condition. The condition itself is unit 13.09.03; the construction here proves it for a specific class of states. The uniqueness-up-to-smooth-corrections principle of 13.09.03 is what makes the construction physically robust: different choices of pseudo-differential parametrix $ϵ$ give Hadamard states differing by smooth functions, all physically equivalent for the purpose of computing renormalised observables.
Klein-Gordon equation on a globally hyperbolic spacetime 13.09.02 supplies the Cauchy-evolution map that lifts initial data from $Σ$ to bi-distributions on $M \times M$ . The well-posedness theorem of 13.09.02 is what makes the Gérard-Wrochna construction unambiguous: the initial-data covariance $η_{ϵ}$ on $Σ \times Σ$ propagates uniquely to a two-point function $ω_{2}^{ϵ}$ on $M \times M$ satisfying $P ω_{2}^{ϵ} = 0$ in each argument, and the propagation is continuous and microlocally well-behaved.
Propagation of singularities along Hamiltonian flow 02.14.03 is the microlocal-analysis input that transports the Cauchy-surface wave-front-set support from $T^{*} Σ$ to the future-pointing bicharacteristic relation on $T^{*} M$ . Without propagation of singularities, the wave-front-set lifting from initial data to the full bi-distribution would be unconstrained, and the Radzikowski criterion verification would fail. Hörmander's theorem applied to $P = □_{g} + m^{2} + V$ is what makes the lifting precise.
Globally hyperbolic Lorentzian manifolds 13.09.01 supplies the geometric arena. The Bernal-Sánchez splitting gives the canonical Cauchy hypersurface $Σ$ on which the construction lives, and the induced Riemannian metric $h$ on $Σ$ on which the spatial Klein-Gordon operator $D_{h}$ is built. Without the smooth-splitting theorem, the choice of Cauchy hypersurface would not be canonical and the construction would depend on auxiliary choices in a way that obscured its functoriality.
CCR algebra, Weyl algebra, and quasi-free states 12.14.01 is the algebraic-QFT framework. The Gérard-Wrochna covariance $η_{ϵ}$ defines a quasi-free state on the CCR algebra of the Klein-Gordon field; the Araki-Woods 1963 GNS construction realises it as a Fock-space representation with explicit one-particle structure (the one-particle Hamiltonian is $D_{h}$ , with the Seeley pseudo-differential representative).
Wick polynomials in curved spacetime via Hadamard parametrix subtraction [13.09.06, pending] is the immediate downstream renormalisation application. The Brunetti-Fredenhagen-Köhler 1996 + Hollands-Wald 2001 construction of renormalised Wick polynomials $: ϕ^{n} (x) :_{ω}$ on a curved background subtracts the Hadamard parametrix from a normal-ordered combination of two-point functions; the Gérard-Wrochna state supplies an explicit Hadamard two-point function with controllable symbol-class error bounds, making the subtraction effective rather than just existence-only.
FLRW cosmology 13.08.01 is the canonical cosmological application. The Bunch-Davies state on de Sitter, and the adiabatic vacua on general FLRW spacetimes (Junker-Schrohe 2002), are instances of the Gérard-Wrochna construction with the de-Sitter-invariant or FLRW-symmetric choice of pseudo-differential parametrix. The explicit two-point function — in terms of hypergeometric functions on de Sitter, in terms of conformal-time Bessel-function expansions on FLRW — comes from applying the Seeley square-root construction to the spatial Klein-Gordon operator on the maximal-symmetric spatial slice.
Black holes and Hawking radiation [13.07.02, pending] is the canonical black-hole application. The Hartle-Hawking state on the Kruskal extension of Schwarzschild is the Gérard-Wrochna construction with the bifurcate-Killing-horizon-respecting choice of pseudo-differential parametrix; the Killing structure of the Schwarzschild Cauchy slice picks out a canonical $ϵ$ whose Cauchy-evolved two-point function exhibits the Hawking-temperature thermal spectrum under analytic continuation through the bifurcate horizon. The Gérard-Wrochna 2017 in/out-state refinement extends this to asymptotically static spacetimes, giving rigorous scattering theory and curved-spacetime particle creation.
Locally covariant QFT (Brunetti-Fredenhagen-Verch 2003) framework: the Gérard-Wrochna construction is naturally compatible with the locally covariant functor structure. The Sahlmann-Verch 2001 ^{[Sahlmann-Verch 2001 in 13.09.04]} theorem on the existence of locally covariant Hadamard states is made explicit by the Gérard-Wrochna construction: assigning to each globally hyperbolic spacetime its canonical Seeley pseudo-differential parametrix on the Bernal-Sánchez Cauchy slice gives a natural assignment of Hadamard states compatible with isometric embeddings (up to smooth corrections).
Dirac fields on curved spacetime (Gérard-Stoskopf 2021) is the vector-bundle extension. The pseudo-differential construction extends mutatis mutandis to free Dirac fields, with the spatial Dirac operator $D_{h}^{Dirac}$ replacing the spatial Klein-Gordon operator $D_{h}$ ; the Seeley square root is replaced by the absolute-value operator $∣ D_{h}^{Dirac} ∣$ , the positive/negative spectral projection encodes the particle/antiparticle decomposition, and the resulting Hadamard state on the spinor CCR/CAR algebra is the natural Dirac analogue of the Gérard-Wrochna construction. The same propagation-of-singularities argument applies to the spinor d'Alembertian, giving the spinor Radzikowski wave-front-set criterion.

Historical & philosophical context Master

The pseudo-differential approach to curved-spacetime QFT emerged in the 1990s from two convergent strands: the microlocal reformulation of the Hadamard condition by Radzikowski 1996 Comm. Math. Phys. 179 529 ^{[Radzikowski 1996]} and the adiabatic-vacuum-approximation programme initiated by Lüders-Roberts 1990 Comm. Math. Phys. 134 29 and pursued in detail by Wolfgang Junker in his 1996 Rev. Math. Phys. 8 1091 dissertation work ^{[Junker 1996]}. Radzikowski's microlocal turn made it possible to state the Hadamard condition as a wave-front-set support condition, opening the door to the pseudo-differential calculus of Hörmander vol. III; Junker's adiabatic-vacuum work made it possible to construct approximate Hadamard states via finite-order pseudo-differential parametrices of the spatial Klein-Gordon square root.

Wolfgang Junker and Elmar Schrohe in 2002 Ann. Henri Poincaré 3 1113 ^{[Junker-Schrohe 2002]} systematised the adiabatic-vacuum construction into a hierarchy of "adiabatic vacua of order $N$ ", with explicit symbol-class error control on the failure of the Hadamard property at each finite $N$ . The Junker-Schrohe framework was the practical computational tool for cosmological applications throughout the 2000s — adiabatic vacua of moderate order $N$ (say $N = 4$ or $N = 6$ ) give physically admissible states for the FLRW perturbation programme, with explicit error bounds that can be tracked through the renormalisation calculations.

Christian Gérard and Michał Wrochna in 2014 Comm. Math. Phys. 325 713 ^{[Gérard-Wrochna 2014]} closed the construction at infinite order: using the Seeley 1967 Proc. Symp. Pure Math. 10 ^{[Seeley 1967]} complex-power machinery applied to the spatial Klein-Gordon operator on the Cauchy hypersurface, they constructed a pseudo-differential parametrix $ϵ \in Ψ^{1} (Σ)$ with $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ smoothing exactly. The resulting Gérard-Wrochna state is exactly Hadamard in the Radzikowski sense, not just approximate to order $N$ . The construction bypassed the FNW 1981 deformation argument entirely — no auxiliary ultrastatic reference, no smooth one-parameter family of metrics — and gave the first explicit construction of Hadamard initial data on a generic globally hyperbolic spacetime.

The 2014 paper was followed by Gérard-Wrochna 2017 Ann. Henri Poincaré 18 2715 ^{[Gérard-Wrochna 2017]}, which extended the construction to asymptotically static spacetimes and established the Hadamard property of the in/out states needed for rigorous scattering theory on a curved background. This was the first treatment of curved-spacetime particle creation (the asymptotically-static analogue of the Hawking effect) at the level of rigorous algebraic QFT, with the Bogoliubov transformation between in and out states made explicit at the symbol-calculus level. The extension to Dirac fields was completed by Gérard-Stoskopf 2021 Lett. Math. Phys. 111 95 ^{[Gérard-Stoskopf 2021]}, demonstrating that the pseudo-differential approach is universal across covariant Klein-Gordon-type operators on vector bundles.

The construction was consolidated as the canonical modern approach in Christian Gérard's 2019 EMS textbook Microlocal Analysis of Quantum Fields on Curved Spacetimes ^{[Gérard 2019]}. Chapter 7 of the book is devoted to the pseudo-differential construction of Hadamard states; chapter 8 covers the FNW deformation as the historical / conceptual prototype that the pseudo-differential approach replaces. The textbook ordering — pseudo-differential construction first, deformation argument as historical context — reflects the contemporary view that the Gérard-Wrochna construction is the working modern theorem and FNW is the conceptual foundation.

The mathematical lineage of the pseudo-differential approach runs through Robert Seeley's 1967 paper on complex powers, through Hörmander's Analysis of Linear Partial Differential Operators volumes I and III (the canonical microlocal-analysis references), through the adiabatic-vacuum work of Lüders-Roberts and Junker in the 1990s, to the Gérard-Wrochna closure in 2014. The physics-side motivation came from the Hawking radiation calculation (Hawking 1975) and the curved-spacetime QFT programme initiated by Wald 1977 and Fulling-Sweeny-Wald 1978; the technical-side motivation came from the desire for an effective construction of Hadamard states that could be plugged into the Hollands-Wald 2001 renormalised Wick-polynomial programme. The pseudo-differential approach fulfils both needs: it gives the explicit construction the physics needed and the rigorous Hadamard property the mathematics required.

Bibliography Master

Foundational microlocal-analysis machinery:

Seeley, R. T., "Complex powers of an elliptic operator", Proc. Symp. Pure Math. 10 (1967), 288-307. [The construction of complex powers $A^{z}$ of an elliptic positive self-adjoint pseudo-differential operator via the resolvent contour integral; the foundation for the pseudo-differential square root $ϵ$ of the spatial Klein-Gordon operator.]
Hörmander, L., The Analysis of Linear Partial Differential Operators (Springer, 1985), Vol. III. [§18 pseudo-differential calculus on a manifold; §26 propagation of singularities — the microlocal-analysis engine of the Gérard-Wrochna construction.]

Adiabatic-vacuum / pseudo-differential precursors:

Lüders, C. & Roberts, J. E., "Local quasiequivalence and adiabatic vacuum states", Comm. Math. Phys. 134 (1990), 29-63. [The adiabatic-vacuum-approximation programme on FLRW spacetimes; antecedent to the systematic Junker-Schrohe construction.]
Junker, W., "Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime", Rev. Math. Phys. 8 (1996), 1091-1159. [Modern pseudo-differential / adiabatic-vacuum existence theorem on general globally hyperbolic spacetimes; the bridge between FNW deformation and Gérard-Wrochna direct construction.]
Junker, W. & Schrohe, E., "Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties", Ann. Henri Poincaré 3 (2002), 1113-1181. [Systematic theory of adiabatic vacua of order $N$ with explicit symbol-class error control; the finite-order framework whose $N \to \infty$ limit is the Gérard-Wrochna construction.]

The Gérard-Wrochna construction:

Gérard, C. & Wrochna, M., "Construction of Hadamard states by pseudo-differential calculus", Comm. Math. Phys. 325 (2014), 713-755. [The originating direct pseudo-differential construction; the central reference of the present unit.]
Gérard, C. & Wrochna, M., "Hadamard property of the in and out states for Klein-Gordon fields on asymptotically static spacetimes", Ann. Henri Poincaré 18 (2017), 2715-2756. [The refined construction on asymptotically static spacetimes; in/out-state Hadamard property used for rigorous scattering theory.]
Gérard, C. & Stoskopf, T., "Hadamard property of the in and out states for Dirac fields on asymptotically static spacetimes", Lett. Math. Phys. 111 (2021), Article 95. [Extension to Dirac fields; demonstrates the universality of the pseudo-differential approach across covariant Klein-Gordon-type operators on vector bundles.]

Modern textbook consolidation:

Gérard, C., Microlocal Analysis of Quantum Fields on Curved Spacetimes, ESI Lectures in Mathematics and Physics (EMS, 2019). [Ch. 7-8 the canonical reference for the Gérard-Wrochna construction and the FNW deformation as historical context.]

Companion algebraic-QFT-on-curved-spacetimes references:

Wald, R. M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, 1994). [Physicist-side framing of the state-selection question that the Gérard-Wrochna construction answers explicitly.]
Bär, C., Ginoux, N. & Pfäffle, F., Wave Equations on Lorentzian Manifolds and Quantization (EMS, 2007). [Cauchy problem and Green's-operator framework; free PDF at arXiv:0806.1036.]
Hollands, S. & Wald, R. M., "Local Wick polynomials and time ordered products of quantum fields in curved spacetime", Comm. Math. Phys. 223 (2001), 289-326. [Downstream renormalisation framework that uses the Gérard-Wrochna state as the Hadamard reference for parametrix subtraction.]
Hollands, S. & Wald, R. M., "Quantum fields in curved spacetime", Phys. Rep. 574 (2015), 1-35. [Free preprint at arXiv:1401.2026; modern review.]

Cosmological / black-hole applications:

Bunch, T. S. & Davies, P. C. W., "Quantum field theory in de Sitter space: renormalization by point-splitting", Proc. R. Soc. A 360 (1978), 117-134. [The Bunch-Davies state on de Sitter, the canonical cosmological instance of the construction.]
Allen, B., "Vacuum states in de Sitter space", Phys. Rev. D 32 (1985), 3136-3149. [Uniqueness of the de-Sitter-invariant Hadamard state; what the Gérard-Wrochna construction reproduces with the de-Sitter-symmetric parametrix.]
Kay, B. S. & Wald, R. M., "Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon", Phys. Rep. 207 (1991), 49-136. [Hartle-Hawking state uniqueness on bifurcate-Killing-horizon spacetimes; what the Gérard-Wrochna construction reproduces with the horizon-respecting parametrix.]

Microlocal-analysis foundations of the Hadamard programme:

Radzikowski, M. J., "Micro-local approach to the Hadamard condition in quantum field theory on curved space-time", Comm. Math. Phys. 179 (1996), 529-553. [Wave-front-set characterisation of Hadamard states; the target criterion verified by the Gérard-Wrochna construction.]
Brunetti, R., Fredenhagen, K. & Köhler, M., "The microlocal spectrum condition", Comm. Math. Phys. 180 (1996), 633-652. [Extension to all $n$ -point functions; automatically satisfied for quasi-free states from the Gérard-Wrochna two-point function via Wick-rule reconstruction.]

Prerequisites

13.09.04
02.14.02

Tier anchors

beginner: Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (Chicago, 1994), Ch. 4 (physicist-side framing of the state-selection question on a curved background, set up so that the modern microlocal answer of Gérard-Wrochna is the natural next step)
intermediate: Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 7 (pseudo-differential construction of Hadamard states); Gérard & Wrochna, *Comm. Math. Phys.* 325 (2014) 713, §3-§4 (the originating construction); Bär, Ginoux & Pfäffle, *Wave Equations on Lorentzian Manifolds and Quantization* (EMS, 2007), Ch. 3-4 (Cauchy problem and quantisation companion; free PDF arXiv:0806.1036)
master: Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 7-8; Gérard & Wrochna, *Comm. Math. Phys.* 325 (2014) 713 (direct pseudo-differential construction); Gérard & Wrochna, *Ann. Henri Poincaré* 18 (2017) 2715 (in/out-state Hadamard property on asymptotically static spacetimes); Junker & Schrohe, *Ann. Henri Poincaré* 3 (2002) 1113 (adiabatic-vacuum precursor); Gérard & Stoskopf, *Lett. Math. Phys.* 111 (2021) 95 (curved-spacetime extensions); Seeley, *Proc. Symp. Pure Math.* 10 (1967) 288 (complex powers of an elliptic operator); Hörmander, *The Analysis of Linear Partial Differential Operators*, Vol. III (Springer, 1985), §18 (pseudo-differential calculus on a manifold) and §26 (propagation of singularities)

References

Gérard, C. & Wrochna, M. — 'Construction of Hadamard states by pseudo-differential calculus', Comm. Math. Phys. 325 (2014) 713 · §3 (pseudo-differential approximation of the square root of the Cauchy-data operator); §4 (Hadamard-form two-point function from the approximation); the originating direct construction that bypasses the FNW deformation argument
Gérard, C. & Wrochna, M. — 'Hadamard property of the in and out states for Klein-Gordon fields on asymptotically static spacetimes', Ann. Henri Poincaré 18 (2017) 2715 · The refined pseudo-differential construction on asymptotically static spacetimes; in/out-state Hadamard property used for rigorous scattering theory on a curved background
Gérard, C. & Stoskopf, T. — 'Hadamard property of the in and out states for Dirac fields on asymptotically static spacetimes', Lett. Math. Phys. 111 (2021) 95 · Extension of the Gérard-Wrochna 2017 construction to Dirac fields; demonstrates the universality of the pseudo-differential approach across covariant Klein-Gordon-type operators on vector bundles
Junker, W. & Schrohe, E. — 'Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties', Ann. Henri Poincaré 3 (2002) 1113 · §3-§5 (adiabatic vacuum states of order N via pseudo-differential approximation of order N); the precursor framework whose Junker-Schrohe N → ∞ limit is the Gérard-Wrochna construction
Junker, W. — 'Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime', Rev. Math. Phys. 8 (1996) 1091 · Ch. 3-4 (pseudo-differential construction); Ch. 5 (adiabatic-vacuum approximation on cosmological FLRW); the modern existence proof via direct construction on general globally hyperbolic spacetimes
Seeley, R. T. — 'Complex powers of an elliptic operator', Proc. Symp. Pure Math. 10 (1967) 288 · §1-§4 (construction of $A^z$ for elliptic $A \in \Psi^m$ via the resolvent; $A^{1/2}$ as the positive self-adjoint square root with $\Psi^{m/2}$ symbol); the foundational machinery for the pseudo-differential square root $\epsilon$ of the spatial Klein-Gordon operator $D_h$
Gérard, C. — Microlocal Analysis of Quantum Fields on Curved Spacetimes (EMS Press, 2019) · Ch. 7 (Hadamard states via pseudo-differential calculus); §7.1 (the principle of the construction); §7.2 (Cauchy-surface covariance with pseudo-differential symbols); §7.3 (verification of the Hadamard wave-front-set condition); the canonical modern textbook entry
Hörmander, L. — The Analysis of Linear Partial Differential Operators (Springer, 1985) · Vol. III §18 (pseudo-differential calculus on a manifold; symbol classes $S^m_{1,0}$, composition, principal symbol, parametrix); Vol. III §26 (propagation of singularities along the bicharacteristic flow); the microlocal-analysis engine of the construction
Bär, C., Ginoux, N. & Pfäffle, F. — Wave Equations on Lorentzian Manifolds and Quantization (EMS, 2007) · Ch. 3 (Cauchy problem and Green's operators on a globally hyperbolic background); Ch. 4 (CCR algebra and quasi-free states); free PDF at arXiv:0806.1036
Wald, R. M. — Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, 1994) · Ch. 4 (Hadamard form, vacuum-state question; physicist-side framing of the state-selection question that the Gérard-Wrochna construction answers explicitly)
Radzikowski, M. J. — 'Micro-local approach to the Hadamard condition in quantum field theory on curved space-time', Comm. Math. Phys. 179 (1996) 529 · The wave-front-set characterisation of Hadamard states cited as the equivalence to be verified by the Gérard-Wrochna construction
Hollands, S. & Wald, R. M. — 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime', Comm. Math. Phys. 223 (2001) 289 · Downstream application: locally covariant Wick polynomials and time-ordered products require an explicitly constructed Hadamard state; the pseudo-differential construction supplies this in computable form
Bernal, A. N. & Sánchez, M. — 'Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes', Comm. Math. Phys. 257 (2005) 43 · Theorem 1.1 (smooth metric splitting $M \cong \mathbb{R} \times \Sigma$ with $g = -\beta^2 dt^2 + h_t$); used to identify the Cauchy hypersurface on which the pseudo-differential construction lives

Estimated time

beginner: 18m
intermediate: 42m
master: 70m