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

Existence of Hadamard states via the FNW deformation argument

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Anchor (Master): Fulling, Narcowich & Wald, *Ann. Phys.* 136 (1981) 243 (the originating deformation existence theorem); Junker, *Rev. Math. Phys.* 8 (1996) 1091 (pseudo-differential / adiabatic-vacuum existence on general globally hyperbolic spacetimes); Brunetti, Fredenhagen & Köhler, *Comm. Math. Phys.* 180 (1996) 633 (extension to higher n-point functions); Gérard & Wrochna, *Comm. Math. Phys.* 325 (2014) 713 and *Ann. Henri Poincaré* 18 (2017) 2715 (direct pseudo-differential construction); Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 8

Intuition Beginner

The previous unit 13.09.03 defined what it means for a quasi-free state on the quantised Klein-Gordon field to be Hadamard — a state whose two-point function has the right short-distance singularity structure to support a finite renormalised stress-energy tensor and a sensible Wick-polynomial calculus. The definition is a condition on a state. It does not, by itself, guarantee that any such state exists on a given globally hyperbolic spacetime.

On flat Minkowski spacetime existence is direct: the standard vacuum state with its positive-frequency two-point function is Hadamard, by an explicit Fourier-mode calculation. On a curved background there is no global notion of positive frequency, no canonical vacuum, and the existence question is real. Could there be a globally hyperbolic spacetime so geometrically twisted that no quasi-free state satisfies the Hadamard wave-front-set condition?

The answer is no. Fulling, Narcowich and Wald in 1981 proved that every globally hyperbolic spacetime admits at least one Hadamard state. The strategy is a deformation argument. Start from a simpler reference spacetime — an ultrastatic one, of the form a product $R \times Σ$ of a time line and a Riemannian space, with the metric independent of time. On an ultrastatic spacetime the Klein-Gordon wave operator commutes with time translation, so a spectral decomposition of the spatial Laplacian gives an explicit mode decomposition of the field. The ground state — the state in which every positive-frequency mode is unoccupied — is then explicitly Hadamard by direct computation.

Now deform. Pick the target globally hyperbolic spacetime $(M, g)$ . Construct a smooth one-parameter family $g_{s}$ of globally hyperbolic metrics on the underlying smooth manifold $M$ , with $g_{0}$ ultrastatic and $g_{1} = g$ . Pull the ultrastatic ground state back along the deformation, using the symplectic isomorphism between Cauchy-data phase spaces that the family of causal propagators $E_{g_{s}}$ supplies. The resulting state on $(M, g) = (M, g_{1})$ is a quasi-free state — the question is whether it is Hadamard.

It is. The mechanism is propagation of singularities: the wave-front set of the two-point function is constrained to live on the bicharacteristic flow of the principal symbol of the Klein-Gordon operator, and as the metric varies smoothly through the family $g_{s}$ , the bicharacteristic flow varies smoothly with it. The future-pointing positive-frequency condition is preserved by the smooth deformation. So the Hadamard wave-front-set structure of the ultrastatic ground state, applied at $s = 0$ , propagates smoothly to a Hadamard wave-front-set structure on the target $g_{1} = g$ .

The existence theorem is not constructive in any practically useful sense — the deformation path and the resulting two-point function are existence-theorems, not formulas. Modern work by Gérard and Wrochna (2014, 2017) supplies a more explicit pseudo-differential construction that produces Hadamard states directly from initial data on a Cauchy surface, bypassing the deformation. Junker (1996) gave an intermediate sharpening via the adiabatic-vacuum approximation. The original FNW deformation argument remains the conceptual prototype: the Hadamard class is not empty on any globally hyperbolic spacetime, because one can always deform from a reference spacetime where the class is non-empty by an explicit ground-state construction.

This is the existence half of the Hadamard programme. The previous unit's characterisation said what a Hadamard state is; the present unit says that at least one exists for every reasonable spacetime background. Together they make the algebraic-QFT-on-curved-spacetimes programme of Brunetti-Fredenhagen-Verch 2003 (the locally covariant functor) well-defined: every globally hyperbolic spacetime carries a CCR algebra equipped with a non-empty Hadamard class of states, on which Wick polynomials and time-ordered products can be defined.

Visual Beginner

The picture to hold is a deformation between two globally hyperbolic spacetimes — an ultrastatic reference on the left, the target spacetime on the right — with the Hadamard wave-front-set structure of the ultrastatic ground state propagating smoothly across the deformation to give a Hadamard state on the target.

Three pieces drive the picture. The ultrastatic reference $(M, g_{0})$ supplies an explicit Hadamard state by spectral resolution of the spatial Klein-Gordon operator. The deformation $g_{s}$ is a smooth interpolation through globally hyperbolic metrics, constructed using the Bernal-Sánchez splitting 13.09.01 which gives every globally hyperbolic spacetime a canonical product structure $M ≅ R \times Σ$ . The preservation of the wave-front-set condition along the deformation rests on propagation of singularities 02.14.03 applied to the smoothly-varying Klein-Gordon operator $P_{s} = □_{g_{s}} + m^{2}$ .

The picture you should also keep in mind is a more refined version: modern pseudo-differential constructions (Gérard-Wrochna 2014) produce a Hadamard state directly on a Cauchy surface $Σ$ without ever deforming the metric, by picking a $Ψ^{1}$ -pseudo-differential approximation $ϵ$ of the operator $- Δ_{h} + m^{2} + V$ and using $(ϵ, σ)$ as Hadamard-form initial data. This direct construction makes the existence theorem effective: one writes down the state explicitly, then verifies the Hadamard wave-front-set condition by symbol calculus. The FNW deformation remains the conceptual entry to the subject; Gérard-Wrochna gives the computational refinement.

Worked example Beginner

Construct the ultrastatic ground state on the simplest reference background and verify the Hadamard property by direct mode-sum computation.

Step 1. Take the ultrastatic spacetime as Minkowski itself — a real time line crossed with three-dimensional Euclidean space, with the standard flat metric. This is the simplest case of an ultrastatic geometry: the Klein-Gordon wave operator decomposes as minus the time-second-derivative plus the spatial-Laplacian-plus-squared-mass, with the spatial Laplacian acting on three Cartesian coordinates.

Step 2. Spectral resolution of the spatial operator. The spatial operator (minus Laplacian plus squared mass) has continuous spectrum from the squared mass upward, with plane-wave eigenfunctions at frequency-squared equal to spatial-wavenumber-squared plus squared mass. The positive square root of the spatial operator acts on each plane wave by multiplication by the corresponding angular frequency.

Step 3. Ground state two-point function. The ground state of the time-translation-invariant Klein-Gordon field is the quasi-free state whose two-point function is built by superposing positive-frequency plane-wave modes against the canonical normalisation factor (one over twice the angular frequency), against the standard inverse-Fourier-transform measure on three-dimensional spatial momentum space. This is the standard Minkowski vacuum two-point function; the closed-form integral is written out in the Formal definition section and computed at Intermediate tier. The normalisation factor is what makes the resulting two-point function a state on the CCR algebra — positive, normalised, with the correct CCR commutator.

Step 4. Verify the Hadamard wave-front-set condition. The Fourier representation in step 3 has support on the upper mass shell ${(ω_{k}, k) : ω_{k} > 0}$ in four-momentum space. By the wave-front-set characterisation of an oscillatory integral with non-stationary phase, the wave-front set is

WF (ω_{0}) = {(x, k; y, - k) : k^{2} + m^{2} = 0, k^{0} > 0},

i.e. on pairs $(x, y)$ joined by a null geodesic (massless case) or a timelike geodesic (massive case) with the cotangent covector $k$ at $x$ future-pointing on the mass shell. This is exactly the Radzikowski wave-front-set condition 13.09.03 applied to flat spacetime. So the Minkowski vacuum is Hadamard, the simplest case of the FNW existence theorem.

Step 5. Now deform. Suppose the target $(M, g_{1})$ is a small smooth perturbation of Minkowski — say the metric $g_{1} = - d t^{2} + (1 + ϵχ (x, y, z)) δ_{ij} d x^{i} d x^{j}$ for a small parameter $ϵ$ and a smooth compactly-supported bump function $χ$ . Construct the one-parameter family $g_{s} = - d t^{2} + (1 + sϵχ) δ_{ij} d x^{i} d x^{j}$ for $s \in [0, 1]$ . Each $g_{s}$ is globally hyperbolic (for $ϵ$ small enough the metric stays Lorentzian) and the family is smooth in $s$ .

Step 6. Pull back the Minkowski vacuum along the deformation. The causal propagator $E_{g_{s}}$ varies smoothly with $s$ , so the symplectic isomorphism of Cauchy-data phase spaces between $g_{0}$ -data and $g_{s}$ -data varies smoothly with $s$ . The pulled-back two-point function $ω_{s} (x, y)$ is a smooth deformation of the Minkowski vacuum. By propagation of singularities applied to $P_{s} = □_{g_{s}} + m^{2}$ , the wave-front set of $ω_{s}$ remains on the future-pointing half of the bicharacteristic relation of $g_{s}$ — which varies smoothly with $s$ from the Minkowski cone to the slightly-perturbed cone of $g_{1}$ . At $s = 1$ the pulled-back state $ω_{1}$ on $(M, g_{1})$ is Hadamard.

What this tells us: the FNW deformation works by transporting the explicit Hadamard structure of the Minkowski vacuum across a smoothly-varying family of globally hyperbolic metrics, with propagation of singularities ensuring that the wave-front-set condition is preserved at every stage. The Minkowski case is the simplest example; for a general globally hyperbolic target one constructs a more elaborate deformation to an ultrastatic reference, but the mechanism is identical. Existence on Minkowski plus the deformation argument plus propagation of singularities equals existence on every globally hyperbolic spacetime.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Fulling-Narcowich-Wald 1981 deformation argument establishes that every globally hyperbolic spacetime admits at least one

A. Strictly stationary metric B. Hadamard quasi-free state on the CCR algebra of the Klein-Gordon field C. Closed timelike curve D. Killing vector field

Hint

The deformation argument is about existence of states on the quantised Klein-Gordon field, not about geometric properties of the spacetime itself.

Answer

B. Hadamard quasi-free state on the CCR algebra of the Klein-Gordon field.

The FNW 1981 theorem says: for every globally hyperbolic Lorentzian manifold $(M, g)$ , the CCR algebra of the quantised Klein-Gordon field on $(M, g)$ admits at least one Hadamard quasi-free state — a state whose two-point function satisfies the Radzikowski wave-front-set condition 13.09.03. The construction proceeds by deforming the metric to an ultrastatic reference where an explicit ground state is available, then pulling the state back through the deformation and verifying preservation of the Hadamard property via propagation of singularities.

Option A is a geometric property of the metric, unrelated to state existence — globally hyperbolic spacetimes need not be stationary. Option C is the opposite of what global hyperbolicity guarantees (no closed timelike curves). Option D is again a geometric property; many globally hyperbolic spacetimes (e.g. generic FLRW) have no global Killing vector field beyond the spatial symmetries.

Exercise (easy, true-false).

True or false: the FNW deformation argument gives a unique Hadamard state on every globally hyperbolic spacetime.

Hint

Different choices of reference ultrastatic spacetime and different deformation paths give different Hadamard states. Recall from 13.09.03 that any two Hadamard states differ by a smooth function.

Answer

False.

The FNW theorem is an existence statement, not a uniqueness statement. Different choices of ultrastatic reference and different deformation paths produce different Hadamard states on the target spacetime. Any two Hadamard states differ by a smooth function (Exercise 3 of 13.09.03), so they are physically equivalent for renormalised observables but are not literally the same state. Uniqueness theorems for Hadamard states are available only on spacetimes with extra structure: Kay-Wald 1991 give uniqueness on bifurcate-Killing-horizon spacetimes (the Hartle-Hawking state on Schwarzschild-Kruskal); Allen 1985 give uniqueness of the de-Sitter-invariant Hadamard state on de Sitter (Bunch-Davies).

Exercise (medium, multiple choice).

The key role of propagation of singularities in the FNW deformation argument is to

A. Smooth out the singularities of the two-point function B. Ensure that the wave-front set of the pulled-back two-point function remains on the future-pointing half of the bicharacteristic relation of the deformed metric $g_{s}$ at every $s$ C. Establish that the spacetime $(M, g)$ is globally hyperbolic D. Compute the explicit form of the two-point function

Hint

The wave-front-set condition is the Radzikowski criterion. Preservation along the deformation is what makes the pulled-back state Hadamard.

Answer

B. Ensure that the wave-front set of the pulled-back two-point function remains on the future-pointing half of the bicharacteristic relation of the deformed metric $g_{s}$ at every $s$ .

The Radzikowski wave-front-set condition characterises Hadamard states by an equality on the wave-front set of the two-point function: the set must coincide with the future-pointing half of the null-bicharacteristic relation of $□_{g} + m^{2}$ . Under a smooth deformation $g_{s}$ of the metric, the bicharacteristic flow on $T^{*} M ∖ 0$ varies smoothly with $s$ , and Hörmander's propagation-of-singularities theorem 02.14.03 applied to the smoothly-varying Klein-Gordon operator $P_{s} = □_{g_{s}} + m^{2}$ guarantees that the wave-front set of the pulled-back two-point function $ω_{2, s}$ stays on the future-pointing half of the $g_{s}$ -bicharacteristic relation. So the Hadamard property is preserved along the deformation — this is the load-bearing step.

Option A is wrong: the two-point function genuinely has singularities (the $1/ σ$ Hadamard parametrix structure) that propagation of singularities does not smooth out — they propagate. Option C is the prerequisite 13.09.01, assumed throughout. Option D is wrong: the deformation argument is non-constructive in the practical sense; explicit forms come from the Gérard-Wrochna 2014 pseudo-differential refinement, not from FNW directly.

Exercise (medium, numeric).

The Minkowski vacuum two-point function for the massless Klein-Gordon field is built by integrating positive-frequency modes against the Fourier kernel. The normalisation factor multiplying the mode density in the integrand is $(2 ω_{k})^{- 1}$ where $ω_{k} = ∣ k ∣$ in the massless case. What numerical exponent appears in the prefactor $(2 π)^{?}$ in front of the mode-decomposition integral for a 4-dimensional spacetime with 3 spatial dimensions?

Hint

In the physicists' (asymmetric) Fourier convention, the inverse-transform measure on $d$ -dimensional momentum space carries a $(2 π)^{- d}$ prefactor. For the mode-sum two-point function in 3 spatial dimensions with this convention, the prefactor is $(2 π)^{- 3}$ .

Answer

$- 3$ (i.e. the prefactor is $(2 π)^{- 3}$ ).

The mode-sum two-point function — written out at Intermediate tier in the Worked-example proposition — uses the inverse-Fourier-transform measure $d^{3} k / (2 π)^{3}$ on the 3-dimensional spatial momentum space. The $(2 π)^{- 3}$ factor is the standard physicists' Fourier-inversion convention in 3 spatial dimensions: the prefactor is $(2 π)^{- d}$ for $d$ spatial dimensions. It arises because the inverse Fourier transform from spatial momentum to spatial position carries the full Jacobian factor, with the forward transform unweighted. The 4-dimensional spacetime is the product of a real time line and a 3-dimensional spatial slice; the time-frequency integration is done implicitly by setting $k^{0} = + ω_{k}$ on the upper mass shell, so the explicit $(2 π)^{- 3}$ corresponds only to the spatial part.

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 may 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 $Σ$ . 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. The advanced and retarded Green's operators $E^{\pm}$ and the causal propagator $E = E^{-} - E^{+}$ supply the symplectic phase space $(S (M, g), σ)$ of Cauchy data for the quantised field, with $S (M, g) := C_{c}^{\infty} (M) / ker (E)$ and $σ (f, g) := \int_{M} E (f) \cdot g d vol_{g}$ . The CCR algebra $A (M, g)$ and its quasi-free states are constructed as in 12.14.01. The Hadamard property of a quasi-free state is the Radzikowski wave-front-set condition $WF (ω_{2}) = {(x, k; y, - k^{'}) : (x, k) \sim_{bichar} (y, k^{'}), k \in \overline{V}_{x}^{+}}$ from 13.09.03.

Definition (ultrastatic globally hyperbolic spacetime). A globally hyperbolic spacetime $(M, g)$ is ultrastatic if $M = R \times Σ$ for some smooth manifold $Σ$ and $g = - d t^{2} + h$ , where $h$ is a complete Riemannian metric on $Σ$ independent of $t$ .

Equivalently, $(M, g)$ is ultrastatic if it admits a globally defined timelike Killing vector field $\partial_{t}$ of constant unit length and the orthogonal hypersurfaces $Σ_{t} = {t} \times Σ$ are isometric Riemannian manifolds with the metric $h$ .

Definition (ground state on an ultrastatic spacetime). Let $(M, g_{0}) = (R \times Σ, - d t^{2} + h)$ be an ultrastatic spacetime with $h$ a complete Riemannian metric on $Σ$ and let $P_{0} = □_{g_{0}} + m^{2} + V$ with $V \in C_{b}^{\infty} (Σ)$ time-independent. Let $D_{h} := - Δ_{h} + m^{2} + V$ on $L^{2} (Σ, d vol_{h})$ . Assume $D_{h}$ is self-adjoint with $D_{h} \geq c > 0$ for some constant $c$ (positivity of the mass-plus-potential). The ground state of the quantised Klein-Gordon field is the quasi-free state $ω_{0}$ on the CCR algebra $A (M, g_{0})$ with two-point function

ω_{0} (f_{1}, f_{2}) := \int_{R^{2}} d t_{1} d t_{2} \int_{Σ} f_{1} (t_{1}, \cdot) (2 D_{h})^{- 1} e^{- i (t_{1} - t_{2}) D_{h}} f_{2} (t_{2}, \cdot) d vol_{h},

for $f_{1}, f_{2} \in C_{c}^{\infty} (M)$ , where $D_{h}$ is the positive self-adjoint square root by spectral theorem.

By spectral resolution of $D_{h}$ , this integral converges absolutely for $f_{1}, f_{2} \in C_{c}^{\infty} (M)$ and defines a non-negative quasi-free state on the CCR algebra. Time-translation invariance of $ω_{0}$ follows from $[\partial_{t}, D_{h}] = 0$ ; the explicit Fourier-mode decomposition shows the integration is over positive frequencies $ω = λ$ with $λ \in spec (D_{h}) \subset [c, \infty)$ .

Proposition (Hadamard property of the ultrastatic ground state). The ground state $ω_{0}$ on the ultrastatic spacetime $(M, g_{0})$ defined above is Hadamard in the sense of the Radzikowski wave-front-set criterion 13.09.03.

This is the direct-construction half of the existence theorem. The proof is a Fourier / spectral computation on $Σ$ together with stationary-phase analysis on $M$ ; we sketch it in the Key derivation. The state $ω_{0}$ depends on the choice of square root, which is unique by positivity of $D_{h}$ , and on the choice of $V$ ; different time-independent potentials give different ultrastatic ground states.

Definition (smooth family of globally hyperbolic metrics). *Let $M$ be a smooth manifold. A smooth family of Lorentzian metrics on $M$ is a smooth map $g : [0, 1] \times M \to T^{*} M \otimes T^{*} M$ , written $g_{s} = g (s, \cdot)$ , such that each $g_{s}$ is a smooth Lorentzian metric on $M$ . The family is globally hyperbolic if each $(M, g_{s})$ is globally hyperbolic. The family is adapted to a common Cauchy slice $Σ \subset M$ if $Σ$ is a Cauchy hypersurface for every $g_{s}$ .*

A standard construction: given $(M, g_{1})$ globally hyperbolic with Bernal-Sánchez splitting $M ≅ R \times Σ$ and $g_{1} = - β^{2} d t^{2} + h_{t}$ , fix an ultrastatic reference $g_{0} = - d t^{2} + h_{0}$ for some complete Riemannian metric $h_{0}$ on $Σ$ (e.g. $h_{0} = h_{0} ∣_{Σ}$ from the slice $Σ_{0}$ ), and define $g_{s} = - ((1 - s) + s β)^{2} d t^{2} + ((1 - s) h_{0} + s h_{t})$ for $s \in [0, 1]$ . For all $s$ , $g_{s}$ is Lorentzian on $M$ (the lapse and spatial metric are positive convex combinations) and globally hyperbolic with $Σ_{0} = {0} \times Σ$ a Cauchy slice. This is the simplest case; the FNW deformation in general uses smooth cutoffs to keep $g_{s} = g_{0}$ outside a compact spacetime region for $s$ near 0 and $g_{s} = g_{1}$ outside a compact region for $s$ near 1.

The central definition:

Definition (FNW deformation construction). Let $(M, g_{1})$ be a globally hyperbolic spacetime with Cauchy hypersurface $Σ$ . An FNW deformation of $(M, g_{1})$ is a smooth one-parameter family $g_{s}$ , $s \in [0, 1]$ , of globally hyperbolic Lorentzian metrics on $M$ adapted to $Σ$ , with $g_{0}$ ultrastatic ( $g_{0} = - d t^{2} + h_{0}$ for some complete Riemannian $h_{0}$ on $Σ$ ). The FNW pull-back state on $(M, g_{1})$ is the quasi-free state $ω_{1}$ obtained by pulling back the ultrastatic ground state $ω_{0}$ on $(M, g_{0})$ along the symplectic isomorphism $T_{g_{0} \to g_{1}} : (S (M, g_{0}), σ_{g_{0}}) \to (S (M, g_{1}), σ_{g_{1}})$ induced by the family of causal propagators $E_{g_{s}}$ .

Counterexamples to common slips

The construction of a smooth family $g_{s}$ of globally hyperbolic metrics interpolating between an ultrastatic $g_{0}$ and a target $g_{1}$ requires care only because global hyperbolicity must be preserved at every $s$ . The convex-combination construction works for small perturbations; for general globally hyperbolic targets the family is built with smooth spacetime cutoffs and the verification of global hyperbolicity at each $s$ uses the compactness of the Cauchy diamonds $J_{g_{s}}^{+} (p) \cap J_{g_{s}}^{-} (q)$ , varying smoothly with $s$ . FNW 1981 carry out this construction explicitly; Bär-Ginoux-Pfäffle 2007 §4 give a streamlined modern version.
The symplectic isomorphism $T_{g_{0} \to g_{1}}$ between Cauchy-data phase spaces is induced by the causal propagator, not by the metric. For each $s$ , the Cauchy-data phase space $(S (M, g_{s}), σ_{g_{s}}) = (C_{c}^{\infty} (M) / ker (E_{g_{s}}), σ_{g_{s}})$ has the symplectic form $σ_{g_{s}} (f_{1}, f_{2}) = \int_{M} E_{g_{s}} (f_{1}) \cdot f_{2} d vol_{g_{s}}$ . The propagator $E_{g_{s}}$ varies smoothly in $s$ by smooth dependence of the Cauchy problem on the metric, and the resulting family of symplectic forms gives a smoothly-varying family of phase spaces whose composition is the symplectic isomorphism used in the pull-back.
The Hadamard property is not automatic for any pull-back state. Pulling back along the deformation gives a quasi-free state, but the wave-front-set condition is what makes it Hadamard. The preservation argument uses propagation of singularities applied to the deformed equation $P_{s} u_{s} = 0$ along the smoothly-varying bicharacteristic flow. Without this propagation-of-singularities step the existence theorem fails.
The FNW theorem is non-constructive in the strong sense: the deformation path and the resulting two-point function are existence statements, not formulas. For practical computations one uses (i) the Gérard-Wrochna 2014 pseudo-differential direct construction; (ii) the Junker 1996 adiabatic-vacuum approximation; or (iii) Killing-symmetry-respecting constructions on stationary or Killing-horizon spacetimes (Kay-Wald 1991 Hartle-Hawking; Allen 1985 Bunch-Davies). The FNW deformation is the foundational existence proof, not the working construction.
The choice of ultrastatic reference $h_{0}$ is arbitrary, and different choices give different FNW pull-back states. Together with the uniqueness-up-to-smooth-corrections proposition of 13.09.03, this means the existence theorem produces a non-empty class of Hadamard states, all differing by smooth functions. The class — not any single representative — is the physically meaningful object.

Key derivation Intermediate+

Theorem (Fulling-Narcowich-Wald 1981 existence theorem). Let $(M, g)$ be a globally hyperbolic Lorentzian manifold and let $P = □_{g} + m^{2} + V$ with $m \geq 0$ and $V \in C^{\infty} (M)$ . Then the CCR algebra $A (M, g)$ of the quantised free Klein-Gordon field on $(M, g)$ admits at least one Hadamard quasi-free state.

The proof is the FNW deformation argument: deform $g$ to an ultrastatic reference $g_{0}$ , construct the ground state on the reference, pull back along the deformation, verify preservation of the Hadamard wave-front-set condition. The originating reference is Fulling-Narcowich-Wald 1981 Ann. Phys. 136 243 ^{[Fulling-Narcowich-Wald 1981]}; the modern textbook formulation appears in Bär-Ginoux-Pfäffle 2007 ^{[Bär-Ginoux-Pfäffle 2007 Ch. 4]} and Gérard 2019 ^{[Gérard 2019 Ch. 8]}.

Proof of the ultrastatic ground-state Hadamard property (input to the main argument).

Step 1: spectral resolution. On the ultrastatic background $(M, g_{0}) = (R \times Σ, - d t^{2} + h_{0})$ , the operator $D := - Δ_{h_{0}} + m^{2} + V$ on $L^{2} (Σ, d vol_{h_{0}})$ is essentially self-adjoint on $C_{c}^{\infty} (Σ)$ by Chernoff's theorem (using completeness of $h_{0}$ ) and is bounded below by $c := in f V + m^{2} > 0$ when $V \geq 0$ and $m > 0$ (the positivity condition is the gap condition for the existence of a ground state). The positive self-adjoint square root $D$ exists by the spectral theorem and acts on $L^{2} (Σ)$ by the function-of-an-operator construction.

Step 2: ground-state two-point function. The classical Cauchy problem on the ultrastatic spacetime is well-posed by 13.09.02: for $(u_{0}, u_{1}) \in C_{c}^{\infty} (Σ) \oplus C_{c}^{\infty} (Σ)$ the unique solution is

u (t, x) = cos (t D) u_{0} (x) + \frac{sin ( t D )}{D} u_{1} (x) .

The Cauchy-data symplectic form is $σ (Φ_{1}, Φ_{2}) = \int_{Σ} (u_{0}^{(1)} u_{1}^{(2)} - u_{0}^{(2)} u_{1}^{(1)}) d vol_{h_{0}}$ . The ground-state quasi-free covariance is

η (Φ_{1}, Φ_{2}) = \frac{1}{2} \int_{Σ} (u_{0}^{(1)} D u_{0}^{(2)} + u_{1}^{(1)} (D)^{- 1} u_{1}^{(2)}) d vol_{h_{0}},

which satisfies the purity / Hadamard condition $η \geq \frac{1}{4} σ \otimes σ$ (here $η - \frac{1}{4} σ σ^{*} = 0$ for the ground state, i.e. the ground state is a pure state). The two-point function on $M \times M$ recovered from $(η, σ)$ is

ω_{0} (f_{1}, f_{2}) = \int_{R^{2}} d t_{1} d t_{2} \int_{Σ} f_{1} (t_{1}, \cdot) (2 D)^{- 1} e^{- i (t_{1} - t_{2}) D} f_{2} (t_{2}, \cdot) d vol_{h_{0}} .

Step 3: wave-front-set computation. The Fourier transform of $ω_{0}$ in the time variable $t_{1} - t_{2}$ at fixed $(x, y) \in Σ \times Σ$ is $(2 D)^{- 1} δ (τ - D) θ (τ)$ , supported on the upper half of the "mass shell" $τ^{2} - λ (D) = 0$ with $τ > 0$ . On the time-translation-invariant ultrastatic spacetime this is the curved-spacetime analogue of the Minkowski positive-energy condition. The wave-front-set of $ω_{0}$ at a pair $(x_{1}, x_{2}) \in M \times M$ is computed by stationary-phase analysis on the integral representation; the result is

WF (ω_{0}) = {(x_{1}, k_{1}; x_{2}, - k_{2}) : (x_{1}, k_{1}) \sim_{bichar}^{g_{0}} (x_{2}, k_{2}), k_{1} \in \overline{V}_{x_{1}}^{+} (g_{0})},

where $\sim_{bichar}^{g_{0}}$ is the bicharacteristic relation of $g_{0}$ and $\overline{V}_{x_{1}}^{+} (g_{0})$ is the closed future light cone in $T_{x_{1}}^{*} M$ for $g_{0}$ . This is exactly the Radzikowski wave-front-set condition 13.09.03 applied at $g_{0}$ . $□$

Proof of the deformation existence theorem.

Step 1: construction of the deformation. Given the target $(M, g_{1})$ globally hyperbolic, apply the Bernal-Sánchez splitting 13.09.01 to write $M ≅ R \times Σ$ with $g_{1} = - β^{2} d t^{2} + h_{t}$ . Fix an arbitrary complete Riemannian metric $h_{0}$ on $Σ$ (e.g. the slice metric $h_{t} ∣_{t = 0}$ ); define the ultrastatic reference $g_{0} = - d t^{2} + h_{0}$ on $R \times Σ$ . Construct a smooth one-parameter family interpolating between them:

g_{s} = - β_{s} (t, x)^{2} d t^{2} + h_{s} (t, x), β_{s} = (1 - χ (s)) + χ (s) β, h_{s} = (1 - χ (s)) h_{0} + χ (s) h_{t},

with a smooth cutoff $χ : [0, 1] \to [0, 1]$ , $χ (0) = 0$ , $χ (1) = 1$ . For each $s \in [0, 1]$ , $g_{s}$ is Lorentzian (positive convex combinations of positive lapses and positive-definite spatial metrics), and a verification using the convex-combination property of the Cauchy-diamond compactness shows $g_{s}$ is globally hyperbolic with $Σ_{0}$ a Cauchy slice. (For generic $(M, g_{1})$ , the construction may require smooth cutoffs to keep $g_{s}$ ultrastatic outside a compact spacetime region for $s$ near 0; this is the FNW 1981 Ann. Phys. construction.)

Step 2: smooth family of causal propagators. For each $s$ , the Klein-Gordon operator $P_{s} = □_{g_{s}} + m^{2} + V$ has a unique pair of advanced and retarded fundamental solutions $E_{s}^{\pm}$ and a causal propagator $E_{s} = E_{s}^{-} - E_{s}^{+}$ by 13.09.02. Smooth dependence of the Cauchy problem on the metric (proved by the same energy-estimate argument of 13.09.02 applied uniformly in $s$ ) gives that $s \mapsto E_{s}$ is a smooth map into the appropriate Fréchet space of distributions on $M \times M$ . The Cauchy-data symplectic forms $σ_{s} (f, g) = \int_{M} E_{s} (f) \cdot g d vol_{g_{s}}$ vary smoothly in $s$ .

Step 3: symplectic isomorphism of phase spaces. The Cauchy-data phase spaces $(S_{s}, σ_{s})$ are connected by a smooth family of symplectic isomorphisms $T_{s} : S_{0} \to S_{s}$ obtained as follows. For each $s$ , the Cauchy-data evaluation map $ρ_{s} : C_{sc}^{\infty} (M) \to C^{\infty} (Σ) \oplus C^{\infty} (Σ)$ sending a solution $u$ to its Cauchy data $(u ∣_{Σ}, n_{s}^{μ} \nabla_{μ} u ∣_{Σ})$ (with $n_{s}$ the $g_{s}$ -future-directed unit normal) is a bijection onto smooth Cauchy data; the inverse is solution of the Cauchy problem with that data. The composition $ρ_{s}^{- 1} \circ ρ_{0} : C_{sc, 0}^{\infty} (M) \to C_{sc, s}^{\infty} (M)$ takes a $g_{0}$ -solution to the $g_{s}$ -solution with the same Cauchy data on $Σ$ . This descends to a symplectic isomorphism $T_{s} : (S_{0}, σ_{0}) \to (S_{s}, σ_{s})$ .

Step 4: pull-back state. Pull back the ultrastatic ground state $ω_{0}$ on $(M, g_{0})$ to a quasi-free state $ω_{s}$ on $(M, g_{s})$ by the pull-back of covariances: $η_{s} (Φ_{1}, Φ_{2}) := η_{0} (T_{s}^{- 1} Φ_{1}, T_{s}^{- 1} Φ_{2})$ . The pair $(η_{s}, σ_{s})$ satisfies the quasi-free state conditions ( $η_{s}$ symmetric, $η_{s} \geq \frac{1}{4} σ_{s} σ_{s}^{*}$ , etc.) by transport from $(η_{0}, σ_{0})$ . The corresponding two-point function $ω_{2, s}$ on $M \times M$ is a distribution depending smoothly on $s$ .

Step 5: preservation of the Hadamard wave-front-set condition. The crux. By Step 3 of the ultrastatic-ground-state proof, $WF (ω_{2, 0}) = WF (ω_{0})$ is the future-pointing half of the bicharacteristic relation of $g_{0}$ . For $s > 0$ , the pulled-back two-point function $ω_{2, s}$ satisfies the Klein-Gordon equation $P_{s} ω_{2, s} = 0$ in each argument (by construction: $ω_{2, s}$ is built from the symplectic data at $s$ which respects the $g_{s}$ -Klein-Gordon dynamics). By Hörmander's propagation-of-singularities theorem 02.14.03 applied to the real-principal-type operator $P_{s}$ , the wave-front set of $ω_{2, s}$ is invariant under the Hamiltonian flow of $p_{s} (x, ξ) = g_{s}^{μν} (x) ξ_{μ} ξ_{ν}$ in each argument. The smooth dependence of $p_{s}$ on $s$ (via the smooth dependence of $g_{s}$ ) means the bicharacteristic flow of $g_{s}$ varies smoothly in $s$ .

The future-pointing positive-frequency condition $k \in \overline{V}_{x}^{+} (g_{s})$ is open in $T^{*} M ∖ 0$ and varies smoothly with $s$ (the closed future light cone of $g_{s}$ moves smoothly as $g_{s}$ varies). Since $WF (ω_{2, 0})$ is contained in the future-pointing half of the $g_{0}$ -bicharacteristic relation, and the wave-front set of $ω_{2, s}$ is the image of $WF (ω_{2, 0})$ under a smooth family of symplectic transformations whose principal symbol is the $g_{s}$ -bicharacteristic flow, the wave-front set of $ω_{2, s}$ is contained in the future-pointing half of the $g_{s}$ -bicharacteristic relation for every $s$ .

By the Radzikowski theorem 13.09.03 (the wave-front-set characterisation of Hadamard states), $ω_{s}$ is Hadamard for every $s \in [0, 1]$ . Setting $s = 1$ gives $ω_{1}$ a Hadamard state on $(M, g_{1})$ . Existence is established. $■$

Bridge. The FNW existence theorem builds toward 13.09.06, where Wick polynomials and time-ordered products on a curved background are defined via Hadamard-parametrix subtraction (Brunetti-Fredenhagen-Köhler 1996; Hollands-Wald 2001). The construction requires a Hadamard state to subtract against — the FNW existence theorem of the present unit is exactly what guarantees that such a state exists on every globally hyperbolic spacetime, making the Wick-polynomial construction well-defined. Appears again in 13.07.02, where the Hartle-Hawking state on the Kruskal extension of Schwarzschild is constructed by a Killing-horizon-respecting refinement of the FNW argument: instead of an arbitrary ultrastatic reference, one uses the Hartle-Hawking-vacuum-like state on the Killing-horizon spacetime, with the deformation respecting the bifurcate-Killing-horizon symmetry. The pattern — an existence theorem proved by deformation from an explicit reference — recurs in the construction of the Bunch-Davies state on de Sitter (Allen 1985; Bunch-Davies 1978) and the adiabatic vacua on FLRW (Lüders-Roberts 1990; Junker 1996), each of which is a deformation-existence result with a specific reference spacetime tailored to the cosmological / black-hole symmetries.

Exercises Intermediate+

Exercise 1 (easy, symbolic).

Write out the ground-state two-point function $ω_{0} (x, y)$ on the simplest ultrastatic spacetime $(M, g_{0}) = (R \times R^{3}, - d t^{2} + δ_{ij} d x^{i} d x^{j})$ for the massless Klein-Gordon field. Identify the integration measure and the positive-frequency selection.

Hint

On flat space $R^{3}$ , $- Δ = - \sum_{i} \partial_{i}^{2}$ , so $- Δ + m^{2}$ acts by Fourier transform as multiplication by $ω_{k} = ∣ k ∣^{2} + m^{2}$ . In the massless case, $ω_{k} = ∣ k ∣$ .

Answer

In the massless case on Minkowski $R \times R^{3}$ ,

ω_{0} (x, y) = \int_{R^{3}} \frac{d ^{3} k}{( 2 π ) ^{3} \cdot 2∣ k ∣} e^{- i ∣ k ∣ (t_{x} - t_{y}) + i k \cdot (x - y)} .

The integration measure is the standard Lorentz-invariant measure on the upper massless mass shell $k^{0} = + ∣ k ∣$ in four-momentum space, with the factor $(2∣ k ∣)^{- 1}$ corresponding to the spectral-theorem inverse of $D = ∣ k ∣$ . The positive-frequency selection is the restriction $k^{0} = + ∣ k ∣$ (upper sheet, not lower) — this is the spectral-resolution-of- $D$ statement made concrete: $e^{- i (t_{x} - t_{y}) D}$ in the time variable gives the positive-frequency phase $e^{- i ω_{k} (t_{x} - t_{y})}$ , never $e^{+ i ω_{k} (t_{x} - t_{y})}$ . The result reproduces the standard Minkowski vacuum two-point function from the worked example. By Step 3 of the ultrastatic-ground-state proof, this state has wave-front set on the future-pointing half of the bicharacteristic relation of the Minkowski metric — the Radzikowski wave-front-set condition holds. The Minkowski vacuum is the simplest case of the FNW theorem.

Exercise 2 (easy, multiple choice).

The construction of a smooth family $g_{s}$ of globally hyperbolic metrics interpolating between an ultrastatic reference $g_{0}$ and a target $g_{1}$ uses the Bernal-Sánchez splitting because

A. The splitting provides a canonical product structure $M ≅ R \times Σ$ on which the convex combination $g_{s} = - ((1 - s) + s β)^{2} d t^{2} + ((1 - s) h_{0} + s h_{t})$ makes sense B. The splitting forces all globally hyperbolic spacetimes to be ultrastatic C. The splitting eliminates the need for the propagation-of-singularities theorem D. The splitting is what makes the spacetime globally hyperbolic in the first place

Hint

The Bernal-Sánchez theorem says every globally hyperbolic spacetime can be written as $R \times Σ$ with metric $- β^{2} d t^{2} + h_{t}$ . This is the canonical product structure on which one interpolates.

Answer

A. The splitting provides a canonical product structure $M ≅ R \times Σ$ on which the convex combination makes sense.

The Bernal-Sánchez 2005 theorem says every globally hyperbolic spacetime admits a smooth diffeomorphism $Ψ : R \times Σ \sim M$ in which the metric takes the form $g = - β^{2} d t^{2} + h_{t}$ with $β : M \to R_{> 0}$ smooth and $h_{t}$ a smoothly $t$ -varying Riemannian metric on $Σ$ . Once the target $g_{1} = - β^{2} d t^{2} + h_{t}$ is in this form, an ultrastatic reference $g_{0} = - d t^{2} + h_{0}$ can be defined on the same product manifold $R \times Σ$ , and the convex combination $g_{s} = - ((1 - s) + s β)^{2} d t^{2} + ((1 - s) h_{0} + s h_{t})$ interpolates between them through Lorentzian metrics on $R \times Σ$ . The product structure is canonical (independent of choices up to the diffeomorphism), so the interpolation is well-defined.

Option B is wrong: the splitting writes the metric in an adapted gauge, but does not make it ultrastatic — the ultrastatic case is the very special subset where $β \equiv 1$ and $h_{t}$ is independent of $t$ . Option C is wrong: propagation of singularities is the load-bearing analytic input to the preservation step, separate from the geometric interpolation. Option D inverts cause and effect — global hyperbolicity is the prerequisite that makes the Bernal-Sánchez splitting available, not the other way around.

Exercise 3 (medium, symbolic).

Show that the convex-combination family $g_{s} = (1 - s) g_{0} + s g_{1}$ of two Lorentzian metrics $g_{0}, g_{1}$ in mostly-plus signature with the same time orientation need not be Lorentzian for all $s \in (0, 1)$ . Give a counterexample, then explain why the more careful construction $g_{s} = - β_{s}^{2} d t^{2} + h_{s}$ in the Bernal-Sánchez gauge avoids this failure.

Hint

Convex combinations of Lorentzian metrics in different orientations can be degenerate or even Riemannian. Consider what happens if $g_{0}$ and $g_{1}$ have time directions that are not aligned in a global coordinate.

Answer

Counterexample. On $R^{1, 1}$ , let $g_{0} = - d t^{2} + d x^{2}$ (standard Minkowski) and $g_{1} = + d t^{2} - d x^{2} = - g_{0}$ . Both are Lorentzian, but $g_{1/2} = \frac{1}{2} (g_{0} + g_{1}) = 0$ is the zero tensor — degenerate, not Lorentzian. So convex combinations fail Lorentzian-ness even for elementary examples once the time-direction orientation is not consistently maintained.

Why the Bernal-Sánchez gauge construction works. In the gauge $g_{0} = - d t^{2} + h_{0}$ and $g_{1} = - β^{2} d t^{2} + h_{t}$ with $β > 0$ and $h_{0}, h_{t}$ Riemannian, the convex-combination construction takes the explicit form

g_{s} = - ((1 - s) + s β)^{2} d t^{2} + ((1 - s) h_{0} + s h_{t}) .

The lapse coefficient $((1 - s) + s β)^{2}$ is positive for $s \in [0, 1]$ and any $β > 0$ (positive convex combination); the spatial metric $(1 - s) h_{0} + s h_{t}$ is positive-definite for $s \in [0, 1]$ as a convex combination of positive-definite metrics. So $g_{s}$ has signature $(-, +, \dots, +)$ throughout the interpolation — manifestly Lorentzian. The trick is that the gauge isolates the time direction $d t$ and the spatial directions $h$ separately, with explicit positive-coefficient interpolation in each; the naive convex combination $(1 - s) g_{0} + s g_{1}$ does not separate the directions and can fail.

The Bernal-Sánchez gauge thus does double duty: it gives a canonical product structure on the spacetime (the topological part of the FNW argument), and it gives explicit positive-coefficient interpolation formulas that preserve Lorentzian signature (the geometric part).

Exercise 4 (medium, symbolic).

Show that the FNW pull-back state $ω_{1}$ on $(M, g_{1})$ depends on the choice of ultrastatic reference $h_{0}$ on $Σ$ . Specifically, show that two different reference Riemannian metrics $h_{0}^{(A)}$ and $h_{0}^{(B)}$ on the same $Σ$ give pull-back states $ω_{1}^{(A)}, ω_{1}^{(B)}$ that differ by a smooth function on $M \times M$ — but are not literally the same state.

Hint

Use the uniqueness-up-to-smooth-corrections result from 13.09.03 Exercise 3: any two Hadamard states differ by a smooth function. The FNW pull-back gives a Hadamard state for any choice of reference, so applying that uniqueness result to the two pull-backs gives the smooth-difference statement.

Answer

Each FNW pull-back $ω_{1}^{(A)}, ω_{1}^{(B)}$ is a Hadamard quasi-free state on $(M, g_{1})$ by the FNW existence theorem (Steps 1-5 of the proof) applied to each reference choice. By the uniqueness-up-to-smooth-corrections proposition of 13.09.03 (proved in its Full proof set), any two Hadamard quasi-free states on the same CCR algebra have two-point functions differing by a smooth function:

ω_{1}^{(A)} (x, y) - ω_{1}^{(B)} (x, y) \in C^{\infty} (M \times M) .

This smooth difference is real — the two states are physically equivalent for the purpose of computing renormalised observables (the Hadamard subtraction $ω_{2} - H$ gives the same singular structure up to the smooth correction), but they are not the same distribution. The freedom is the choice of "vacuum" on a curved background: there is no canonical Hadamard state, only a class of them parametrised by the smooth corrections.

This is the curved-spacetime analogue of the flat-space Bogoliubov-coefficient freedom in choosing different "vacua" of an accelerating observer (Rindler vacuum) or a moving frame (Doppler-shifted vacuum) — different but unitarily equivalent (up to smooth corrections) realisations of the same CCR algebra. The Wightman vacuum on Minkowski is uniquely picked out by Poincaré invariance; on a generic curved spacetime no symmetry picks out a unique Hadamard state.

Exercise 5 (medium, symbolic).

Verify that the ultrastatic ground-state quasi-free covariance

η (Φ_{1}, Φ_{2}) = \frac{1}{2} \int_{Σ} (u_{0}^{(1)} D u_{0}^{(2)} + u_{1}^{(1)} (D)^{- 1} u_{1}^{(2)}) d vol_{h_{0}}

with $D = - Δ_{h_{0}} + m^{2} + V$ , $V \geq 0$ , $m > 0$ , satisfies the purity condition $η = \frac{1}{4} σ \otimes σ$ where $σ (Φ_{1}, Φ_{2}) = \int_{Σ} (u_{0}^{(1)} u_{1}^{(2)} - u_{0}^{(2)} u_{1}^{(1)}) d vol_{h_{0}}$ is the Cauchy-data symplectic form. (Purity is the algebraic-QFT condition that the ground state is a pure state, not a mixed thermal state.)

Hint

Compute the matrix elements of $η$ and of $\frac{1}{4} σ σ^{*}$ in the basis $(u_{0}, 0), (0, u_{1})$ of Cauchy data on $Σ$ . The purity condition reduces to a quadratic identity in $D$ .

Answer

Pick Cauchy data $Φ_{1} = (u_{0}^{(1)}, u_{1}^{(1)})$ and $Φ_{2} = (u_{0}^{(2)}, u_{1}^{(2)})$ in $C_{c}^{\infty} (Σ) \oplus C_{c}^{\infty} (Σ)$ . The symplectic form gives

σ (Φ_{1}, Φ_{2}) = \int_{Σ} (u_{0}^{(1)} u_{1}^{(2)} - u_{0}^{(2)} u_{1}^{(1)}) d vol_{h_{0}} .

The covariance is the symmetric form

η (Φ_{1}, Φ_{2}) = \frac{1}{2} \int_{Σ} (u_{0}^{(1)} D u_{0}^{(2)} + u_{1}^{(1)} (D)^{- 1} u_{1}^{(2)}) d vol_{h_{0}} .

The purity condition $η = \frac{1}{4} σ σ^{*}$ as quadratic forms on Cauchy data unpacks to: for any $Φ$ , $η (Φ, Φ) \cdot η (Φ^{'}, Φ^{'}) \geq \frac{1}{4} σ (Φ, Φ^{'})^{2}$ with equality for some $Φ^{'}$ depending on $Φ$ . Pick the test data $Φ^{'} = (D^{- 1/2} u_{1}, - D^{1/2} u_{0})$ for $Φ = (u_{0}, u_{1})$ . Compute:

σ (Φ, Φ^{'}) = \int_{Σ} (u_{0} \cdot (- D^{1/2} u_{0}) - D^{- 1/2} u_{1} \cdot u_{1}) d vol_{h_{0}} = - \int_{Σ} (u_{0} D^{1/2} u_{0} + u_{1} D^{- 1/2} u_{1}) d vol_{h_{0}} = - 2 η (Φ, Φ) .

So $σ (Φ, Φ^{'})^{2} = 4 η (Φ, Φ)^{2}$ , and $η (Φ, Φ) \cdot η (Φ^{'}, Φ^{'}) = η (Φ, Φ)^{2}$ (a short check shows $η (Φ^{'}, Φ^{'}) = η (Φ, Φ)$ using the spectral identity $D^{1/2} (D^{- 1/2})^{*} = 1$ ). The purity-saturation condition $4 η (Φ, Φ) η (Φ^{'}, Φ^{'}) = σ (Φ, Φ^{'})^{2}$ becomes $4 η (Φ, Φ)^{2} = 4 η (Φ, Φ)^{2}$ , which holds. The general inequality $η \geq \frac{1}{4} σ σ^{*}$ is saturated for this choice of $Φ^{'}$ , so the ground state is pure. This matches the Araki-Woods 1963 characterisation: a quasi-free state with $η = \frac{1}{4} σ σ^{*}$ has a Fock-representation GNS realisation with cyclic vacuum vector — the pure-state algebraic-QFT condition.

Exercise 6 (medium, short-answer).

Explain why the Gérard-Wrochna 2014 pseudo-differential construction of Hadamard states bypasses the FNW deformation argument. What is the explicit input to the Gérard-Wrochna construction, and why does it give a Hadamard state directly?

Hint

The Gérard-Wrochna construction picks a Cauchy hypersurface $Σ$ and constructs a Hadamard state from a pseudo-differential operator $ϵ$ on $Σ$ that approximates $D_{h}$ in the appropriate sense.

Answer

The Gérard-Wrochna 2014 Comm. Math. Phys. 325 713 construction takes the following input: a smooth spacelike Cauchy hypersurface $Σ \subset (M, g)$ with induced metric $h$ and induced operator $D_{h} = - Δ_{h} + m^{2} + V$ on $L^{2} (Σ, d vol_{h})$ . The construction picks a $Ψ^{1}$ -pseudo-differential operator $ϵ$ on $Σ$ that is a parametrix for $D_{h}$ — i.e. $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ is smoothing — and has the property that the principal symbol of $ϵ$ is the positive square root of the principal symbol of $D_{h}$ .

From the pair $(ϵ, σ)$ on Cauchy data, the Hadamard two-point function is constructed by the explicit formula

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

Why this gives a Hadamard state directly: the wave-front-set condition of the Radzikowski criterion translates, on the Cauchy hypersurface $Σ$ , into a condition on the principal symbol of the Cauchy-data covariance operator. The Gérard-Wrochna choice of $ϵ$ has principal symbol exactly $p_{h} (x, ξ)$ where $p_{h}$ is the principal symbol of $D_{h}$ on $Σ$ , and this is the positive square root that produces the future-pointing positive-frequency wave-front-set on $M$ when the Cauchy data are evolved forward by the Klein-Gordon equation. The pseudo-differential approximation $ϵ^{2} \approx D_{h}$ modulo smoothing is what guarantees that the construction is well-defined (the operator $ϵ^{- 1}$ exists modulo smoothing, since the principal symbol is invertible).

The construction bypasses the deformation argument because no auxiliary ultrastatic spacetime is used; everything happens on the target $(M, g)$ with the Cauchy slice $Σ$ as the working data. The trade-off is that the construction requires the pseudo-differential calculus on $Σ$ (and propagation of singularities to extend the Hadamard property off $Σ$ to $M$ ), while the FNW deformation is more elementary — it uses only the explicit ultrastatic-ground-state spectral construction plus smooth-family-of-globally-hyperbolic-metrics geometry. For practical computations (especially explicit two-point functions on cosmological or black-hole backgrounds) Gérard-Wrochna is preferred; for the conceptual existence theorem the FNW deformation is the foundational argument.

Exercise 7 (hard, symbolic).

Sketch the propagation-of-singularities argument that preserves the wave-front-set Hadamard condition along the FNW deformation. Identify the role of (i) smooth dependence of the principal symbol $p_{s} (x, ξ) = g_{s}^{μν} (x) ξ_{μ} ξ_{ν}$ on $s$ ; (ii) Hörmander's theorem for the deformed equation $P_{s} u_{s} = 0$ ; (iii) openness of the future-pointing condition $k \in \overline{V}_{x}^{+} (g_{s})$ .

Hint

The wave-front set of the pulled-back state at each $s$ is constrained by the Hamiltonian flow of $p_{s}$ in each argument. Combine this with the bound from the $s = 0$ case (where the wave-front set is in the future-pointing half of the $g_{0}$ -bicharacteristic relation).

Answer

Setup. The pulled-back two-point function $ω_{2, s}$ satisfies $P_{s} ω_{2, s} = 0$ in each argument, by construction (the pull-back is along symplectic isomorphisms that preserve the Klein-Gordon dynamics in each argument). So $ω_{2, s}$ is a bi-distribution on $M \times M$ that solves the homogeneous $g_{s}$ -Klein-Gordon equation in each argument and at $s = 0$ has wave-front set in the future-pointing half of the $g_{0}$ -bicharacteristic relation.

Step (i): smooth dependence of $p_{s}$ . The principal symbol $p_{s} (x, ξ) = g_{s}^{μν} (x) ξ_{μ} ξ_{ν}$ depends smoothly on $s$ as a function on $T^{*} M$ , because the metric components $g_{s}^{μν}$ depend smoothly on $s$ by construction of the family. The Hamiltonian flow $Φ_{s}^{τ}$ of $p_{s}$ on $T^{*} M ∖ 0$ depends smoothly on $s$ by standard smooth-dependence-on-parameters of Hamilton's equations. The closed future light cone $\overline{V}_{x}^{+} (g_{s})$ in $T_{x}^{*} M$ is the closed set ${k : g_{s}^{μν} (x) k_{μ} k_{ν} \leq 0, k \cdot \partial_{t} (g_{s}) \leq 0}$ (with appropriate time-orientation choice), which varies smoothly with $s$ in the closed-set Hausdorff topology.

Step (ii): Hörmander's theorem. For each $s$ , $P_{s}$ is a real-principal-type pseudo-differential operator of order 2 on $M$ with real principal symbol $p_{s}$ . By Hörmander's propagation-of-singularities theorem 02.14.03, for any solution $u_{s}$ of $P_{s} u_{s} = f_{s}$ , $WF (u_{s}) ∖ WF (f_{s})$ is contained in the characteristic variety $p_{s}^{- 1} (0)$ and is invariant under the Hamiltonian flow of $p_{s}$ on $T^{*} M ∖ 0$ . Applied to the two-point function $ω_{2, s}$ in each argument: $WF (ω_{2, s})$ is invariant under the bicharacteristic flow of $g_{s}$ on each cotangent argument.

Step (iii): openness of the future-pointing condition. The condition $k \in int \overline{V}_{x}^{+} (g_{s})$ (interior of the closed future light cone) is open in $(s, x, k)$ jointly. The future-pointing half of the $g_{s}$ -bicharacteristic relation,

C_{s}^{+} := {(x, k; y, - k^{'}) : (x, k) \sim_{bichar}^{g_{s}} (y, k^{'}), k \in \overline{V}_{x}^{+} (g_{s})},

is invariant under bicharacteristic flow in each argument (by step (ii)), depends smoothly on $s$ (by step (i) plus openness), and equals the wave-front set of $ω_{2, 0}$ at $s = 0$ (by the ultrastatic-ground-state Hadamard property).

Conclusion. Let $C_{s} := WF (ω_{2, s})$ . By Hörmander's theorem, $C_{s} \subseteq p_{s}^{- 1} (0) \times p_{s}^{- 1} (0)$ in $T^{*} (M \times M)$ and is invariant under bicharacteristic flow of $g_{s}$ in each argument. By the continuity of the family $s \mapsto ω_{2, s}$ (in the Fréchet topology of distributions on $M \times M$ ), and the upper-semicontinuity of the wave-front set under continuous limits with bounded singular orders, $C_{s}$ varies upper-semicontinuously in $s$ . The set $C_{0}$ is exactly the future-pointing half $C_{0}^{+}$ . Combining: $C_{s}$ is contained in the $g_{s}$ -future-pointing half $C_{s}^{+}$ for $s \in [0, 1]$ by continuous deformation from $C_{0} = C_{0}^{+}$ .

Equality $C_{s} = C_{s}^{+}$ comes from the CCR-commutator lower bound on the wave-front set of any quasi-free state's two-point function (Proposition: CCR-commutator-condition $\Rightarrow$ wave-front-set lower bound, in the Full proof set of 13.09.03): $WF (ω_{2, s}) \supseteq$ future-pointing half of the bicharacteristic relation, since $Im (ω_{2, s}) = - E_{g_{s}} /2$ has full bicharacteristic wave-front set and the symmetric part cancels only the past-pointing half. Combined with the upper bound, $C_{s} = C_{s}^{+}$ for every $s$ — the Hadamard wave-front-set condition is preserved by the deformation. At $s = 1$ , $ω_{1}$ is Hadamard on $(M, g_{1})$ , establishing the existence theorem. $□$

Exercise 8 (hard, short-answer).

Explain why the FNW existence theorem implies that the Brunetti-Fredenhagen-Verch 2003 locally covariant QFT functor — assigning to each globally hyperbolic spacetime a $C^{*}$ -algebra of observables together with a non-empty class of Hadamard states — is well-defined on the entire category of globally hyperbolic spacetimes with isometric embeddings.

Hint

Local covariance requires the assignment $ghs \mapsto (A, H)$ to be a functor; the Hadamard class $H$ must be non-empty on every object. FNW supplies non-emptiness.

Answer

The Brunetti-Fredenhagen-Verch 2003 Comm. Math. Phys. 237 31 framework axiomatises curved-spacetime QFT as a covariant functor from the category $GlobHyp$ of globally hyperbolic Lorentzian spacetimes with isometric embeddings (preserving causal structure and orientation) to the category $CAlg$ of unital $C^{*}$ -algebras with unit-preserving $*$ -homomorphisms. The functor assigns to each $(M, g) \in GlobHyp$ a $C^{*}$ -algebra of observables $A (M, g)$ — concretely, the CCR / Weyl algebra of the quantised free scalar field on $(M, g)$ in our setting — and to each isometric embedding $ι : (M_{1}, g_{1}) ↪ (M_{2}, g_{2})$ a unit-preserving $*$ -homomorphism $A (ι) : A (M_{1}, g_{1}) \to A (M_{2}, g_{2})$ extending the natural pull-back of CCR generators.

Three axioms make this a well-defined locally covariant QFT: (i) isotony (the functor sends inclusions to subalgebra inclusions), (ii) covariance (composition of embeddings goes to composition of $*$ -homomorphisms), (iii) time-slice (the embedding of a Cauchy-time-slice extension is an isomorphism). All three hold for the free Klein-Gordon functor on globally hyperbolic spacetimes by the Cauchy problem and the symplectic formalism of 13.09.02 and 12.14.01.

A locally covariant state on this functor is, in addition, an assignment $(M, g) \mapsto ω_{(M, g)} \in S (A (M, g))$ compatible with the morphism action. The physical states are required to be Hadamard — the BFK 1996 Comm. Math. Phys. 180 633 microlocal-spectrum condition on all $n$ -point functions, which reduces to the Radzikowski wave-front-set condition for the two-point function. Non-emptiness of the Hadamard class on every $(M, g)$ is exactly the FNW existence theorem of the present unit.

Without FNW the locally covariant functor framework would be vacuous on spacetimes where no Hadamard state existed — the physical-state class would be empty, the Wick-polynomial construction would have nothing to subtract against, and renormalised observables would be undefined. FNW guarantees the Hadamard class is non-empty on every globally hyperbolic spacetime, so the locally covariant functor framework is genuinely defined throughout $GlobHyp$ . The full programme — Hollands-Wald 2001 Wick polynomials, BFV 2003 local covariance, perturbative algebraic QFT on curved spacetimes — rests on this existence result.

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 CCR-algebra layer, no Hadamard-parametrix construction, and no theory of smoothly-varying families of globally hyperbolic metrics as of 2026-05. The closest layers are Geometry.Manifold.SmoothManifoldWithCorners (smooth manifolds), Geometry.Manifold.MetricSpace (positive-definite Riemannian metrics), Mathlib.Analysis.CStarAlgebra.Basic ( $C^{*}$ -algebras), Mathlib.Analysis.NormedSpace.Spectrum (spectral theorem for unbounded operators), Mathlib.Analysis.Distribution (partial distribution theory), and the symmetric tensor algebra for Fock-space substrates.

The full chain of formalisation gaps identified in 13.09.01, 13.09.02, 02.14.01, 02.14.03, 12.14.01, and 13.09.03 must be filled before the FNW deformation existence theorem can be stated in Lean. Above those layers, the present unit additionally requires (i) a notion of a smooth one-parameter family of globally hyperbolic metrics, with smooth dependence of the Cauchy problem and the causal propagator on the metric; (ii) the explicit spectral construction of the ultrastatic ground state via $D_{h}$ on $L^{2} (Σ)$ ; (iii) the symplectic isomorphism of Cauchy-data phase spaces induced by the family of causal propagators; (iv) propagation of singularities along a smoothly-varying family of bicharacteristic flows. Each of these is a substantial Mathlib contribution in its own right. The closest current Lean 4 work is the SpaceTime project (M. Larson and collaborators, outside Mathlib) with a partial Lorentzian-metric layer; nothing in the algebraic-QFT-on-curved-spacetimes 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 ultrastatic-ground-state construction, the deformation construction in the Bernal-Sánchez gauge, and the preservation-of-Hadamard-property argument via propagation of singularities. The Master-tier locally-covariant-existence statement (Sahlmann-Verch 2001) and the pseudo-differential construction (Gérard-Wrochna 2014/2017) are flagged for external review by a microlocal-QFT specialist.

Advanced results Master

Four structural developments extend the FNW existence theorem to the depth required by the modern curved-spacetime QFT programme.

The Junker 1996 adiabatic-vacuum / pseudo-differential refinement. Junker 1996 Rev. Math. Phys. 8 1091 ^{[Junker 1996]} gave the first modern sharpening of FNW. Strategy: pick a Cauchy hypersurface $Σ$ and decompose the Klein-Gordon Cauchy problem into mode-by-mode harmonic oscillators using a smooth-mode approximation of $D_{h}$ on $Σ$ . The adiabatic vacuum of order $N$ is the quasi-free state whose Cauchy-data covariance approximates the would-be exact-ground-state covariance to order $N$ in a smooth-variation parameter; for finite $N$ the state is not exactly Hadamard, but Junker showed that the resulting state has wave-front-set differing from the Hadamard wave-front-set by a $Ψ^{- N}$ -smoothing correction. Taking $N \to \infty$ gives a genuine Hadamard state, supplied by an explicit (rather than deformation-existence) construction. Junker's framework was the prototype for the Gérard-Wrochna 2014 pseudo-differential construction that completed the programme.

The Brunetti-Fredenhagen-Köhler 1996 extension to all $n$ -point functions. A quasi-free state is determined by its two-point function, but a generic state on the CCR algebra has a full hierarchy of $n$ -point functions $ω_{n} (x_{1}, \dots, x_{n})$ . For renormalised perturbative QFT (Wick polynomials, time-ordered products) one needs a Hadamard property on every $ω_{n}$ . Brunetti, Fredenhagen and Köhler 1996 Comm. Math. Phys. 180 633 ^{[Brunetti-Fredenhagen-Köhler 1996]} introduced the microlocal-spectrum condition $μ SC$ on the wave-front sets of $n$ -point functions: $ω$ is microlocally Hadamard if for every $n$ , $WF (ω_{n})$ is contained in a specific subset of $T^{*} (M^{n})$ characterised by instanton graphs — graphs with $n$ external vertices and internal edges realised by null geodesics carrying parallel-transported covectors, with the covector orientation at each external vertex determined by a consistent positive-frequency assignment. The condition reduces to the Radzikowski condition for $n = 2$ and to the Wightman spectrum condition on Minkowski. Existence of states satisfying $μ SC$ on all $n$ -point functions: follows from FNW existence for the two-point function (the present theorem) together with Wick's-rule reconstruction of higher $n$ -point functions from the two-point function (for quasi-free states) plus the BFK 1996 verification that the Wick reconstruction preserves $μ SC$ . For non-quasi-free states $μ SC$ is a genuine additional condition.

The Sahlmann-Verch 2001 locally-covariant existence theorem. Sahlmann-Verch 2001 Rev. Math. Phys. 13 1203 ^{[Sahlmann-Verch 2001]} codified the FNW existence theorem in the locally-covariant-functor language of Brunetti-Fredenhagen-Verch 2003. The statement: the assignment $(M, g) \mapsto H (M, g) \subseteq S (A (M, g))$ taking each globally hyperbolic spacetime to its non-empty class of Hadamard states is a subfunctor of the locally covariant QFT functor — covariant under isometric embeddings, and non-empty on every object. The non-emptiness is exactly the FNW existence theorem of the present unit; the covariance is the natural pull-back of Hadamard states along isometric embeddings, which preserves the wave-front-set condition by functoriality of the wave-front set under embeddings. Sahlmann and Verch also extended the existence theorem to vector-valued fields (Dirac, electromagnetic, Yang-Mills) by adapting the FNW deformation to the appropriate covariant Klein-Gordon-type operators on the relevant vector bundles.

The Gérard-Wrochna 2014 / 2017 direct pseudo-differential construction. The most explicit modern existence construction comes from Gérard and Wrochna ^{[Gérard-Wrochna 2014]}^{[Gérard-Wrochna 2017]}. Strategy: on a globally hyperbolic spacetime $(M, g)$ with Cauchy hypersurface $Σ$ and induced Cauchy-data operator $D_{h} = - Δ_{h} + m^{2} + V$ , choose a $Ψ^{1}$ -pseudo-differential operator $ϵ \in Ψ^{1} (Σ)$ that is a parametrix for $D_{h}$ — i.e. $ϵ^{2} - D_{h} \in Ψ^{- \infty} (Σ)$ — with positive principal symbol $σ_{1} (ϵ) (x, ξ) = p_{h} (x, ξ)$ . From the pair $(ϵ, σ_{h})$ on Cauchy data, construct the quasi-free state with covariance $η (Φ_{1}, Φ_{2}) = \frac{1}{2} \int_{Σ} (u_{0}^{(1)} ϵ u_{0}^{(2)} + u_{1}^{(1)} ϵ^{- 1} u_{1}^{(2)}) d vol_{h}$ . Gérard-Wrochna 2014 show that this state is Hadamard, i.e. its two-point function on $M \times M$ satisfies the Radzikowski wave-front-set condition. The construction bypasses the deformation entirely — no auxiliary ultrastatic spacetime, no smooth one-parameter family of metrics — and gives explicit Hadamard initial data. The trade-off: requires the full pseudo-differential calculus on $Σ$ as input. The Gérard-Wrochna 2017 Ann. Henri Poincaré refinement extends the construction to asymptotically static spacetimes and gives Hadamard property of the in/out states needed for scattering theory on a curved background.

Synthesis. The FNW deformation existence theorem is the foundational existence result of the Hadamard-state programme. It establishes that the class of Hadamard states on every globally hyperbolic spacetime is non-empty, making the algebraic-QFT-on-curved-spacetimes framework well-defined. The theorem builds toward the BFK 1996 extension to all $n$ -point functions, the Sahlmann-Verch 2001 locally-covariant formulation, the Brunetti-Fredenhagen-Verch 2003 locally-covariant QFT functor, the Hollands-Wald 2001 Wick-polynomial construction (which uses the existence of a Hadamard state as a hypothesis), the Junker 1996 / Gérard-Wrochna 2014 pseudo-differential refinements, and applications to the Hartle-Hawking state on Schwarzschild-Kruskal and the Bunch-Davies state on de Sitter. Every concrete curved-spacetime QFT construction — every renormalised stress-energy tensor, every Wick polynomial, every black-hole-radiation calculation, every cosmological-perturbation power spectrum — depends on the existence of a Hadamard reference state. FNW supplies that existence in full generality.

Full proof set Master

Proposition (smooth dependence of the Cauchy problem on the metric). Let $(M, g_{s})_{s \in [0, 1]}$ be a smooth one-parameter family of globally hyperbolic metrics on a smooth manifold $M$ adapted to a common Cauchy hypersurface $Σ \subset M$ . For $(u_{0}, u_{1}) \in C_{c}^{\infty} (Σ) \oplus C_{c}^{\infty} (Σ)$ , let $u_{s}$ denote the unique solution of $P_{s} u_{s} = 0$ with $u_{s} ∣_{Σ} = u_{0}$ , $n_{s}^{μ} \nabla_{μ} u_{s} ∣_{Σ} = u_{1}$ , where $P_{s} = □_{g_{s}} + m^{2} + V$ . Then the map $s \mapsto u_{s}$ is smooth as a map $[0, 1] \to C^{\infty} (M)$ in the Fréchet topology.

Proof. For each $s$ , $u_{s} \in C^{\infty} (M)$ exists and is unique by the Cauchy theorem of 13.09.02. The map $s \mapsto P_{s}$ is smooth in $s$ as a map into the space of second-order linear differential operators on $M$ with smooth coefficients. The energy estimate of 13.09.02 (Step 1 of its key derivation) gives a uniform-in- $s$ bound on $u_{s}$ in $H^{k}$ norms over any compact spacetime region, with the bound depending continuously on $s$ through the metric components. Differentiating the energy estimate in $s$ gives bounds on the $s$ -derivatives $\partial_{s}^{k} u_{s}$ in $H^{k}$ norms uniformly on compact subsets, hence in $C^{\infty} (M)$ in the Fréchet topology by standard Sobolev-embedding arguments. The map $s \mapsto u_{s}$ is therefore $C^{\infty}$ as a map $[0, 1] \to C^{\infty} (M)$ . $□$

Proposition (smooth dependence of the causal propagator on the metric). Under the hypotheses of the previous proposition, the family of causal propagators $E_{s} : C_{c}^{\infty} (M) \to C^{\infty} (M)$ depends smoothly on $s$ .

Proof. For each $s$ , $E_{s} = E_{s}^{-} - E_{s}^{+}$ where $E_{s}^{\pm}$ is defined (Step 5 of the key derivation of 13.09.02) as the unique solution operator with zero Cauchy data on a Cauchy hypersurface to the past (resp. future) of the source. The construction is a smooth functional of $g_{s}$ via smooth dependence of the Cauchy problem, by the previous proposition. So the map $s \mapsto E_{s}$ is smooth as a map into the appropriate Fréchet space of operators $C_{c}^{\infty} (M) \to C^{\infty} (M)$ . The corresponding kernels $E_{s} (x, y) \in D^{'} (M \times M)$ depend smoothly on $s$ as distributions on $M \times M$ . $□$

Proposition (Hadamard wave-front-set on the ultrastatic ground state). Let $(M, g_{0}) = (R \times Σ, - d t^{2} + h_{0})$ be ultrastatic globally hyperbolic with $h_{0}$ complete on $Σ$ , and let $D = - Δ_{h_{0}} + m^{2} + V$ on $L^{2} (Σ, d vol_{h_{0}})$ with $V \in C_{b}^{\infty} (Σ)$ , $V \geq 0$ , $m > 0$ . The ultrastatic ground state $ω_{0}$ with two-point function

ω_{0} (f_{1}, f_{2}) = \int_{R^{2}} d t_{1} d t_{2} \int_{Σ} f_{1} (t_{1}, \cdot) (2 D)^{- 1} e^{- i (t_{1} - t_{2}) D} f_{2} (t_{2}, \cdot) d vol_{h_{0}}

has wave-front set

WF (ω_{0}) = {(x_{1}, k_{1}; x_{2}, - k_{2}) : (x_{1}, k_{1}) \sim_{bichar}^{g_{0}} (x_{2}, k_{2}), k_{1} \in \overline{V}_{x_{1}}^{+} (g_{0})},

the future-pointing half of the null-bicharacteristic relation of $g_{0}$ .

Proof. Apply Fourier transform in the time variables $t_{1}, t_{2}$ at fixed $x_{1}, x_{2} \in Σ$ . The integrand reduces (after orthogonality of the time-Fourier modes) to an integral over the spectral parameter $τ$ of the operator family $D$ :

ω_{0} (τ_{1}, τ_{2}, x_{1}, x_{2}) = δ (τ_{1} - τ_{2}) θ (τ_{1}) \cdot δ (τ_{1} - D) (x_{1}, x_{2}) \cdot (2 τ_{1})^{- 1},

where $δ (τ - D)$ is the spectral projection of $D$ at eigenvalue $τ$ , acting on $L^{2} (Σ)$ with integral kernel given by the spectral density. The support of $ω_{0}$ in $(τ_{1}, τ_{2})$ is on $τ_{1} = τ_{2} > 0$ — the upper sheet of the "frequency mass shell" — by the spectral support of $D$ on $[c, \infty)$ with $c = in f V + m^{2} > 0$ (and the positive-square-root choice that makes $D$ positive semi-definite).

The spectral-density kernel $δ (τ - D) (x_{1}, x_{2})$ for fixed $τ > 0$ has wave-front set, by the propagation-of-singularities theorem applied to $D$ on $Σ$ , on pairs $(x_{1}, ζ_{1}; x_{2}, - ζ_{2})$ with $p_{h} (ζ_{i}) = τ^{2}$ (i.e. on the spatial mass shell $∣ ζ ∣^{2} + V + m^{2} = τ^{2}$ ) and $(x_{1}, ζ_{1}) \sim_{bichar}^{h_{0}} (x_{2}, ζ_{2})$ with bicharacteristic on $Σ$ given by the geodesic flow of $h_{0}$ .

Combining with the upper-time-frequency condition $τ > 0$ gives the wave-front set of $ω_{0}$ on $M \times M$ (with $M = R \times Σ$ ):

WF (ω_{0}) = {(x_{1}, (τ, ζ_{1}); x_{2}, (- τ, - ζ_{2})) : τ^{2} = ∣ ζ ∣^{2} + V + m^{2}, τ > 0, (x_{1}, ζ_{1}) \sim^{h_{0}} (x_{2}, ζ_{2})} .

The condition $τ^{2} = ∣ ζ ∣^{2} + V + m^{2}$ is the on-mass-shell condition for the $g_{0}$ -Klein-Gordon operator (with $g_{0}^{μν} (τ, ζ)_{μ} (τ, ζ)_{ν} = - τ^{2} + ∣ ζ ∣^{2} + V + m^{2} = 0$ ). The bicharacteristic-on- $Σ$ relation $\sim^{h_{0}}$ lifts to the bicharacteristic-on- $M$ relation $\sim_{bichar}^{g_{0}}$ for the ultrastatic metric by adding the constant time-translation component. The future-pointing condition $τ > 0$ is the closed-future-light-cone condition $k = (τ, ζ) \in \overline{V}_{x}^{+} (g_{0})$ .

So $WF (ω_{0})$ is exactly the future-pointing half of the $g_{0}$ -bicharacteristic relation. The Radzikowski wave-front-set condition holds; $ω_{0}$ is Hadamard. $□$

Proposition (preservation of Hadamard wave-front-set along smooth deformation). Let $(g_{s})_{s \in [0, 1]}$ be a smooth family of globally hyperbolic metrics on $M$ adapted to a common Cauchy hypersurface $Σ$ , and let $ω_{2, s}$ be the family of pulled-back two-point functions from an ultrastatic ground state $ω_{0}$ on $(M, g_{0})$ . Then $ω_{2, s}$ is a Hadamard quasi-free state for every $s \in [0, 1]$ .

Proof. By the ultrastatic-ground-state proposition, $WF (ω_{2, 0}) = C_{0}^{+}$ where $C_{s}^{+} := {(x, k; y, - k^{'}) : (x, k) \sim^{g_{s}} (y, k^{'}), k \in \overline{V}_{x}^{+} (g_{s})}$ . By Hörmander's theorem applied to the homogeneous equation $P_{s} ω_{2, s} = 0$ in each argument, $WF (ω_{2, s})$ is contained in the characteristic variety $p_{s}^{- 1} (0) \times p_{s}^{- 1} (0)$ of $P_{s} \otimes P_{s}$ on $T^{*} (M \times M) ∖ 0$ and is invariant under the bicharacteristic flow of $g_{s}$ in each argument. Combined with the smooth dependence of $p_{s}$ on $s$ (and hence smooth dependence of the bicharacteristic flow), the set $WF (ω_{2, s})$ deforms continuously from $C_{0}^{+}$ at $s = 0$ .

Upper bound: by upper semi-continuity of the wave-front set under continuous deformations of distributions (with bounded singular orders, which holds because $ω_{2, s}$ has uniform-in- $s$ Hadamard singularity-degree estimates by the parametrix construction), $WF (ω_{2, s}) \subseteq C_{s}^{+}$ for every $s$ .

Lower bound: by the CCR-commutator-condition proposition of 13.09.03 (Full proof set), $WF (ω_{2, s}) \supseteq$ future-pointing half of the bicharacteristic relation of $g_{s}$ , i.e. $WF (ω_{2, s}) \supseteq C_{s}^{+}$ .

Combining: $WF (ω_{2, s}) = C_{s}^{+}$ , the Radzikowski wave-front-set condition for $g_{s}$ . By the Radzikowski theorem 13.09.03, $ω_{2, s}$ is Hadamard for every $s \in [0, 1]$ . $□$

Proposition (FNW existence theorem). For every globally hyperbolic Lorentzian manifold $(M, g)$ and every Klein-Gordon operator $P = □_{g} + m^{2} + V$ with $V \in C^{\infty} (M)$ , $m > 0$ , the CCR algebra $A (M, g)$ admits at least one Hadamard quasi-free state.

Proof. Apply the Bernal-Sánchez splitting 13.09.01 to write $M ≅ R \times Σ$ with $g = - β^{2} d t^{2} + h_{t}$ for $β > 0$ smooth and $h_{t}$ smoothly $t$ -varying Riemannian on $Σ$ . Pick an ultrastatic reference $g_{0} = - d t^{2} + h_{0}$ where $h_{0}$ is any complete Riemannian metric on $Σ$ (e.g. $h_{0} = h_{t = 0}$ ). Construct the smooth family $g_{s} = - ((1 - χ (s)) + χ (s) β)^{2} d t^{2} + ((1 - χ (s)) h_{0} + χ (s) h_{t})$ for $s \in [0, 1]$ with $χ : [0, 1] \to [0, 1]$ smooth, $χ (0) = 0$ , $χ (1) = 1$ . Each $g_{s}$ is globally hyperbolic with Cauchy slice $Σ_{0} = {0} \times Σ$ .

By the ultrastatic-ground-state proposition, the ground state $ω_{0}$ on $(M, g_{0})$ is Hadamard. By smooth dependence of the causal propagator on the metric, the family of symplectic isomorphisms $T_{s} : (S_{0}, σ_{0}) \to (S_{s}, σ_{s})$ varies smoothly in $s$ and gives a family of pulled-back quasi-free states $ω_{s}$ . By the preservation-along-deformation proposition, $ω_{s}$ is Hadamard for every $s \in [0, 1]$ . Setting $s = 1$ gives a Hadamard quasi-free state $ω_{1}$ on $(M, g_{1}) = (M, g)$ . $□$

Connections Master

Globally hyperbolic Lorentzian manifolds 13.09.01 supplies the geometric arena and the Bernal-Sánchez splitting $M ≅ R \times Σ$ used to construct the smooth deformation $g_{s}$ between the target and the ultrastatic reference. Without the smooth metric splitting, the FNW construction would be much harder — the Geroch 1970 continuous-Cauchy-surface version supplies only a topological product structure, insufficient for the smooth interpolation of the metric.
Klein-Gordon equation on a globally hyperbolic spacetime 13.09.02 supplies the analytic infrastructure: smooth dependence of the Cauchy problem on the metric (an extension of the energy-estimate argument from a single metric to a smooth family), the causal propagators $E_{s}$ varying smoothly with the metric, and the symplectic isomorphism $T_{s}$ of Cauchy-data phase spaces induced by the family of propagators. The unique pull-back of the ultrastatic ground state along this symplectic isomorphism is the FNW pull-back state.
Hadamard states via the wave-front-set criterion 13.09.03 is the immediate downstream input. The Radzikowski wave-front-set characterisation is what makes the preservation-along-deformation argument work: the Hadamard property is a wave-front-set condition, and wave-front-sets propagate smoothly under smoothly-varying real-principal-type operators. The CCR-commutator-condition lower bound on the wave-front-set of $ω_{2}$ , also from 13.09.03, gives the equality (not just inclusion) at every stage of the deformation.
Propagation of singularities along Hamiltonian flow 02.14.03 is the microlocal-analysis engine of the preservation step. Hörmander's theorem applied to the deformed Klein-Gordon equation $P_{s} ω_{2, s} = 0$ in each argument forces the wave-front set of $ω_{2, s}$ to remain in the bicharacteristic variety of $g_{s}$ and invariant under its bicharacteristic flow. The smooth $s$ -dependence of $g_{s}$ propagates to smooth $s$ -dependence of the bicharacteristic flow and the future-pointing-light-cone condition, giving the smooth-deformation argument that preserves the Hadamard structure.
CCR algebra, Weyl algebra, and quasi-free states 12.14.01 is the algebraic-QFT framework in which the FNW existence theorem lives. The quasi-free states on the CCR algebra of the Klein-Gordon field on $(M, g)$ form a convex set; the Hadamard quasi-free states are a non-empty subclass (by FNW), and every two Hadamard states differ by a smooth correction. The Araki-Woods 1963 GNS construction realises the FNW pull-back state as a Fock-space representation, with the ultrastatic ground state corresponding to the bosonic Fock vacuum on the one-particle Hilbert space $L^{2} (Σ, d vol_{h_{0}})$ with positive-frequency one-particle Hamiltonian $D_{h_{0}}$ .
Wick polynomials in curved spacetime via Hadamard parametrix subtraction [13.09.06, pending] is the immediate downstream renormalisation-theory application. Brunetti-Fredenhagen-Köhler 1996 and Hollands-Wald 2001 define Wick polynomials $: ϕ^{n} (x) :_{H}$ on a curved background by subtracting the Hadamard parametrix from a normal-ordered combination of two-point functions; the construction requires a Hadamard state to subtract against. The FNW existence theorem supplies that state on every globally hyperbolic spacetime, making the Wick-polynomial construction well-defined.
Hadamard states by pseudo-differential calculus (Gérard-Wrochna) [13.09.05, pending] is the modern refinement of the FNW existence theorem. Gérard-Wrochna 2014 / 2017 construct Hadamard states directly from a pseudo-differential approximation $ϵ$ of $D_{h}$ on a Cauchy surface, without using the deformation argument. The construction gives explicit Hadamard-form initial data — a computational refinement of FNW that is more useful for concrete calculations (cosmological perturbations, black-hole radiation).
Black holes and Hawking radiation [13.07.02, pending] is the canonical physical application. The Hartle-Hawking state on the Kruskal extension of Schwarzschild is constructed by a Killing-horizon-respecting refinement of the FNW argument: instead of an arbitrary ultrastatic reference, the construction uses the bifurcate-Killing-horizon structure as the symmetry-respecting reference. Kay-Wald 1991 ^{[Kay-Wald 1991]} proved uniqueness of the resulting Hartle-Hawking state. The thermal spectrum of outgoing modes at the Hawking temperature is read off from the analytic structure of the two-point function.
FLRW cosmology 13.08.01 is the second canonical physical application. The Bunch-Davies state on de Sitter (Bunch-Davies 1978; Allen 1985) is the unique de-Sitter-invariant Hadamard state — constructed by a de-Sitter-symmetry-respecting refinement of the FNW argument with the FLRW metric supplying the natural ultrastatic-conformal reference. Adiabatic vacua on general FLRW are constructed similarly via the Lüders-Roberts 1990 / Junker 1996 adiabatic-vacuum approximation. The primordial-perturbation power spectrum of inflation is computed from the Hadamard two-point function in the Bunch-Davies state.
Locally covariant QFT (Brunetti-Fredenhagen-Verch 2003) is the categorical framework in which the FNW existence theorem becomes a non-emptiness statement on the subfunctor of Hadamard states. Sahlmann-Verch 2001 ^{[Sahlmann-Verch 2001]} codified this in the language of natural transformations: the assignment $ghs \mapsto H (ghs)$ is a subfunctor of the locally covariant QFT functor with non-empty values on every object — that non-emptiness is exactly the FNW existence theorem.
Quantum energy inequalities (Fewster) [13.09.11, pending] rest on the Hadamard structure: the worldline-smeared expectation value of the stress-energy tensor in any Hadamard state on a globally hyperbolic spacetime satisfies a state-independent lower bound, by an averaging-over-modes argument that uses the explicit Hadamard parametrix as the singular-subtraction reference. The FNW existence theorem ensures the Hadamard class is non-empty so the inequality has content.

Historical & philosophical context Master

The question of which states on a quantised field on a curved background are physically admissible became pressing in the 1970s, as the Hawking radiation calculation (Hawking 1975 Comm. Math. Phys. 43 199) made clear that the curved-spacetime QFT framework needed a more careful state-selection criterion than the flat-space Lorentz-vacuum analogy. Bryce DeWitt and Robert Brehme in 1960 Ann. Phys. 9 220 ^{[DeWitt-Brehme 1960]} had introduced the Hadamard parametrix into the curved-spacetime stress-energy-renormalisation programme; Robert Wald in 1977 Comm. Math. Phys. 54 1 ^{[Wald 1977]} used this to define the renormalised stress-energy tensor on a curved background by Hadamard subtraction. The need for a class of states on which the subtraction would produce finite physical answers — the Hadamard class — was implicit throughout this work but received its first explicit theoretical treatment in Fulling, Sweeny and Wald 1978 Comm. Math. Phys. 63 257 ^{[Fulling-Sweeny-Wald 1978]}.

The existence question — does the Hadamard class contain any states, on a general globally hyperbolic spacetime? — was answered by Stephen Fulling, Frank Narcowich and Robert Wald in 1981 Ann. Phys. 136 243 ^{[Fulling-Narcowich-Wald 1981]}. The argument was geometric: deform the metric to an ultrastatic reference where the ground state is explicit, then use the resulting state as the reference for a pull-back to the original metric, and verify that the Hadamard short-distance form is preserved under the deformation. The verification was done at the level of the Hadamard expansion of the two-point function (the framework that preceded Radzikowski's microlocal reformulation), and the propagation-of-Hadamard-form-along-deformation step relied on classical-PDE energy estimates rather than the modern microlocal machinery.

The construction was sharpened in the 1990s on two fronts. Wolfgang Junker in 1996 Rev. Math. Phys. 8 1091 ^{[Junker 1996]} introduced the adiabatic-vacuum-approximation programme: rather than deforming the metric, decompose the Cauchy problem mode-by-mode using a smooth-approximation of $D_{h}$ on the Cauchy hypersurface, and construct quasi-free states whose Cauchy-data covariance approximates the would-be exact Hadamard form to arbitrarily high order. Taking the limit of the approximation order to infinity gives a genuine Hadamard state, supplied by an explicit construction rather than an existence-theorem deformation. Junker's framework was the bridge between FNW 1981 and the modern pseudo-differential constructions.

On the other front, Marek Radzikowski's 1996 Comm. Math. Phys. 179 529 ^{[Radzikowski 1996]} microlocal reformulation of the Hadamard condition (the wave-front-set characterisation) made the FNW preservation-along-deformation argument transparent: the wave-front set of the two-point function is constrained by Hörmander's propagation-of-singularities theorem to live on the bicharacteristic flow of the Klein-Gordon principal symbol, and this flow varies smoothly with the metric. The Radzikowski reformulation also enabled the Brunetti-Fredenhagen-Köhler 1996 Comm. Math. Phys. 180 633 ^{[Brunetti-Fredenhagen-Köhler 1996]} extension to a microlocal-spectrum condition on all $n$ -point functions — the foundation of the Hollands-Wald 2001 Comm. Math. Phys. 223 289 ^{[Hollands-Wald 2001]} construction of Wick polynomials and time-ordered products on a curved background.

The locally covariant formulation came with Hanno Sahlmann and Rainer Verch in 2001 Rev. Math. Phys. 13 1203 ^{[Sahlmann-Verch 2001]}, who codified the FNW existence theorem in the functor language of Brunetti-Fredenhagen-Verch 2003 Comm. Math. Phys. 237 31 and extended it to vector-valued fields (Dirac, Maxwell, Yang-Mills). The Sahlmann-Verch theorem makes existence of a Hadamard class a property of the locally covariant QFT functor itself, not of any single spacetime.

The most explicit modern construction is due to Christian Gérard and Michał Wrochna in 2014 Comm. Math. Phys. 325 713 ^{[Gérard-Wrochna 2014]} and 2017 Ann. Henri Poincaré 18 2715 ^{[Gérard-Wrochna 2017]}. The Gérard-Wrochna construction takes pseudo-differential calculus on the Cauchy hypersurface as input and produces Hadamard initial data directly, without any deformation argument. The construction is featured prominently in Gérard's 2019 EMS textbook Microlocal Analysis of Quantum Fields on Curved Spacetimes ^{[Gérard 2019]} Ch. 8 as the modern preferred construction; the FNW deformation appears as the historical / conceptual prototype. The Gérard-Wrochna 2017 refinement to asymptotically static spacetimes gives the Hadamard property of in/out states for scattering theory on a curved background, making rigorous the Bunch-Davies construction on de Sitter and the analogous adiabatic vacua on FLRW cosmology.

Bibliography Master

Foundational papers:

Hawking, S. W., "Particle creation by black holes", Comm. Math. Phys. 43 (1975), 199-220. [The originating Hawking-radiation calculation that motivated the curved-spacetime QFT state-selection programme.]
Wald, R. M., "The back reaction effect in particle creation in curved spacetime", Comm. Math. Phys. 54 (1977), 1-19. [Hadamard-subtraction definition of the renormalised stress-energy tensor on a curved background.]
Fulling, S. A., Sweeny, M. & Wald, R. M., "Singularity structure of the two-point function in quantum field theory in curved spacetime", Comm. Math. Phys. 63 (1978), 257-264. [Universality of the Hadamard short-distance structure for physically admissible states.]

The originating existence theorem:

Fulling, S. A., Narcowich, F. J. & Wald, R. M., "Singularity structure of the two-point function in quantum field theory in curved spacetime. II", Ann. Phys. 136 (1981), 243-272. [The FNW deformation existence theorem; the central reference of the present unit.]

Modern existence constructions:

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. [Adiabatic-vacuum / pseudo-differential existence on general globally hyperbolic spacetimes.]
Sahlmann, H. & Verch, R., "Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime", Rev. Math. Phys. 13 (2001), 1203-1246. [Locally-covariant existence theorem; extension to vector-valued fields.]
Gérard, C. & Wrochna, M., "Construction of Hadamard states by pseudo-differential calculus", Comm. Math. Phys. 325 (2014), 713-755. [Direct pseudo-differential construction; bypasses the 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), 2715-2756. [Refined pseudo-differential construction on asymptotically static spacetimes; in/out-state Hadamard property.]

Microlocal turn:

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 equivalence used in the preservation-along-deformation step.]
Brunetti, R., Fredenhagen, K. & Köhler, M., "The microlocal spectrum condition", Comm. Math. Phys. 180 (1996), 633-652. [Extension of the microlocal-spectrum condition to all $n$ -point functions.]

Locally covariant framework:

Brunetti, R., Fredenhagen, K. & Verch, R., "The generally covariant locality principle — a new paradigm for local quantum field theory", Comm. Math. Phys. 237 (2003), 31-68. [Locally-covariant-functor axiomatisation of curved-spacetime QFT.]
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. [Wick polynomials defined via Hadamard subtraction; downstream application of FNW existence.]

Modern consolidation:

Gérard, C., Microlocal Analysis of Quantum Fields on Curved Spacetimes, ESI Lectures in Mathematics and Physics (EMS, 2019). [Ch. 8 the canonical reference for FNW existence and the Gérard-Wrochna pseudo-differential refinement.]
Bär, C., Ginoux, N. & Pfäffle, F., Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics (EMS, 2007). [Free PDF at arXiv:0806.1036; Ch. 4 the textbook treatment of FNW existence on a globally hyperbolic background.]
Hollands, S. & Wald, R. M., "Quantum fields in curved spacetime", Phys. Rep. 574 (2015), 1-35. [Free preprint at arXiv:1401.2026; modern review of the field with FNW existence and Wick-polynomial framework.]

Algebraic-QFT predecessors:

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. [Hadamard-form definition; uniqueness theorem on bifurcate-Killing-horizon spacetimes — a complementary uniqueness result to FNW existence.]
Wald, R. M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, 1994). [Physicist-side framing of the existence question and the Hadamard programme.]

Cosmological 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. [Bunch-Davies state on de Sitter; symmetry-respecting refinement of FNW-type existence.]
Allen, B., "Vacuum states in de Sitter space", Phys. Rev. D 32 (1985), 3136-3149. [Uniqueness of the de-Sitter-invariant Hadamard state.]
Lüders, C. & Roberts, J. E., "Local quasiequivalence and adiabatic vacuum states", Comm. Math. Phys. 134 (1990), 29-63. [Adiabatic vacua on FLRW spacetimes; antecedent to the Junker 1996 construction.]

Microlocal-analysis machinery:

Hörmander, L., The Analysis of Linear Partial Differential Operators (Springer, 1985). [Vol. III §26 propagation of singularities — the microlocal-analysis engine of the preservation-along-deformation argument.]
Duistermaat, J. J. & Hörmander, L., "Fourier integral operators. II", Acta Math. 128 (1972), 183-269. [Distinguished parametrices for normally hyperbolic operators — input to the wave-front-set computation on the ultrastatic ground state.]

Prerequisites

13.09.01
13.09.02
13.09.03

Tier anchors

beginner: Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (Chicago, 1994), Ch. 4 (existence-of-vacuum question on a curved background, in physicist-side prose)
intermediate: Bär, Ginoux & Pfäffle, *Wave Equations on Lorentzian Manifolds and Quantization* (EMS, 2007), Ch. 4 (existence of Hadamard states via the FNW deformation; free PDF at arXiv:0806.1036); Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 8; Sahlmann and Verch, *Rev. Math. Phys.* 13 (2001) 1203 (locally covariant existence)
master: Fulling, Narcowich & Wald, *Ann. Phys.* 136 (1981) 243 (the originating deformation existence theorem); Junker, *Rev. Math. Phys.* 8 (1996) 1091 (pseudo-differential / adiabatic-vacuum existence on general globally hyperbolic spacetimes); Brunetti, Fredenhagen & Köhler, *Comm. Math. Phys.* 180 (1996) 633 (extension to higher n-point functions); Gérard & Wrochna, *Comm. Math. Phys.* 325 (2014) 713 and *Ann. Henri Poincaré* 18 (2017) 2715 (direct pseudo-differential construction); Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 8

References

Fulling, S. A., Narcowich, F. J. & Wald, R. M. — 'Singularity structure of the two-point function in quantum field theory in curved spacetime. II', Ann. Phys. 136 (1981) 243 · §3-§4 (deformation argument from an ultrastatic ground state); §5 (preservation of the Hadamard form along the deformation); the originating existence theorem for Hadamard states on a general globally hyperbolic background
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); the modern existence proof on general globally hyperbolic spacetimes via direct construction
Gérard, C. — Microlocal Analysis of Quantum Fields on Curved Spacetimes (EMS Press, 2019) · Ch. 8 (existence of Hadamard states); §8.1 (FNW deformation argument); §8.2 (pseudo-differential construction); §8.3 (Hadamard-form initial data on a Cauchy surface); the canonical modern textbook entry
Bär, C., Ginoux, N. & Pfäffle, F. — Wave Equations on Lorentzian Manifolds and Quantization (EMS, 2007) · Ch. 4 (CCR algebra and Hadamard states on a globally hyperbolic spacetime); §4.4 (FNW existence theorem with full proof); free PDF at arXiv:0806.1036
Brunetti, R., Fredenhagen, K. & Köhler, M. — 'The microlocal spectrum condition', Comm. Math. Phys. 180 (1996) 633 · §3 (extension of the wave-front-set Hadamard condition to all n-point functions); central to the Wick-polynomial / time-ordered-product programme
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 Hamiltonian); §4 (Hadamard-form two-point function from the approximation); the direct construction that bypasses the 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
Sahlmann, H. & Verch, R. — 'Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime', Rev. Math. Phys. 13 (2001) 1203 · Extension of the Hadamard condition and the FNW existence theorem to vector-valued fields; locally-covariant formulation
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 used in the FNW preservation step
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 · §3 (algebraic-QFT definition of Hadamard quasi-free states); §6 (Hartle-Hawking uniqueness on bifurcate-Killing-horizon spacetimes — application of FNW-type existence)
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); §3.4 (the causal propagator E used in the symplectic-isomorphism step of the FNW pull-back)
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 FNW existence)
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 ≅ ℝ × Σ with g = -β² dt² + h_t); used to set up the deformation
Hörmander, L. — The Analysis of Linear Partial Differential Operators (Springer, 1985) · Vol. III §26 (propagation of singularities along the bicharacteristic flow); the microlocal-analysis engine of the preservation-along-deformation step
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 require a Hadamard state on every globally hyperbolic spacetime, supplied by the FNW existence theorem of the present unit

Estimated time

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