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

Hadamard states via the wave-front-set criterion

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Anchor (Master): Radzikowski, *Comm. Math. Phys.* 179 (1996) 529; Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 7; Brunetti, Fredenhagen & Köhler, *Comm. Math. Phys.* 180 (1996) 633; Hollands and Wald, *Comm. Math. Phys.* 223 (2001) 289

Intuition Beginner

In flat-space quantum field theory there is one canonical vacuum — the unique Lorentz-invariant state on the Minkowski free field, the state whose two-point function is positive-frequency and whose modes have positive energy. On a curved background that uniqueness collapses. There is no global time-translation symmetry, no preferred Killing vector, no canonical definition of "positive frequency." Different observers split the field into creation and annihilation operators along different choices of time coordinate, and what one observer calls the vacuum another observer sees as a thermal bath of particles. This is the source of the Unruh effect and of Hawking radiation: there is no observer-independent answer to "is there a particle here?" on a curved spacetime.

The honest replacement for "the vacuum" on a curved background is therefore not a single state but a class of states sharing the right short-distance behaviour. The class is called the Hadamard states, and the right short-distance behaviour is the one a smooth Lorentzian geometry forces on any quantum field that resembles the Minkowski vacuum near every event.

Two events that are very close in the geometry should see the two-point function blow up the way it blows up on Minkowski near coincident points — like $1/ σ$ where $σ$ is the squared geodesic distance, plus a logarithmic correction, plus a smooth remainder. This is the Hadamard form. States that obey it are physically admissible; states that do not are pathological — they would assign infinite expectation values to physical observables that should be finite, such as the renormalised stress-energy tensor.

The Hadamard form was the working definition through the 1980s. Marek Radzikowski in 1996 rewrote it microlocally. The wave-front set of the two-point function — the position-direction record of where the distribution fails to be smooth — is exactly the set of pairs of points joined by a null geodesic, with the cotangent covector at each point being the parallel-transport of the other along the geodesic, and with the covector at the first point pointing into the future light cone.

This is a compact geometric condition. It picks out the positive-frequency half of the null-bicharacteristic relation of the Klein-Gordon operator, the relation supplied by the wave-front set of the causal propagator $E = E^{-} - E^{+}$ from 13.09.02. A state is Hadamard iff its two-point function lives microlocally on this future-pointing half.

The microlocal reformulation does several things at once. It makes the Hadamard condition coordinate-free, local in spacetime, and stable under perturbations that preserve smoothness. It opens the door to the Brunetti-Fredenhagen-Köhler extension to all $n$ -point functions, which is the technical engine of the Hollands-Wald 2001 construction of Wick polynomials and time-ordered products on a curved background.

It also makes the Fulling-Narcowich-Wald 1981 deformation argument work: starting from an ultrastatic spacetime with its ground state and deforming to a general globally hyperbolic background, the propagation-of-singularities theorem guarantees the Hadamard wave-front set propagates with it. And it gives the geometric mechanism behind Hawking radiation: the Hartle-Hawking state on the Kruskal extension of Schwarzschild is the unique Hadamard state respecting the bifurcate Killing horizon, and its restriction to the exterior region is thermal at the Hawking temperature.

Every concrete construction of a quantum field theory on a curved background — every renormalised stress-energy tensor, every black-hole-radiation calculation, every cosmological-perturbation power spectrum — is anchored in the Hadamard condition. The Radzikowski criterion is the microlocal formulation of that condition; it is the foundational definition of the modern subject.

Visual Beginner

The picture to hold is two spacetime points $x$ and $y$ connected by a null geodesic $γ$ , with a cotangent covector $k_{x}$ at $x$ pointing into the future light cone and a covector $k_{y}$ at $y$ that is the parallel-transport of $k_{x}$ along $γ$ . The wave-front set of a Hadamard two-point function $ω_{2} (x, y)$ lives exactly on the set of such pairs, with the convention $k_{y} \mapsto - k_{y}$ in the cotangent pair $(x, k_{x}; y, - k_{y})$ to encode the antisymmetry between source and target.

The picture has three pieces. The null geodesic $γ$ encodes the bicharacteristic relation of the Klein-Gordon operator $□_{g} + m^{2}$ — its principal symbol is the squared length of the cotangent covector with respect to the metric, and the Hamiltonian flow of this principal symbol traces out null geodesics on $T^{*} M$ . The parallel transport of $k_{x}$ to $k_{y}$ along $γ$ encodes the connecting cotangent action. The future-pointing requirement $k_{x} \in \overline{V}_{x}^{+}$ encodes positive-frequency selection: of the two halves of the bicharacteristic relation (future-pointing and past-pointing), the Hadamard two-point function lives on only the future-pointing half. The past-pointing half is the wave-front set of the conjugate $\overline{ω_{2}}$ , or equivalently of the two-point function with arguments swapped.

For the Minkowski vacuum the picture simplifies to plane waves on the future mass shell paired with plane waves on the past mass shell; for a black-hole background it picks out the null geodesics of the Schwarzschild metric and the future-pointing covectors along them. The geometric mechanism is the same in every case: positive-frequency propagation along the null flow of the metric's principal symbol.

Worked example Beginner

Compute the two-point function of the Minkowski vacuum and verify it satisfies the Radzikowski wave-front-set condition in the simplest case.

Step 1. The free scalar field on Minkowski is built by superposing plane waves of every spatial momentum $k$ , with each plane wave a quantum harmonic oscillator of frequency $ω_{k} = ∣ k ∣^{2} + m^{2}$ . The mode coefficients are creation and annihilation operators $a_{k}^{†}, a_{k}$ obeying the standard commutation relation $[a_{k}, a_{k^{'}}^{†}] = δ^{(3)} (k - k^{'})$ .

Step 2. The Minkowski vacuum is the state annihilated by every $a_{k}$ . Its two-point function is built by Fourier inversion: the inverse Fourier transform of the spectral density $(2 π)^{- 3} (2 ω_{k})^{- 1}$ supported on the upper sheet of the mass shell $k^{0} = + ω_{k} > 0$ in four-momentum space. The support on the upper sheet — not the lower one — is the positive-frequency selection that defines the vacuum.

Step 3. Compute the wave-front set. By definition the wave-front set of $ω_{2}^{vac}$ is the conic set of position-direction pairs at which the Fourier transform of a localisation fails to decay rapidly. The Fourier representation in step 2 shows directly: $ω_{2}^{vac}$ is the inverse Fourier transform in $x - y$ of the distribution $(2 π)^{- 3} (2 ω_{k})^{- 1} δ (k^{0} - ω_{k}) θ (k^{0})$ on momentum space.

The wave-front set of an oscillatory integral with such a phase localises on the support of the momentum-space density times the bicharacteristic relation of the phase. The phase has stationary points where the gradient with respect to $k$ vanishes, which never happens on a non-degenerate mass shell — so the WF analysis gives directly $$ \mathrm{WF}(\omega_2^{\mathrm{vac}}) = \big{(x, k; y, -k) : k^2 + m^2 = 0,\ k^0 > 0\big}, $$ with $x - y$ on the light cone (for $m = 0$ ) or in its closure (for $m > 0$ ).

The covectors at the two points are equal and future-pointing: $k$ at $x$ and $- k$ at $y$ . Parallel transport in flat spacetime is the identity map, so this is exactly the Radzikowski form for Minkowski.

Step 4. Check the bicharacteristic structure. In flat spacetime two points $x, y$ are bicharacteristically related to a covector $k$ iff $y - x$ is parallel to $g^{μν} k_{ν}$ — that is, iff $y$ lies on the null geodesic through $x$ with cotangent $k$ (or on the timelike line through $x$ if $m > 0$ ). For the massless case this is exactly the lightlike-separation condition. The wave-front set from step 3 sits on precisely these pairs with $k^{0} > 0$ — the future-pointing half of the null bicharacteristic relation. The Radzikowski criterion holds.

Step 5. The Hadamard parametrix in this case. The short-distance behaviour of $ω_{2}^{vac}$ for $∣ x - y ∣$ small is, by an explicit integral (Wightman 1956 / Bär-Ginoux-Pfäffle 2007), $$ \omega_2^{\mathrm{vac}}(x, y) = \frac{1}{4\pi^2},\frac{1}{(x - y)^2 + i\epsilon (x^0 - y^0)} + \text{smooth remainder for } m = 0, $$ where $(x - y)^{2} = - (x^{0} - y^{0})^{2} + ∣ x - y ∣^{2}$ is the Minkowski squared interval and $ϵ \to 0^{+}$ enforces the future-pointing positive-frequency boundary condition.

The leading singularity is $1/ ((x - y)^{2} + i ϵ)$ — exactly the Minkowski case of the Hadamard parametrix $H (x, y) = (8 π^{2})^{- 1} (U / σ_{ϵ})$ with $U \equiv 1$ in flat space and $σ = \frac{1}{2} (x - y)^{2}$ Synge's world function. For $m > 0$ a logarithmic term $V lo g σ$ appears, with $V$ determined by a transport equation along null geodesics.

What this tells us: the Minkowski vacuum two-point function is Hadamard in the original Hadamard-form sense (step 5) and in the modern Radzikowski wave-front-set sense (steps 3-4), and the two are equivalent. Every Hadamard state on a curved background is — locally — a deformation of this Minkowski picture. The leading $1/ σ$ singularity and the future-pointing positive-frequency selection are the two pieces of structure the curved-spacetime version preserves; the smooth remainder $W$ encodes the curvature-dependent state-dependent freedom.

Check your understanding Beginner

Exercise (easy, multiple choice).

A state $ω$ on the CCR algebra of the Klein-Gordon field on a globally hyperbolic spacetime is called Hadamard if

A. Its two-point function $ω_{2}$ is a smooth function on $M \times M$ B. Its two-point function $ω_{2}$ has the same singularity structure as the Hadamard parametrix near coincident points, up to a smooth remainder C. Its two-point function $ω_{2}$ vanishes outside the diagonal D. Its two-point function $ω_{2}$ is identically zero

Hint

Recall that "Hadamard form" refers to a specific short-distance singularity structure built from Synge's world function $σ$ , not to smoothness or vanishing.

Answer

B. Its two-point function $ω_{2}$ has the same singularity structure as the Hadamard parametrix near coincident points, up to a smooth remainder.

The Hadamard parametrix $H (x, y) = (8 π^{2})^{- 1} (U (x, y) / σ_{ϵ} (x, y)) + V (x, y) lo g σ_{ϵ} (x, y) + W (x, y)$ — with Synge's world function $σ$ and curvature-determined coefficients $U, V$ — captures the universal short-distance behaviour of any physically admissible two-point function on a curved background. A Hadamard state is one whose two-point function differs from $H$ by a smooth function: $ω_{2} - H \in C^{\infty}$ on a neighbourhood of the diagonal.

Option A is wrong because $ω_{2}$ is a genuine distribution with a $1/ σ$ -type singularity on the light cone; smoothness would mean the state has no short-distance physics. Option C is the opposite of correct: the two-point function is supported on $M \times M$ with the singularity concentrated near the light cone. Option D is the zero state, which fails positivity.

Exercise (easy, true-false).

True or false: the Radzikowski 1996 wave-front-set characterisation says that a state is Hadamard iff the wave-front set of its two-point function is contained in the past-pointing half of the null-bicharacteristic relation.

Hint

Read the criterion carefully. Which half of the bicharacteristic relation — future-pointing or past-pointing — does the cotangent covector at the first argument live in?

Answer

False.

The Radzikowski criterion says the wave-front set lives on the future-pointing half, not the past-pointing half: $WF (ω_{2}) = {(x, k; y, - k^{'}) : (x, k) \sim_{bichar} (y, k^{'}), k \in \overline{V}_{x}^{+}}$ , with $k$ in the closed future light cone $\overline{V}_{x}^{+} \subset T_{x}^{*} M$ . This is the positive-frequency selection — the modern microlocal restatement of the flat-space positive-energy condition that fixes the Minkowski vacuum. The past-pointing half is the wave-front set of the conjugate or argument-swapped two-point function $ω_{2} (y, x) = \overline{ω_{2} (x, y)}$ .

Exercise (medium, multiple choice).

Which 1996 theorem established the equivalence between the Kay-Wald 1991 Hadamard-form definition of an admissible quasi-free state and the wave-front-set criterion?

A. The Bernal-Sánchez splitting theorem B. The Radzikowski microlocal-spectrum theorem C. The Stone-von Neumann uniqueness theorem D. Hörmander's product theorem for wave-front sets

Hint

The author's name is also the standard short label for the theorem; the journal is Comm. Math. Phys. 179.

Answer

B. The Radzikowski microlocal-spectrum theorem.

Marek Radzikowski in Comm. Math. Phys. 179 (1996) 529 proved that for a quasi-free state on the CCR algebra of the Klein-Gordon field on a globally hyperbolic spacetime, the Hadamard-form condition on $ω_{2}$ (Kay-Wald 1991) is equivalent to the microlocal-spectrum condition $WF (ω_{2}) = {(x, k; y, - k^{'}) : (x, k) \sim_{bichar} (y, k^{'}), k \in \overline{V}_{x}^{+}}$ .

Option A (Bernal-Sánchez 2005) gives the smooth-splitting refinement of Geroch's theorem for globally hyperbolic spacetimes 13.09.01, unrelated to Hadamard states. Option C (Stone-von Neumann) is the uniqueness of the Weyl-form representation of the CCR on finitely many degrees of freedom, unrelated to the Hadamard condition. Option D (Hörmander's product theorem) is the wave-front-set-sumset rule for distributional products, a microlocal-analysis tool used inside Radzikowski's proof but not the equivalence statement itself.

Exercise (medium, numeric).

The Hadamard parametrix $H (x, y) = (8 π^{2})^{- 1} (U (x, y) / σ_{ϵ} (x, y)) + V (x, y) lo g σ_{ϵ} (x, y) + W (x, y)$ has a leading singular piece $U / σ_{ϵ}$ . In Minkowski spacetime with the massless Klein-Gordon equation, the coefficient $U (x, y)$ takes a constant value. What is that value?

Hint

The Hadamard coefficient $U (x, y)$ is the square root of the Van Vleck-Morette determinant. In flat spacetime the Van Vleck-Morette determinant equals 1.

Answer

$U (x, y) = 1$ .

In flat spacetime the Van Vleck-Morette determinant — which measures the volume distortion of the geodesic exponential map — is identically 1, since the exponential map is the identity (geodesics from $x$ are straight lines). The leading Hadamard coefficient $U (x, y) = det Δ (x, y)$ (with $Δ$ the Van Vleck-Morette determinant) is therefore 1 on Minkowski. The leading parametrix term reduces to $(8 π^{2})^{- 1} / σ_{ϵ} (x, y) = (4 π^{2})^{- 1} / ((x - y)^{2} + i ϵ (x^{0} - y^{0}))$ , matching the leading $1/ (x - y)^{2}$ singularity of the Minkowski vacuum two-point function from the worked example. On a curved background $U (x, y)$ deviates from 1 to first order in the curvature; the deviation encodes how Riemannian normal coordinates fail to be Cartesian.

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 $g = - β^{2} d t^{2} + h_{t}$ . The Klein-Gordon operator $P = □_{g} + m^{2} + V$ acts on smooth scalar fields with $m \geq 0$ and a smooth potential $V$ . The advanced and retarded Green's operators $E^{\pm}$ and the causal propagator $E = E^{-} - E^{+}$ from 13.09.02 supply the symplectic phase space of the quantised field. The CCR algebra $A (M, g)$ and its quasi-free states are constructed as in 12.14.01.

A quasi-free state $ω$ on $A (M, g)$ is, by definition, a state determined entirely by its two-point function $ω_{2} \in D^{'} (M \times M)$ via the Wick rule $ω (ϕ (f_{1}) \dots ϕ (f_{2 n})) = \sum_{π} \prod_{(i, j) \in π} ω_{2} (f_{i}, f_{j})$ , summed over pair partitions, with odd $n$ -point functions vanishing. Positivity of the state translates into a positivity condition on $ω_{2}$ , and compatibility with the CCR commutator $[ϕ (f), ϕ (g)] = i E (f, g) 1$ translates into the antisymmetric-imaginary-part condition $ω_{2} (x, y) - ω_{2} (y, x) = i E (x, y)$ as distributions on $M \times M$ .

Define Synge's world function $σ : U \to R$ on a normal-convex neighbourhood $U$ of the diagonal in $M \times M$ by $$ \sigma(x, y) := \tfrac{1}{2}, \big(\text{signed squared geodesic length of the unique geodesic from } x \text{ to } y\big). $$ $σ > 0$ for spacelike-separated $x, y$ ; $σ < 0$ for timelike-separated; $σ = 0$ on the light cone. The Van Vleck-Morette determinant $Δ (x, y) := ∣ g (x) ∣^{- 1/2} ∣ g (y) ∣^{- 1/2} det (- \partial_{α}^{x} \partial_{β}^{y} σ (x, y))$ measures the focusing / defocusing of geodesics near the diagonal; in flat spacetime $Δ \equiv 1$ . Both $σ$ and $Δ$ are smooth on $U$ off the diagonal and extend smoothly to the diagonal with $σ ∣_{Δ} = 0$ , $Δ ∣_{Δ} = 1$ .

The Hadamard coefficients $U, V \in C^{\infty} (U)$ are determined by transport equations along null geodesics emanating from $x$ . The coefficient $U$ is fixed by $U (x, y) = Δ (x, y)$ , equivalently by the transport equation $2 σ^{; μ} U_{; μ} + (□_{g} σ - 4) U = 0$ along null geodesics. The coefficient $V$ is determined by a recursive sequence of transport equations $V_{n}$ with $V = \sum_{n} V_{n} σ^{n}$ as a formal series, where the $V_{n}$ satisfy $$ 2(n+1)\sigma^{;\mu}V_{n+1;\mu} + ((n+1)\Box_g\sigma - 4(n+1) + (n+1)(n+2)), V_{n+1} + (\Box_g + m^2 + V_{\mathrm{pot}})V_n = 0, $$ with $V_{0}$ fixed by $V_{0} ∣_{Δ} = - \frac{1}{2} (□_{g} + m^{2} + V_{pot}) U ∣_{Δ}$ (cf. Bär-Ginoux-Pfäffle 2007 ^{[Bär-Ginoux-Pfäffle 2007]} §2; Friedlander 1975 ^{[Friedlander 1975]} Ch. 5; Hadamard 1923 ^{[Hadamard 1923]}). The construction depends only on the metric $g$ and the operator $P$ , not on any choice of state.

The Hadamard parametrix at order $N$ is $$ H_N(x, y) := \frac{1}{8\pi^2},\bigg(\frac{U(x, y)}{\sigma_\epsilon(x, y)}\bigg) + \frac{1}{8\pi^2},\bigg(\sum_{n=0}^N V_n(x, y) \sigma_\epsilon(x, y)^n\bigg)\log\sigma_\epsilon(x, y) + W_N(x, y), $$ where $σ_{ϵ} (x, y) := σ (x, y) + i ϵ (T (x) - T (y)) + ϵ^{2}$ with $T$ a smooth global time function (the $i ϵ$ prescription selects the future-pointing positive-frequency boundary condition), and $W_{N} \in C^{N} (U)$ is a remainder term whose detailed form is state-dependent. The parametrix is universal in the sense that it depends only on $(M, g, P)$ and the integer $N$ — not on the quantum state.

Definition (Kay-Wald 1991 Hadamard state). A quasi-free state $ω$ on the CCR algebra of the Klein-Gordon field on $(M, g)$ is Hadamard if, for every $N \in N$ , the difference $ω_{2} - H_{N}$ extends to a $C^{N}$ -function on a neighbourhood of the diagonal in $M \times M$ ^{[Kay-Wald 1991]}.

Equivalently, the singular structure of $ω_{2}$ near coincident points matches the parametrix $H$ to arbitrarily high order — the state-dependent and the universal pieces decouple. The Kay-Wald definition was the operative one through the 1980s; it has the disadvantages of being awkwardly nonlocal (it asks for behaviour on neighbourhoods of the diagonal) and difficult to manipulate under operations such as restriction to subregions or pullback by isometries.

The microlocal reformulation is the central definition of the modern subject:

Definition (Radzikowski 1996 microlocal-spectrum condition). A quasi-free state $ω$ on the CCR algebra of the Klein-Gordon field on $(M, g)$ satisfies the microlocal-spectrum condition (or is microlocally Hadamard) if the wave-front set of its two-point function is $$ \mathrm{WF}(\omega_2) = \big{((x, k_x), (y, -k_y)) \in T^(M \times M) \setminus 0 :\ (x, k_x) \sim_{\mathrm{bichar}} (y, k_y),\ k_x \in \overline{V}^+x \big}, $$ *where $\sim{\mathrm{bichar}} $i s t h e n u l l - bi c ha r a c t er i s t i cr e l a t i o n (t h e p o in t s$ x $an d$ y $l i eo na co mm o nn u l l g eo d es i co f$ g $, an d$ k_y $i s t h e p a r a l l e l t r an s p or t o f$ k_x $a l o n g t ha t g eo d es i c) an d$ \overline{V}^+_x \subset T^*_x M $i s t h ec l ose df u t u r e l i g h t co n e in t h eco t an g e n t s p a ce a t$ x$ ^{[Radzikowski 1996]}.

The right-hand side is the future-pointing half of the null-bicharacteristic relation of $P = □_{g} + m^{2}$ on $T^{*} M ∖ 0$ . The wave-front set of the causal propagator $E$ from 13.09.02 is the entire null-bicharacteristic relation (both future- and past-pointing halves); the Hadamard state selects one half by the positive-frequency condition $k_{x} \in \overline{V}_{x}^{+}$ .

Counterexamples to common slips

The wave-front-set condition is an equality, not just an inclusion. Both inclusions require work: $\subseteq$ is the smoothness-of-state-dependent-part statement (no extra singularities beyond the Hadamard parametrix), and $\supseteq$ is the canonical-commutation-relation statement (the imaginary part of $ω_{2}$ is $E /2$ which has the full bicharacteristic wave-front set).
The future-pointing condition $k_{x} \in \overline{V}_{x}^{+}$ is what distinguishes the Hadamard state from its time-reverse. The two-point function of the time-reversed state has wave-front set on the past-pointing half. The original Wightman positivity condition on Minkowski — the spectrum condition that the support of the Fourier transform of $ω_{2}$ in $x - y$ lies in the closed forward light cone — is exactly the flat-space form of this future-pointing condition.
The Kay-Wald definition asks for $ω_{2} - H_{N}$ to extend $C^{N}$ for every $N$ , not for one specific $N$ . The reason is that the Hadamard parametrix $H_{N}$ is itself only defined modulo $C^{N}$ , and improving the order of the parametrix improves the regularity of the remainder. The infinite-order statement "for all $N$ " extracts an essentially unique characterisation of the smooth state-dependent piece.
The Hadamard form is local in the sense that it asks for behaviour on a neighbourhood of every point of the diagonal, but globally many such states exist. The Hadamard condition is not a uniqueness condition — it carves out a class, not a single state. Distinct Hadamard states differ by a smooth function $ω_{2}^{(1)} - ω_{2}^{(2)} \in C^{\infty} (M \times M)$ , which is enough to make all renormalised observables differ by smooth corrections.
The Radzikowski criterion is stated for quasi-free states. For general (non-quasi-free) states one demands the BFK 1996 microlocal-spectrum condition on all $n$ -point functions ^{[Brunetti-Fredenhagen-Köhler 1996]}, which is a strictly stronger condition. Quasi-free Hadamard suffices to determine all $n$ -point functions by Wick's rule and is the practical criterion for free-field constructions.

Key derivation Intermediate+

Theorem (Radzikowski 1996). Let $(M, g)$ be a globally hyperbolic Lorentzian manifold and let $ω$ be a quasi-free state on the CCR algebra of the Klein-Gordon field with operator $P = □_{g} + m^{2} + V$ . The following are equivalent:

(1) $ω$ is Hadamard in the Kay-Wald 1991 sense: for every $N$ , $ω_{2} - H_{N} \in C^{N}$ on a neighbourhood of the diagonal in $M \times M$ .

(2) $ω$ satisfies the microlocal-spectrum condition $$ \mathrm{WF}(\omega_2) = \big{((x, k_x), (y, -k_y)) : (x, k_x) \sim_{\mathrm{bichar}} (y, k_y),\ k_x \in \overline{V}^+_x\big}. $$

The two-direction equivalence is the central technical theorem of curved-spacetime QFT. The proof has two ingredients: a Duistermaat-Hörmander parametrix theorem ^{[Duistermaat-Hörmander 1972]} computing the wave-front set of the Hadamard parametrix $H$ , and the propagation-of-singularities theorem 02.14.03 controlling how the wave-front set of the state-dependent remainder behaves under the Klein-Gordon operator.

Proof of $(1) \Rightarrow (2)$ .

Step 1: wave-front set of the Hadamard parametrix. By the Duistermaat-Hörmander 1972 theory ^{[Duistermaat-Hörmander 1972]} of distinguished parametrices for normally hyperbolic operators on a Lorentzian manifold, the Hadamard parametrix $H$ with the $i ϵ$ prescription $σ_{ϵ} = σ + i ϵ (T (x) - T (y)) + ϵ^{2}$ has wave-front set $$ \mathrm{WF}(H) = \big{((x, k_x), (y, -k_y)) : (x, k_x) \sim_{\mathrm{bichar}} (y, k_y),\ k_x \in \overline{V}^+_x\big}, $$ the future-pointing half of the bicharacteristic relation. The argument is a stationary-phase calculation on the oscillatory-integral representation $H = (8 π^{2})^{- 1} \int e^{i ξ \cdot σ} (U + V lo g ∣ ξ ∣ + \dots) d ξ$ with the $i ϵ$ prescription selecting the upper half-plane in the contour. The choice of $T$ as a smooth time function with $T (x) - T (y) > 0$ for $x \in I^{+} (y)$ encodes the future-pointing condition on $k_{x}$ .

Step 2: state-dependent remainder is smooth. By hypothesis (1), $ω_{2} - H_{N} \in C^{N}$ for every $N$ , hence $ω_{2} - H \in C^{\infty}$ on a neighbourhood of the diagonal in the appropriate sense (taking $N \to \infty$ ). A smooth function has empty wave-front set: $WF (ω_{2} - H) = \emptyset$ near the diagonal. Off the diagonal $ω_{2}$ is determined by the Klein-Gordon equation $P_{x} ω_{2} (x, y) = 0$ (acting on the first argument; the analogous equation $P_{y}$ holds for the second), and propagation of singularities along the bicharacteristic flow extends the near-diagonal wave-front-set statement to all of $M \times M$ outside the diagonal.

Step 3: combine. By the additivity property of the wave-front set under sums of distributions, $WF (ω_{2}) = WF (H + (ω_{2} - H)) \subseteq WF (H) \cup WF (ω_{2} - H) = WF (H)$ . The reverse inclusion $WF (ω_{2}) \supseteq WF (H)$ follows from the canonical-commutator condition $ω_{2} (x, y) - ω_{2} (y, x) = i E (x, y)$ : the imaginary part of $ω_{2}$ is $E /2$ up to the symmetric-real-part rearrangement, and the wave-front set of $E$ from 13.09.02 is the full bicharacteristic relation. Selecting the future-pointing half by the positive-frequency condition forces $WF (ω_{2})$ to contain that half exactly. Equality (2) follows.

Proof of $(2) \Rightarrow (1)$ . Suppose $ω$ satisfies the microlocal-spectrum condition (2). Define the difference $D := ω_{2} - H$ on a normal-convex neighbourhood $U$ of the diagonal. By step 1 above, $WF (H)$ is the future-pointing half of the bicharacteristic relation; by (2), $WF (ω_{2})$ is the same set. Therefore $WF (D) \subseteq WF (ω_{2}) \cup WF (H) =$ future-pointing half. But the same argument applied to $D^{'} := H - ω_{2}$ (which differs from $D$ only by a sign) gives $WF (D^{'}) \subseteq$ future-pointing half, and $WF (- D) = WF (D)$ by the sign-invariance of the wave-front set, so $WF (D) \subseteq$ future-pointing half from both directions — consistent but not yet smoothness.

The full smoothness $D \in C^{\infty}$ comes from the equation $P_{x} D = P_{x} ω_{2} - P_{x} H$ . The first term vanishes since $ω_{2}$ satisfies the Klein-Gordon equation in its first argument. The second term $P_{x} H$ is, by the construction of the Hadamard parametrix, smooth: the parametrix is defined precisely so that $P_{x} H \in C^{\infty}$ , with the smoothness gain coming from the recursive determination of the $V_{n}$ coefficients. Therefore $P_{x} D \in C^{\infty}$ , hence $WF (P_{x} D) = \emptyset$ . By the propagation-of-singularities theorem 02.14.03 applied to $P_{x}$ , $WF (D)$ is invariant under the bicharacteristic flow on $T^{*} M ∖ 0$ in its first argument, and the same with $P_{y}$ for the second argument. Combining: $WF (D)$ must be invariant under bicharacteristic flow in both arguments simultaneously, and any such invariant set inside the future-pointing half is either empty or the full half. If non-empty, the bicharacteristic invariance plus the Klein-Gordon equation forces $D$ to contain a non-zero solution component, contradicting the assumed compatibility with the CCR commutator. So $WF (D) = \emptyset$ , hence $D = ω_{2} - H \in C^{\infty}$ near the diagonal. The full Kay-Wald statement $ω_{2} - H_{N} \in C^{N}$ for every $N$ follows by the order-by-order convergence of the parametrix construction.

$■$

Bridge. The Radzikowski theorem builds toward 13.09.04, where the existence of Hadamard states on every globally hyperbolic spacetime is established by the Fulling-Narcowich-Wald 1981 deformation argument: start from the ground state of the Klein-Gordon field on an ultrastatic spacetime — for which the Hadamard property is direct from the explicit mode decomposition — and deform the metric along a one-parameter family of globally hyperbolic spacetimes. Propagation of singularities along the deformation guarantees the wave-front-set condition is preserved, so the deformed state remains Hadamard. Appears again in 13.09.06, where Wick polynomials are defined on a curved background by Hadamard subtraction $: ϕ^{n} (x) :_{H} := lim_{y \to x} (ω_{2} (x, y) - H (x, y))^{n /2}$ — the universality of $H$ makes this independent of the chosen Hadamard state up to smooth corrections, the foundation of the Hollands-Wald 2001 ^{[Hollands-Wald 2001]} perturbative-QFT framework. The pattern — a state-dependent quantity is finite after subtraction of a universal short-distance parametrix — generalises through the BFK 1996 microlocal-spectrum condition on $n$ -point functions to the full algebraic-QFT-on-curved-spacetimes programme of Brunetti, Dütsch, Fredenhagen, and collaborators, where every locally covariant quantum-field-theory functor takes a globally hyperbolic spacetime to a $C^{*}$ -algebra equipped with a distinguished class of Hadamard states.

Exercises Intermediate+

Exercise 1 (easy, symbolic).

Write out the leading singular term of the Hadamard parametrix $H (x, y)$ in the massless case in Minkowski spacetime $R^{1, 3}$ , identifying $U (x, y) = 1$ and $V \equiv 0$ .

Hint

Synge's world function in Minkowski is $σ (x, y) = \frac{1}{2} ((x^{0} - y^{0})^{2} - ∣ x - y ∣^{2})$ in mostly-minus convention, or $\frac{1}{2} ((x - y)^{2}) = \frac{1}{2} (- (x^{0} - y^{0})^{2} + ∣ x - y ∣^{2})$ in mostly-plus. Substitute into the parametrix formula with $U = 1$ and $V_{n} = 0$ .

Answer

In mostly-plus signature the massless Minkowski Hadamard parametrix is $$ H(x, y) = \frac{1}{8\pi^2},\frac{1}{\sigma_\epsilon(x, y)} + W(x, y), \qquad \sigma(x, y) = \tfrac{1}{2}(x - y)^2 = -\tfrac{1}{2}(x^0 - y^0)^2 + \tfrac{1}{2}|\mathbf{x} - \mathbf{y}|^2. $$ The logarithmic term vanishes because $V \equiv 0$ in the massless case in flat spacetime (the recursive transport equations give $V_{0} = - \frac{1}{2} (□ + 0) U ∣_{Δ} = 0$ and similarly $V_{n} = 0$ for all $n$ , since $□ U = 0$ in flat space with $U \equiv 1$ ). The leading term $1/ (8 π^{2} σ_{ϵ})$ reproduces (up to convention-dependent normalisation factors) the $1/ (x - y)^{2}$ singularity of the Minkowski vacuum two-point function from the worked example. The $i ϵ$ prescription $σ_{ϵ} = σ + i ϵ (x^{0} - y^{0}) + ϵ^{2}$ selects the future-pointing positive-frequency boundary condition.

Exercise 2 (easy, multiple choice).

The Hadamard coefficient $V$ in the parametrix is determined by

A. A boundary condition on $Σ$ B. A choice of state C. A recursive sequence of transport equations along null geodesics emanating from $x$ D. The cosmological constant

Hint

$V$ is a universal quantity, depending only on the spacetime metric $g$ and the Klein-Gordon operator $P$ , not on any state choice.

Answer

C. A recursive sequence of transport equations along null geodesics emanating from $x$ .

The Hadamard coefficient $V (x, y) = \sum_{n} V_{n} (x, y) σ (x, y)^{n}$ is determined by a recursive sequence of ordinary differential equations (transport equations) along null geodesics from $x$ to $y$ . The base case $V_{0}$ is fixed by $V_{0} ∣_{Δ} = - \frac{1}{2} (P U) ∣_{Δ}$ and the higher $V_{n + 1}$ by the recursion involving $□_{g} σ$ , $V_{n}$ , and the operator $P = □_{g} + m^{2} + V_{pot}$ . The construction depends only on the metric and the operator, not on any quantum state — this is what makes the parametrix universal. Option A would be the case for a unique-Cauchy-data construction, not the parametrix. Option B confuses the universal parametrix with the state-dependent remainder $W$ . Option D introduces extra physics irrelevant to the construction.

Exercise 3 (medium, symbolic).

Show that if two quasi-free states $ω, ω^{'}$ on the same CCR algebra are both Hadamard, then their two-point functions differ by a smooth function: $ω_{2} - ω_{2}^{'} \in C^{\infty} (M \times M)$ .

Hint

Use the Kay-Wald definition (1) and the universality of $H$ .

Answer

By the Kay-Wald Hadamard definition, $ω_{2} - H_{N} \in C^{N}$ and $ω_{2}^{'} - H_{N} \in C^{N}$ for every $N$ , near the diagonal. Subtracting: $$ \omega_2 - \omega_2' = (\omega_2 - H_N) - (\omega_2' - H_N) \in C^N $$ for every $N$ , near the diagonal — hence $ω_{2} - ω_{2}^{'} \in C^{\infty}$ on a neighbourhood of the diagonal. Off the diagonal both $ω_{2}$ and $ω_{2}^{'}$ are determined by the Klein-Gordon equation in each argument, so the difference also solves the homogeneous equation in both arguments and inherits smoothness from the Cauchy data on a transverse Cauchy hypersurface. By the elliptic-regularity / propagation-of-singularities argument, $ω_{2} - ω_{2}^{'} \in C^{\infty} (M \times M)$ globally.

The fact that two Hadamard states differ by a smooth function — and only a smooth function — is what makes the Hadamard condition the right invariant for renormalisation: every renormalised observable, defined by subtracting a universal short-distance singular piece, depends on the choice of Hadamard state only through a smooth (finite) correction. The choice of Hadamard state is therefore physically meaningful but does not introduce ultraviolet ambiguities.

Exercise 4 (medium, symbolic).

Verify that the Minkowski vacuum two-point function $ω_{2}^{vac} (x, y) = \int (2 π)^{- 3} (2 ω_{k})^{- 1} e^{- ik \cdot (x - y)} d^{3} k$ for the massless Klein-Gordon field satisfies the future-pointing condition $k_{x} \in \overline{V}_{x}^{+}$ in the Radzikowski criterion.

Hint

Read off the Fourier-transform-space support. The integral is over the upper sheet of the mass shell $k^{0} = + ω_{k}$ .

Answer

The two-point function is the inverse Fourier transform in $(x - y)$ of the measure $(2 π)^{- 3} (2 ω_{k})^{- 1} δ (k^{0} - ω_{k}) θ (k^{0})$ supported on the upper mass shell. By the wave-front-set characterisation of an oscillatory integral with a non-stationary phase, the wave-front set is $$ \mathrm{WF}(\omega_2^{\mathrm{vac}}) = {(x, k; y, -k) : k^2 + m^2 = 0,\ k^0 \geq 0}, $$ which sits exactly on the future mass shell at $x$ paired with the past mass shell at $y$ (the $- k$ at $y$ is past-pointing when $k$ at $x$ is future-pointing). The future-pointing condition $k^{0} \geq 0$ at $x$ holds by construction — the integration is over $k^{0} = + ω_{k} > 0$ .

The flat-space bicharacteristic relation reduces to lightlike-separation: $(x, k) \sim_{bichar} (y, k)$ iff $x - y$ is parallel to $g^{μν} k_{ν} = k^{μ}$ (raising indices in mostly-plus signature) and $k$ is null, i.e. iff $x$ and $y$ are connected by a null geodesic with cotangent $k$ . The wave-front set above sits exactly on these pairs with $k^{0} > 0$ . The Radzikowski criterion holds with $k_{x} \in \overline{V}_{x}^{+}$ .

Exercise 5 (medium, multiple choice).

The Hadamard parametrix $H$ has the property $P_{x} H \in C^{\infty} (M \times M)$ where $P = □_{g} + m^{2} + V$ acts on the first argument. What does this property contribute to the proof of the Radzikowski theorem?

A. It makes $H$ smooth everywhere on $M \times M$ B. It guarantees that the state-dependent remainder $ω_{2} - H$ satisfies the homogeneous Klein-Gordon equation modulo smooth corrections, enabling propagation-of-singularities to propagate the diagonal smoothness statement globally C. It identifies $H$ with the Feynman propagator D. It enforces positivity of the state $ω$

Hint

In the proof of $(2) \Rightarrow (1)$ , $P_{x} H \in C^{\infty}$ is what makes $P_{x} D$ smooth when $D = ω_{2} - H$ . Propagation of singularities then takes it from there.

Answer

B. It guarantees that the state-dependent remainder $ω_{2} - H$ satisfies the homogeneous Klein-Gordon equation modulo smooth corrections, enabling propagation-of-singularities to propagate the diagonal smoothness statement globally.

The parametrix $H$ is constructed precisely so that the recursive transport equations for the Hadamard coefficients $V_{n}$ make $P_{x} H$ smooth — the term $P_{x} H$ is exactly the smoothing error that the parametrix construction is designed to make $C^{\infty}$ . In the proof of $(2) \Rightarrow (1)$ , this is used as follows: define $D = ω_{2} - H$ ; compute $P_{x} D = P_{x} ω_{2} - P_{x} H$ ; the first term vanishes (the state $ω_{2}$ solves the homogeneous Klein-Gordon equation), and the second is $C^{\infty}$ by the parametrix property. Therefore $P_{x} D \in C^{\infty}$ , so $WF (P_{x} D) = \emptyset$ . The propagation-of-singularities theorem 02.14.03 for the real-principal-type operator $P_{x}$ then says $WF (D)$ is invariant under bicharacteristic flow on $T^{*} M ∖ 0$ in its first argument. Combined with the future-pointing-half wave-front-set bound on $D$ from (2), this forces $WF (D) = \emptyset$ and hence $D \in C^{\infty}$ .

Option A is wrong: $H$ has the leading $1/ σ$ singularity, so it is not smooth on $M \times M$ . Option C confuses the Hadamard parametrix with the Feynman propagator (they share the leading singularity but the boundary conditions and the global structure differ). Option D is unrelated; positivity of $ω$ is an independent input to the construction.

Exercise 6 (medium, short-answer).

Explain why the future-pointing condition $k_{x} \in \overline{V}_{x}^{+}$ in the Radzikowski criterion is the curved-spacetime analogue of the Wightman spectrum condition on Minkowski spacetime.

Hint

The Wightman spectrum condition says the Fourier transform of the Minkowski vacuum two-point function in $(x - y)$ is supported in the closed forward light cone of momentum space. Translate that condition into the wave-front-set language.

Answer

The Wightman axioms in flat-space QFT include the spectrum condition: the support of $ω_{2} (p)$ , the Fourier transform of the Minkowski vacuum two-point function in $x - y$ , lies in the closed forward light cone $\overline{V}^{+} = {p : p^{2} + m^{2} \leq 0, p^{0} \geq 0}$ in momentum space. This is the positive-energy condition: physical particle states have non-negative energy and lie on or inside the future mass shell.

In Minkowski with translation invariance, the wave-front set of $ω_{2} (x, y) = f (x - y)$ at any point $(x_{0}, y_{0})$ is determined by the directions in which $\hat{f} (p)$ fails to decay rapidly. The support of $\hat{f}$ in $\overline{V}^{+}$ translates directly to: the wave-front-set covector at the first argument is in $\overline{V}^{+}$ (the closed future light cone in cotangent space). On a curved background there is no global Fourier transform, so the spectrum condition cannot be stated globally. The Radzikowski criterion localises the Wightman condition: at every event $x$ , the cotangent covector at $x$ in the wave-front-set must be future-pointing, $k_{x} \in \overline{V}_{x}^{+}$ . The pairing condition $(x, k_{x}) \sim_{bichar} (y, k_{y})$ with $k_{y}$ the parallel transport of $k_{x}$ replaces the Minkowski translation-invariance of the on-shell condition $p^{2} + m^{2} = 0$ by its general-spacetime analogue.

This is why the Radzikowski criterion is sometimes called the microlocal spectrum condition: it is the wave-front-set localisation of the Wightman spectrum condition, generalised to general globally hyperbolic spacetimes where the global Fourier transform does not exist.

Exercise 7 (hard, symbolic).

Show that the imaginary part of the two-point function of any quasi-free state is determined by the causal propagator: $Im ω_{2} (x, y) = - E (x, y) /2$ as distributions on $M \times M$ (up to sign convention). Use this to verify that the wave-front set of $ω_{2}$ must contain the full bicharacteristic relation, even though the state-dependent real part may have smaller wave-front set.

Hint

The CCR commutator condition $[ϕ (f), ϕ (g)] = i E (f, g) 1$ implies $ω_{2} (x, y) - ω_{2} (y, x) = i E (x, y)$ , by taking expectation values in a state $ω$ .

Answer

For a quasi-free state $ω$ , the CCR commutator $[ϕ (f), ϕ (g)] = i E (f, g) 1$ gives $$ \omega(\phi(f)\phi(g)) - \omega(\phi(g)\phi(f)) = i, E(f, g). $$ The left side is $ω_{2} (f, g) - ω_{2} (g, f)$ . So $ω_{2} (x, y) - ω_{2} (y, x) = i E (x, y)$ as distributions on $M \times M$ . Splitting $ω_{2}$ into its symmetric and antisymmetric parts gives $ω_{2}^{antisym} (x, y) = \frac{1}{2} (ω_{2} (x, y) - ω_{2} (y, x)) = \frac{i}{2} E (x, y)$ , hence the imaginary part of $ω_{2}$ is $- E /2$ (the sign depends on the convention for "imaginary part" — different texts give $\pm E /2$ ).

The wave-front-set consequence: by 13.09.02, the wave-front set of the causal propagator $E$ is the full null-bicharacteristic relation of $□_{g} + m^{2}$ , both future- and past-pointing halves: $$ \mathrm{WF}(E) = {(x, k; y, -k') : (x, k) \sim_{\mathrm{bichar}} (y, k')} = \mathrm{WF}^+(E) \cup \mathrm{WF}^-(E), $$ where $WF^{\pm}$ are the future- and past-pointing halves. The decomposition $ω_{2} = ω_{2}^{sym} + ω_{2}^{antisym}$ gives $WF (ω_{2}^{antisym}) = WF (i E /2) = WF (E)$ . By the Radzikowski criterion, $WF (ω_{2})$ is only the future-pointing half — so the symmetric part must cancel the past-pointing half of the antisymmetric part. This forces $WF (ω_{2}^{sym}) \supseteq WF^{-} (E)$ (with appropriate cotangent-orientation flip), and the sum $ω_{2} = ω_{2}^{sym} + ω_{2}^{antisym}$ recombines to give wave-front set exactly the future-pointing half. The cancellation between the past-pointing pieces of the symmetric and antisymmetric parts is what encodes the positive-frequency selection of the Hadamard state.

Exercise 8 (hard, short-answer).

Sketch the Fulling-Narcowich-Wald 1981 deformation argument for the existence of Hadamard states on a general globally hyperbolic spacetime, identifying the role of propagation-of-singularities at the critical step.

Hint

Start from the ground state on an ultrastatic spacetime (where the Hadamard property is direct from the mode decomposition), deform the metric along a smooth one-parameter family of globally hyperbolic metrics, and use propagation-of-singularities to follow the Hadamard wave-front-set structure along the deformation.

Answer

Step 1: ultrastatic ground state. An ultrastatic globally hyperbolic spacetime is one of the form $(M, g) = (R \times Σ, - d t^{2} + h)$ with $h$ a complete Riemannian metric on $Σ$ independent of $t$ . The Klein-Gordon operator $□_{g} + m^{2}$ commutes with $\partial_{t}$ , so its solutions decompose into positive- and negative-frequency modes by spectral resolution of $- Δ_{h} + m^{2}$ on $L^{2} (Σ, d vol_{h})$ . The ground state is the quasi-free state with two-point function built from positive-frequency modes, $$ \omega_2^{\mathrm{ground}}(x, y) = \langle f(x), (2\sqrt{-\Delta_h + m^2})^{-1} e^{-i(t_x - t_y)\sqrt{-\Delta_h + m^2}} f(y)\rangle_{L^2(\Sigma)}. $$ Direct verification: this state has Hadamard wave-front set on the ultrastatic spacetime, by an explicit Fourier analysis in $t$ that puts the positive-frequency support on the future mass shell.

Step 2: deformation. Given any globally hyperbolic spacetime $(M, g)$ with a Cauchy hypersurface $Σ$ , the Bernal-Sánchez splitting gives $M ≅ R \times Σ$ with metric $g = - β^{2} d t^{2} + h_{t}$ . Construct a smooth one-parameter family $g_{s}$ , $s \in [0, 1]$ , of globally hyperbolic metrics on $M$ with $g_{0} =$ ultrastatic (e.g., $g_{0} = - d t^{2} + h_{0}$ ) and $g_{1} = g$ . The family is built so that the spacetime structure outside a compact spacetime region agrees with $g_{0}$ for $s$ near 0 and with $g$ for $s$ near 1.

Step 3: propagate Hadamard property. The ground state of $g_{0}$ is Hadamard by step 1. Pull this state back along the deformation to a state on $g_{1} = g$ via the symplectic isomorphism of Cauchy-data phase spaces. The pulled-back state's two-point function $ω_{2}^{(1)}$ is the result of applying the propagator of $g_{s}$ at each stage to the initial $ω_{2}^{(0)}$ . The propagator $E_{g_{s}}$ varies smoothly in $s$ , and the Hadamard wave-front-set condition is preserved under the smooth deformation by the propagation-of-singularities theorem 02.14.03: the wave-front-set is invariant under the bicharacteristic flow of $g_{s}$ , and any one-parameter family of bicharacteristic flows preserves the future-pointing positive-frequency half of the relation. So the pulled-back state on $(M, g)$ has Hadamard wave-front-set structure.

Step 4: verify Kay-Wald form. By the Radzikowski theorem (just proved), the wave-front-set condition is equivalent to the Hadamard form, so the state on $(M, g)$ is Hadamard in the Kay-Wald sense.

Conclusion: every globally hyperbolic spacetime admits at least one Hadamard state. The state is highly non-unique (it depends on the choice of ultrastatic reference and the deformation path), and different Hadamard states differ by smooth corrections per Exercise 3. The original FNW 1981 paper ^{[Fulling-Narcowich-Wald 1981]} gave the construction in a slightly different language (before Radzikowski's microlocal reformulation); the modern proof exploits the wave-front-set characterisation as the central step.

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, and no Hadamard-parametrix construction 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.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, and 12.14.01 must be filled before the Radzikowski criterion can be stated in Lean. The Lorentzian-metric layer (the d'Alembertian, time orientation, causal structure), the wave-front-set theory of distributions on a manifold (chart-by-chart definition, propagation of singularities along Hamiltonian flow of a real-principal-type $Ψ$ DO), the CCR / Weyl-algebra layer (symplectic phase space, quasi-free states, GNS representations), and the Hadamard-parametrix construction (Synge world function, Van Vleck-Morette determinant, recursive transport equations for $V_{n}$ ) are each substantial Mathlib contributions in their 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 Kay-Wald definition, the Radzikowski equivalence statement, and the Minkowski vacuum worked example. The Master-tier existence-via-deformation argument and the Hadamard-subtraction renormalisation are flagged for external review by a microlocal-QFT specialist.

Advanced results Master

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

The BFK extension to all $n$ -point functions. A general (non-quasi-free) state $ω$ on the CCR algebra has a hierarchy of $n$ -point functions $ω_{n} (x_{1}, \dots, x_{n}) := ω (ϕ (x_{1}) \dots ϕ (x_{n}))$ . The Radzikowski criterion characterises the two-point function only; for non-quasi-free states one needs a microlocal-spectrum condition on every $ω_{n}$ . Brunetti, Fredenhagen and Köhler 1996 ^{[Brunetti-Fredenhagen-Köhler 1996]} gave the right generalisation: $ω$ is microlocally Hadamard if for every $n$ , $$ \mathrm{WF}(\omega_n) \subseteq \big{(x_1, k_1; \ldots; x_n, k_n) : \sum_i k_i = 0,\ \exists \text{ instanton graph realising the } (x_i, k_i)\big}, $$ where an instanton graph is a graph with $n$ external vertices at the $x_{i}$ 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 in flat spacetime. Its importance is that it admits an iterative-quantisation framework — the Hollands-Wald 2001 ^{[Hollands-Wald 2001]} construction of perturbative Wick polynomials and time-ordered products.

Hadamard subtraction of the stress-energy tensor. Wald 1977 ^{[Wald 1977]} defined the renormalised expectation value of the stress-energy tensor in a state $ω$ by point-splitting and subtraction of the Hadamard parametrix: $$ \langle T_{\mu\nu}(x)\rangle_\omega^{\mathrm{ren}} := \lim_{y \to x}\Big[D_{\mu\nu}^{(x, y)}\big(\omega_2(x, y) - H(x, y)\big)\Big], $$ where $D_{μν}^{(x, y)}$ is a bidifferential operator built from the classical-stress-tensor combination $T_{μν} = \nabla_{μ} ϕ \nabla_{ν} ϕ - \frac{1}{2} g_{μν} (\nabla ϕ)^{2} - \frac{1}{2} g_{μν} m^{2} ϕ^{2}$ acting on the two arguments. The Hadamard condition ensures the limit exists and produces a smooth symmetric covariant 2-tensor satisfying the conservation law $\nabla^{μ} ⟨ T_{μν} ⟩ = 0$ up to a smooth state-independent anomaly term (the conformal anomaly in the conformally-coupled case, an irreducible curvature-dependent contribution that cannot be absorbed by any local counterterm). The construction is independent of the choice of Hadamard state up to a finite smooth correction — different Hadamard states give renormalised stress-energy tensors differing by a smooth state-dependent piece, but no ultraviolet ambiguity.

Wick polynomials on a curved background. The Hadamard subtraction of $T_{μν}$ extends to all polynomial expressions $: ϕ^{n} (x) :$ , $: (\nabla ϕ)^{n} (x) :$ , etc. The Hollands-Wald 2001 ^{[Hollands-Wald 2001]} construction defines the renormalised Wick polynomial by an iterative subtraction: $$ :!\phi^n(x)!:H := \lim{y_1, \ldots, y_n \to x}\Big[\prod_{i < j}(\omega_2(y_i, y_j) - H(y_i, y_j))\Big]^{1/2} \text{ (schematic; see HW 2001 for the precise combinatorics)}, $$ or equivalently by the BFK-style $n$ -point microlocal subtraction. The construction is locally covariant in the sense of Brunetti-Fredenhagen-Verch 2003: it commutes with the embedding of one globally hyperbolic spacetime into another, so the same Wick polynomial expression evaluated on a sub-spacetime agrees with its restriction from the ambient spacetime. Local covariance is the modern axiomatic framework for QFT on curved spacetimes, replacing the flat-space Poincaré covariance.

Application to Hawking radiation. The Hartle-Hawking state on the Kruskal extension of Schwarzschild is constructed as the unique (up to smooth corrections) quasi-free state respecting the bifurcate Killing horizon of the Kruskal manifold and satisfying the wave-front-set condition. The Kay-Wald 1991 uniqueness theorem ^{[Kay-Wald 1991]} establishes this in the abstract: on any spacetime with a bifurcate Killing horizon, there is at most one Hadamard quasi-free state invariant under the Killing flow. Restricting the Hartle-Hawking state to the exterior region $r > 2 GM$ , the expectation values of outgoing-mode number operators read off from the Bogoliubov coefficients between the in-vacuum (at past null infinity) and the Hartle-Hawking restriction give the thermal spectrum at Hawking temperature $T_{H} = (8 π GM)^{- 1}$ : $$ \langle n_\omega\rangle_{\mathrm{out}} = \frac{1}{e^{\hbar\omega / k_B T_{\mathrm{H}}} - 1}. $$ The Bisognano-Wichmann theorem in flat-space algebraic QFT (1975-1976) is the analogue: on Minkowski with a Rindler wedge, the modular flow of the vacuum state on the wedge algebra coincides with the boost subgroup, producing the Unruh temperature for accelerated observers. The Hadamard-state framework gives the conceptual mechanism in both cases: the Hadamard wave-front set of the global state restricted to a subregion looks thermal in the subregion's natural frame.

Synthesis. The Radzikowski wave-front-set criterion is the foundational definition of admissible quantum states on a globally hyperbolic Lorentzian spacetime. It packages the short-distance singularity structure required by renormalisability — the Hadamard form — into a coordinate-free microlocal condition, the positive-frequency-half of the null-bicharacteristic relation. The theorem builds toward the Brunetti-Fredenhagen-Köhler extension to $n$ -point functions, the Wald 1977 / Hollands-Wald 2001 construction of renormalised Wick polynomials and the stress-energy tensor, the Brunetti-Fredenhagen-Verch 2003 locally covariant functor formulation of curved-spacetime QFT, the Fulling-Narcowich-Wald 1981 / Junker 1996 existence proofs of Hadamard states on every globally hyperbolic spacetime via deformation, the Kay-Wald 1991 uniqueness theorem on bifurcate-Killing-horizon spacetimes that powers the Hartle-Hawking-state construction, and the Hawking radiation derivation from the Hadamard structure of the Hartle-Hawking state on Kruskal. Every concrete curved-spacetime QFT calculation — every renormalised expectation value, every black-hole-radiation amplitude, every inflationary perturbation spectrum — is anchored in the Hadamard condition this unit characterises. The bridge from Lorentzian geometry through algebraic QFT to the renormalised observables of curved-spacetime field theory passes through this microlocal criterion.

Full proof set Master

Proposition (wave-front set of the Hadamard parametrix). On a globally hyperbolic Lorentzian manifold $(M, g)$ with normally hyperbolic operator $P = □_{g} + m^{2} + V_{pot}$ , the Hadamard parametrix $H (x, y) = (8 π^{2})^{- 1} (U / σ_{ϵ}) + (8 π^{2})^{- 1} V lo g σ_{ϵ} + W$ with the $i ϵ$ prescription $σ_{ϵ} = σ + i ϵ (T (x) - T (y)) + ϵ^{2}$ has wave-front set $$ \mathrm{WF}(H) = {((x, k_x), (y, -k_y)) : (x, k_x) \sim_{\mathrm{bichar}} (y, k_y),\ k_x \in \overline{V}^+_x}. $$

Proof. The leading term $U / σ_{ϵ}$ is a Feynman-style distribution with $i ϵ$ regularisation in the future direction. Using the integral representation $$ \frac{1}{\sigma + i\epsilon T_\Delta} = -i\int_0^\infty e^{-i\tau(\sigma + i\epsilon T_\Delta)}, d\tau, $$ with $T_{Δ} := T (x) - T (y)$ , the leading term becomes an oscillatory integral with phase $τ σ (x, y)$ and amplitude $- i U (x, y) e^{- ϵ τ T_{Δ}}$ . Stationary-phase analysis: stationary points of $τ σ (x, y)$ with respect to $τ$ require $σ (x, y) = 0$ (i.e., $y$ on the light cone of $x$ ), and stationary points with respect to $(x, y)$ require $\nabla_{x} σ = - k_{x} / τ$ and $\nabla_{y} σ = k_{y} / τ$ for some non-zero $τ$ . Since $\nabla_{x} σ (x, y) = - \overset{γ}{˙} (0)$ where $γ$ is the affinely-parametrised geodesic from $x$ to $y$ , the stationary-phase condition pins $k_{x}$ to be proportional to $\overset{γ}{˙} (0)^{♭}$ — the cotangent of the null geodesic at $x$ . Similarly $k_{y}$ proportional to $\overset{γ}{˙} (1)^{♭}$ — the parallel transport of $k_{x}$ along $γ$ . The orientation of $τ$ (positive in the integral representation) plus the $i ϵ$ prescription $ϵ T_{Δ} > 0$ for $x \in I^{+} (y)$ pins $k_{x}$ to be future-pointing: $k_{x} \in \overline{V}_{x}^{+}$ . The wave-front set of the leading term is therefore the future-pointing half of the null-bicharacteristic relation.

The logarithmic term $V lo g σ_{ϵ}$ has the same singularity structure (logarithmic instead of $1/ σ$ , but living on the same light cone with the same $i ϵ$ prescription); its wave-front set is also the future-pointing half. The smooth remainder $W$ contributes no wave-front-set points. The wave-front set of $H$ is therefore the future-pointing half, as claimed. The detailed Duistermaat-Hörmander 1972 ^{[Duistermaat-Hörmander 1972]} argument formalises this through the distinguished-parametrix classification for normally hyperbolic operators; the four distinguished parametrices (Feynman, anti-Feynman, retarded, advanced) correspond to the four choices of which half of the bicharacteristic relation to include with which sign convention, and the Hadamard parametrix is the Feynman choice. $□$

Proposition (smoothness of state-dependent remainder via propagation of singularities). Let $ω$ be a quasi-free state on the CCR algebra of $P = □_{g} + m^{2} + V_{pot}$ on a globally hyperbolic spacetime $(M, g)$ , and assume the microlocal-spectrum condition $WF (ω_{2}) = WF (H)$ . Then $ω_{2} - H \in C^{\infty}$ on a neighbourhood of the diagonal in $M \times M$ .

Proof. Set $D := ω_{2} - H$ . The state $ω_{2}$ satisfies $P_{x} ω_{2} = 0$ (acting on the first argument), since the CCR algebra is generated by smeared fields $ϕ (f)$ with $f \in C_{c}^{\infty} (M)$ and $ω$ is a quasi-free state on the CCR algebra (which annihilates $P f$ classes). The parametrix $H$ satisfies $P_{x} H \in C^{\infty} (M \times M)$ by construction: the recursive determination of the Hadamard coefficients $V_{n}$ in the transport equations makes the singularities of $P_{x} H$ cancel order-by-order, leaving only a smooth remainder. Therefore $P_{x} D = P_{x} ω_{2} - P_{x} H = - P_{x} H \in C^{\infty}$ , hence $WF (P_{x} D) = \emptyset$ .

By the propagation-of-singularities theorem 02.14.03 applied to the real-principal-type operator $P_{x}$ on $T^{*} M ∖ 0$ , the set $WF (D) ∖ WF (P_{x} D) = WF (D)$ is contained in the characteristic variety $p^{- 1} (0) = {ξ : g^{μν} ξ_{μ} ξ_{ν} = 0}$ (the null-covector cone) and is invariant under the Hamiltonian flow of $p = g^{μν} ξ_{μ} ξ_{ν}$ in the first cotangent argument. The same argument with $P_{y}$ gives bicharacteristic invariance in the second argument.

By hypothesis $WF (D) \subseteq WF (ω_{2}) \cup WF (H) = WF (H)$ — the future-pointing half of the bicharacteristic relation. A subset of this half that is invariant under bicharacteristic flow in both arguments simultaneously and is consistent with the CCR commutator condition $ω_{2} (x, y) - ω_{2} (y, x) = i E (x, y)$ is either empty or contains the full half. If non-empty, $D$ would be a non-zero distributional solution of the wave equation with positive-frequency wave-front set, contradicting the smoothness of $H$ and the fact that $ω_{2}$ is determined modulo smooth corrections by the state. Therefore $WF (D) = \emptyset$ , hence $D \in C^{\infty}$ on a neighbourhood of the diagonal. $□$

Proposition (CCR commutator condition $\Rightarrow$ wave-front set lower bound). For any quasi-free state $ω$ on the CCR algebra, $WF (ω_{2}) \supseteq$ future-pointing half of the bicharacteristic relation of $□_{g} + m^{2}$ .

Proof. The CCR commutator condition $ω_{2} (x, y) - ω_{2} (y, x) = i E (x, y)$ gives $2 Im (ω_{2}) (x, y) = E (x, y)$ as distributions (with appropriate sign convention). By 13.09.02, $WF (E) =$ full bicharacteristic relation (both halves). The wave-front set of the imaginary part lower-bounds the wave-front set of $ω_{2}$ : $WF (ω_{2}) \supseteq WF (Im ω_{2}) = WF (E /2) = WF (E)$ contains both halves. The microlocal-spectrum condition then says $WF (ω_{2})$ is exactly the future-pointing half; the past-pointing half is cancelled by the corresponding part of the symmetric (real) part of $ω_{2}$ , as in Exercise 7. $□$

Proposition (uniqueness of Hadamard states up to smooth corrections). If $ω, ω^{'}$ are two Hadamard quasi-free states on the same CCR algebra, then $ω_{2} - ω_{2}^{'} \in C^{\infty} (M \times M)$ .

Proof. Exercise 3 gives the near-diagonal statement. Off the diagonal, both $ω_{2}$ and $ω_{2}^{'}$ solve the homogeneous Klein-Gordon equation in each argument and have wave-front set on the future-pointing half (by hypothesis). The difference $D = ω_{2} - ω_{2}^{'}$ solves the homogeneous equation in each argument, has wave-front set in the future-pointing half (by additivity of WF), and is smooth near the diagonal. Propagation of singularities along the bicharacteristic flow in each argument from the smooth-near-diagonal data extends the smoothness globally: any non-smoothness of $D$ at a non-diagonal point would propagate (via the bicharacteristic) back to a non-smoothness on the diagonal, contradiction. So $D \in C^{\infty} (M \times M)$ . $□$

Connections Master

Globally hyperbolic Lorentzian manifolds 13.09.01 supplies the geometric arena: the Cauchy hypersurface $Σ$ , the smooth Bernal-Sánchez splitting, the compactness of causal diamonds $J^{+} (p) \cap J^{-} (q)$ . Without global hyperbolicity the Klein-Gordon Cauchy problem fails, the causal propagator $E$ is undefined, the CCR algebra has no well-defined symplectic phase space, and the Radzikowski criterion cannot even be stated. Globally hyperbolic is the geometric prerequisite for everything in this unit.
Klein-Gordon equation on a globally hyperbolic spacetime 13.09.02 supplies the analytic infrastructure: the Klein-Gordon operator $P = □_{g} + m^{2} + V$ , the advanced and retarded fundamental solutions $E^{\pm}$ , the causal propagator $E = E^{-} - E^{+}$ , and the wave-front-set characterisation of $E$ on Minkowski and on general globally hyperbolic backgrounds. The wave-front set of $E$ is the full null-bicharacteristic relation; the Radzikowski criterion selects its future-pointing half as the wave-front set of a Hadamard $ω_{2}$ .
CCR algebra, Weyl algebra, and quasi-free states 12.14.01 supplies the algebraic-QFT framework: the CCR algebra $A (M, g)$ as the abstract $*$ -algebra generated by smeared field operators $ϕ (f)$ with the relation $[ϕ (f), ϕ (g)] = i E (f, g) 1$ , the Weyl-algebra $C^{*}$ -completion, the notion of a quasi-free state as a state determined by its two-point function via Wick's rule, and the GNS construction producing a cyclic Hilbert-space representation. The Hadamard property is a refinement of the quasi-free property; non-quasi-free Hadamard states are characterised by the BFK 1996 microlocal-spectrum condition on all $n$ -point functions.
Propagation of singularities along Hamiltonian flow 02.14.03 is the microlocal-analysis engine of the Radzikowski proof. Hörmander's theorem: for a properly supported $Ψ$ DO $P$ of real principal type, $WF (u) ∖ WF (P u)$ is invariant under the Hamiltonian flow of the principal symbol on $T^{*} M ∖ 0$ . Applied to the Klein-Gordon operator and the difference $D = ω_{2} - H$ , this theorem propagates the diagonal smoothness statement to global smoothness — the load-bearing technical input.
Wave-front set of a distribution 02.14.01 supplies the microlocal-invariant language: the wave-front set $WF (u) \subseteq T^{*} M ∖ 0$ as the closed conic subset whose complement encodes microlocal smoothness, the additivity under sums, the pseudolocality under $Ψ$ DOs, and the conormal-bundle examples that make the geometric picture concrete. The wave-front set is the right notion of singularity to which all the curved-spacetime QFT results refer.
Existence of Hadamard states (FNW deformation) [13.09.04, pending] is the immediate downstream existence theorem. Fulling-Narcowich-Wald 1981 and Junker 1996 constructed Hadamard states on every globally hyperbolic spacetime by deforming the ultrastatic ground state along a one-parameter family of globally hyperbolic metrics, using propagation-of-singularities (the same engine as in the Radzikowski proof) to preserve the wave-front-set condition along the deformation. Existence plus the present unit's characterisation gives the complete Hadamard-state theory.
Wick polynomials in curved spacetime via Hadamard parametrix subtraction [13.09.06, pending] is the downstream renormalisation-theory application. Brunetti-Fredenhagen-Köhler 1996 and Hollands-Wald 2001 defined Wick polynomials $: ϕ^{n} (x) :_{H}$ on a curved background by subtracting the Hadamard parametrix from a normal-ordered combination of two-point functions evaluated at coincident points. The construction is locally covariant: it commutes with the embedding of one globally hyperbolic spacetime into another, the central axiom of the Brunetti-Fredenhagen-Verch 2003 locally-covariant-functor formulation.
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 the unique Hadamard quasi-free state invariant under the Killing flow that respects the bifurcate Killing horizon (Kay-Wald 1991 uniqueness). Restriction to the exterior region gives a thermal spectrum at the Hawking temperature $T_{H} = (8 π GM)^{- 1}$ — the Hadamard wave-front-set structure is what makes the Bogoliubov-coefficient calculation finite and well-defined.
FLRW cosmology 13.08.01 is the second canonical physical application. The mode decomposition of a scalar field on FLRW gives a one-parameter family of Bunch-Davies-style adiabatic vacuum states; the Hadamard property is verified for the Bunch-Davies state in de Sitter and for higher-order adiabatic vacua on general FLRW backgrounds (Lüders-Roberts 1990; Junker 1996). The primordial-perturbation power spectrum of inflation is computed from the Hadamard two-point function in the adiabatic vacuum.
Wightman axioms 08.10.07 is the flat-space analogue of the Hadamard framework. The Wightman spectrum condition — support of $\overset{ω}{^}_{2} (p)$ in the closed forward light cone of momentum space — is the flat-space localisation of the Radzikowski criterion's future-pointing condition $k_{x} \in \overline{V}_{x}^{+}$ . The Reeh-Schlieder theorem, the Bisognano-Wichmann theorem, and the modular-theory characterisations of the Wightman vacuum all have curved-spacetime analogues in the Hadamard-state framework.
Bosonic Fock space and second quantisation 08.10.01 supplies the GNS-representation target of the Hadamard quasi-free state. The Araki-Woods 1963 representation of a quasi-free state on the CCR algebra with positive-definite two-point function $ω_{2}$ as a bosonic Fock-space representation is the curved-spacetime analogue of the Minkowski Fock space; the Hadamard property determines which Fock space the state lives on up to unitary equivalence in the same equivalence class.

Historical & philosophical context Master

The Hadamard form traces back to Jacques Hadamard's 1923 Lectures on Cauchy's Problem in Linear Partial Differential Equations ^{[Hadamard 1923]}, where the parametrix construction for second-order hyperbolic operators in a Riemannian or Lorentzian geometry was developed as a tool for proving local well-posedness. The geometric ingredients — Synge's world function $σ$ , the Van Vleck-Morette determinant $Δ$ , the Hadamard coefficients $U$ and $V$ satisfying transport equations along null geodesics — were classical objects of differential geometry by the 1930s, used in the analytic study of the wave equation by F. G. Friedlander in the 1950s-1970s and consolidated in his 1975 monograph The Wave Equation on a Curved Space-Time ^{[Friedlander 1975]}.

The QFT-side appearance was via the DeWitt-Brehme 1960 paper Radiation damping in a gravitational field in Ann. Phys. 9 ^{[DeWitt-Brehme 1960]}, where the Hadamard parametrix was used to subtract the divergent self-energy of a classical point charge moving in a curved spacetime. This was reformulated for the quantised scalar field by S. A. Fulling, F. J. Narcowich and R. M. Wald in Ann. Phys. 136 (1981) 243 ^{[Fulling-Narcowich-Wald 1981]}, who gave the first systematic treatment of the Hadamard form on a globally hyperbolic spacetime and proved existence of Hadamard states on ultrastatic spacetimes via the explicit mode decomposition. The companion paper Fulling, Sweeny and Wald 1978 ^{[Fulling-Sweeny-Wald 1978]} established the universality of the Hadamard short-distance structure for any physically admissible state.

The Wald 1977 paper Comm. Math. Phys. 54 1 ^{[Wald 1977]} introduced the Hadamard-subtraction definition of the renormalised stress-energy tensor on a curved background — the prototype application of the framework. The state-dependence-modulo-smooth-corrections property of the Hadamard subtraction made the renormalisation independent of any specific choice of vacuum, resolving the ambiguity that had plagued earlier ad-hoc point-splitting prescriptions. The conformal anomaly — an irreducible state-independent curvature-dependent contribution to $\nabla^{μ} ⟨ T_{μν} ⟩$ — was identified in this framework as a genuine feature of the curved-spacetime quantisation, with no flat-space analogue.

The algebraic-QFT-side reformulation came with the Kay-Wald 1991 review Phys. Rep. 207 49 ^{[Kay-Wald 1991]}, where the Hadamard condition was promoted to the definition of an admissible quasi-free state on the CCR algebra on a globally hyperbolic spacetime, together with a uniqueness theorem for the Hartle-Hawking state on the Kruskal extension of Schwarzschild — the algebraic-QFT mechanism behind Hawking radiation. The Kay-Wald definition was awkward: it asked for the Kay-Wald-form on a neighbourhood of every diagonal point, an essentially analytic / function-theoretic condition with no obvious coordinate-free reformulation.

The microlocal turn came in 1996 with two papers by Marek J. Radzikowski, both in Comm. Math. Phys. vol. 179 — the main paper "Micro-local approach to the Hadamard condition in quantum field theory on curved space-time" CMP 179 (1996) 529 ^{[Radzikowski 1996]}, and the companion paper "A local-to-global singularity theorem for quantum field theory on curved space-time" CMP 180 (1996) 1. Radzikowski recognised that Hörmander's microlocal-analysis machinery — wave-front sets, propagation of singularities, the Duistermaat-Hörmander 1972 ^{[Duistermaat-Hörmander 1972]} distinguished-parametrix theory — was exactly the right framework to express the Hadamard condition in a coordinate-free, local-in-spacetime, microlocally-precise form. The microlocal-spectrum condition on the wave-front set of the two-point function turned the Hadamard property into a single closed condition on a closed conic subset of $T^{*} (M \times M) ∖ 0$ , immediately yielding the uniqueness-up-to-smooth-corrections statement, the propagation-along-deformation construction (FNW with a microlocal proof), and the iterative extension to all $n$ -point functions in the Brunetti-Fredenhagen-Köhler 1996 paper Comm. Math. Phys. 180 633 ^{[Brunetti-Fredenhagen-Köhler 1996]}.

The microlocal framework opened the door to the Hollands-Wald 2001 construction of locally covariant Wick polynomials and time-ordered products Comm. Math. Phys. 223 289 ^{[Hollands-Wald 2001]}, the Brunetti-Fredenhagen-Verch 2003 locally-covariant-functor axiomatisation, and the systematic perturbative QFT on curved spacetimes developed by Brunetti, Dütsch, Fredenhagen, Hollands, Wald, and their schools through the 2000s and 2010s. Christian Gérard's 2019 EMS textbook Microlocal Analysis of Quantum Fields on Curved Spacetimes ^{[Gérard 2019]} gave the modern consolidation, with Ch. 7 the canonical reference for the Radzikowski theorem and its existence-via-pseudo-differential-calculus refinement due to Gérard and Wrochna. The parallel classical-PDE-side textbook is Bär, Ginoux and Pfäffle's 2007 EMS lectures Wave Equations on Lorentzian Manifolds and Quantization ^{[Bär-Ginoux-Pfäffle 2007]}, with a freely available arXiv preprint at 0806.1036 that has become the standard free reference.

Bibliography Master

Foundational originator papers:

Hadamard, J., Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale, 1923). [Originating parametrix construction for second-order hyperbolic operators; the geometric ingredients $σ$ , $U$ , $V$ .]
DeWitt, B. S. & Brehme, R. W., "Radiation damping in a gravitational field", Ann. Phys. 9 (1960), 220-259. [QFT-side appearance of the Hadamard parametrix in coincidence-limit subtraction.]
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. [Originating treatment of the Hadamard form on a globally hyperbolic background.]
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. [FNW deformation existence theorem for Hadamard states on globally hyperbolic spacetimes.]

Renormalisation framework:

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.]
Wald, R. M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994). [Canonical physicist-side textbook reference; Ch. 4 on the Hadamard form.]

Algebraic-QFT formulation:

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. [Algebraic-QFT promotion of the Hadamard form to a definition of admissible states; uniqueness of the Hartle-Hawking state.]
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 on the CCR algebra and the Hadamard condition.]

Microlocal turn:

Duistermaat, J. J. & Hörmander, L., "Fourier integral operators. II", Acta Math. 128 (1972), 183-269. [Distinguished parametrices for normally hyperbolic operators; classification of wave-front-set structures.]
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. [The wave-front-set characterisation of Hadamard states; the central theorem of this unit.]
Radzikowski, M. J., "A local-to-global singularity theorem for quantum field theory on curved space-time", Comm. Math. Phys. 180 (1996), 1-22. [Companion paper to the main Radzikowski 1996 paper; local-to-global propagation of the Hadamard property.]
Brunetti, R., Fredenhagen, K. & Köhler, M., "The microlocal spectrum condition", Comm. Math. Phys. 180 (1996), 633-652. [Extension of the wave-front-set criterion to all $n$ -point functions; foundation for Wick polynomials.]
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. [Existence of Hadamard states on general globally hyperbolic spacetimes via pseudodifferential / adiabatic-vacuum construction.]

Modern consolidation:

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. [Locally covariant Wick polynomials defined via Hadamard subtraction; the systematic perturbative-QFT framework on a curved background.]
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; the modern framework in which the Hadamard condition is the privileged input.]
Gérard, C., Microlocal Analysis of Quantum Fields on Curved Spacetimes, ESI Lectures in Mathematics and Physics (EMS, 2019). [Modern textbook entry; Ch. 7 the canonical reference for the Radzikowski theorem and its existence-via-pseudo-differential-calculus refinement (Gérard-Wrochna).]
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.]

Microlocal-analysis machinery:

Hörmander, L., The Analysis of Linear Partial Differential Operators (Springer, 1985). [Vol. I §8 on wave-front sets, Vol. III §18.1 on $Ψ$ DOs, Vol. IV §26 on propagation of singularities; the canonical reference.]
Friedlander, F. G., The Wave Equation on a Curved Space-Time (Cambridge University Press, 1975). [Classical-PDE textbook treatment of the Hadamard parametrix construction.]

Prerequisites

13.09.01
13.09.02
12.14.01
02.14.01

Tier anchors

beginner: Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (Chicago, 1994), Ch. 4 (Hadamard form and the vacuum-state question on a curved background, in physicist-side prose)
intermediate: Kay and Wald, *Phys. Rep.* 207 (1991) 49 (the original definition of an admissible quasi-free Hadamard state and the uniqueness theorem on a bifurcate-Killing-horizon spacetime); Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 7; Bär, Ginoux & Pfäffle, *Wave Equations on Lorentzian Manifolds and Quantization* (EMS, 2007), Ch. 4
master: Radzikowski, *Comm. Math. Phys.* 179 (1996) 529; Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 7; Brunetti, Fredenhagen & Köhler, *Comm. Math. Phys.* 180 (1996) 633; Hollands and Wald, *Comm. Math. Phys.* 223 (2001) 289

References

Radzikowski, M. J. — 'Micro-local approach to the Hadamard condition in quantum field theory on curved space-time', Comm. Math. Phys. 179 (1996) 529 · Theorem 5.1 (microlocal-spectrum characterisation of Hadamard states); §3-§4 (Hadamard parametrix and propagation of singularities); the central theorem this unit develops
Radzikowski, M. J. — 'A local-to-global singularity theorem for quantum field theory on curved space-time', Comm. Math. Phys. 180 (1996) 1 (companion paper) · Local-to-global propagation of the Hadamard property; complements the main 1996 paper
Gérard, C. — Microlocal Analysis of Quantum Fields on Curved Spacetimes (EMS Press, 2019) · Ch. 7 (Hadamard states); §7.1 (Hadamard parametrix); §7.2 (Radzikowski theorem); §7.3 (existence via deformation); the modern textbook entry to the subject
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 (definition of admissible quasi-free Hadamard states via the Hadamard form on the two-point function); §6 (Hartle-Hawking state uniqueness); the algebraic-QFT predecessor to Radzikowski's wave-front-set characterisation
Bär, C., Ginoux, N. & Pfäffle, F. — Wave Equations on Lorentzian Manifolds and Quantization (EMS, 2007) · Ch. 4 (CCR-algebra construction on a globally hyperbolic spacetime; quasi-free states; Hadamard condition); free PDF at arXiv:0806.1036
Brunetti, R., Fredenhagen, K. & Köhler, M. — 'The microlocal spectrum condition', Comm. Math. Phys. 180 (1996) 633 · Extension of Radzikowski's wave-front-set criterion to all $n$-point functions of a Hadamard state; central to the Wick-polynomial / time-ordered-product programme
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 · Existence theorem for Hadamard states on ultrastatic globally hyperbolic spacetimes; the FNW deformation argument that extends existence to general globally hyperbolic backgrounds
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 · Originating treatment of the Hadamard form on a globally hyperbolic background and the iterative construction of the Hadamard coefficients $U$ and $V$ along null geodesics
Hadamard, J. — Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale, 1923) · Original parametrix construction for second-order hyperbolic operators; the geometric ingredients $\sigma$ (Synge world function), $U$, $V$ that bear his name
DeWitt, B. S. & Brehme, R. W. — 'Radiation damping in a gravitational field', Ann. Phys. 9 (1960) 220 · Coincidence-limit subtraction scheme for two-point functions on a curved background; the predecessor of the Hadamard-subtraction renormalisation of the stress-energy tensor
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 · Existence of Hadamard states on general globally hyperbolic spacetimes via the pseudodifferential / adiabatic-vacuum construction
Hollands, S. & Wald, R. M. — 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime', Comm. Math. Phys. 223 (2001) 289 · Local and covariant Wick polynomials defined via Hadamard-subtracted normal ordering; the systematic perturbative-QFT framework on a curved background
Wald, R. M. — 'The back reaction effect in particle creation in curved spacetime', Comm. Math. Phys. 54 (1977) 1 · Hadamard subtraction of the stress-energy tensor: $T_{\mu\nu}^{\mathrm{ren}}(x) = \lim_{y \to x}[\partial_x \partial_y(\omega_2 - H)]$; the renormalisation prescription for the curved-spacetime expectation value of $T_{\mu\nu}$
Wald, R. M. — Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, 1994) · Ch. 4 (Hadamard form; physical motivation; uniqueness of the Hartle-Hawking state); the canonical physicist-side reference
Hörmander, L. — The Analysis of Linear Partial Differential Operators (Springer, 1985) · Vol. I §8 (wave-front sets), Vol. III §18.1 (pseudo-differential operators), Vol. IV §26 (propagation of singularities); the microlocal-analysis machinery that underlies the Radzikowski theorem
Duistermaat, J. J. & Hörmander, L. — 'Fourier integral operators. II', Acta Math. 128 (1972) 183 · Distinguished parametrices for normally hyperbolic operators on a Lorentzian manifold; the Feynman, anti-Feynman, retarded, and advanced parametrices that classify wave-front-set structures of two-point distributions

Estimated time

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