02.14.03 · analysis / microlocal-analysis

Propagation of singularities along Hamiltonian flow

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Anchor (Master): Hörmander 1971 *Acta Math.* 127 §3; Duistermaat-Hörmander 1972 *Acta Math.* 128 §6; Hörmander Vol. III §26; Gérard *Microlocal Analysis of Quantum Fields on Curved Spacetimes* Ch. 4; Sjöstrand-Zworski *Lecture Notes on Resolvents and Trapping*

Intuition Beginner

A wave-equation singularity does not stay put. If you drop a sharp ripple into a string at one point, the ripple splits and travels outward at the wave speed; the singularity moves with the wave. Propagation of singularities is the rigorous version of this picture: the wave-front set of a solution rides along curves in cotangent space called bicharacteristics, which are exactly the trajectories of a classical-mechanical Hamiltonian flow driven by the operator's symbol.

The classical mechanics is forced on you by the geometry. The principal symbol $p (x, ξ)$ of the wave operator $□$ is the quadratic form $- τ^{2} + ∣ ξ ∣^{2}$ in spacetime coordinates $(t, x)$ and dual coordinates $(τ, ξ)$ . Singularities propagate on the set where this symbol vanishes — the characteristic variety — which is exactly the light cone, the surface in cotangent space encoding "directions in which the wave equation can have travelling singularities". On a curved Lorentzian manifold the symbol becomes $g^{μν} ξ_{μ} ξ_{ν}$ , and the same picture holds: singularities propagate along null geodesics, the lightlike trajectories on the manifold.

Why should you care? Because every linear wave-type equation in mathematical physics — electromagnetism, gravitational waves, fields of any spin in curved spacetime — inherits this picture. Hörmander's theorem is the master statement that says "if you know how your operator's symbol looks geometrically, you know how singularities will propagate". The Hadamard-state criterion in algebraic quantum field theory (Radzikowski 1996) is one downstream consequence; the rigorous derivation of geometric optics from the wave equation is another.

Visual Beginner

Picture spacetime with the position $x$ on the horizontal axis and time $t$ on the vertical. A sharp pulse starts at the origin. The wave equation $u_{tt} - u_{xx} = 0$ splits the pulse into a left-moving copy on the line $x = - t$ and a right-moving copy on the line $x = t$ . These are the two null geodesics through the origin in 1+1-dimensional Minkowski spacetime, and they are exactly the trajectories along which the singularity propagates.

In cotangent space, the lifted picture is sharper. Every point of the singularity carries a covector — the direction in which the singularity is visible — and the pair (point, direction) flows along a curve called a bicharacteristic. The bicharacteristic is the trajectory of a Hamiltonian vector field, the same construction that produces classical-mechanical orbits from a Hamiltonian function. Hörmander's theorem says the wave-front set is closed under this flow; geometric optics drops out as the special case where the Hamiltonian is the principal symbol of a wave operator.

Worked example Beginner

Compute the propagation of singularities for the 1+1-dimensional flat wave equation $u_{tt} - u_{xx} = 0$ with the initial pulse $u (0, x) = δ (x)$ and zero initial velocity $u_{t} (0, x) = 0$ .

Step 1. Write down the d'Alembert solution. The general solution of the flat 1+1-d wave equation with the given data is $$ u(t, x) = \tfrac{1}{2} \big( \delta(x - t) + \delta(x + t) \big). $$ The pulse splits into a right-moving half at $x = t$ and a left-moving half at $x = - t$ , each carrying half the strength of the original delta.

Step 2. Identify the singular support in spacetime. The solution $u (t, x)$ is smooth away from the two lines $x = t$ and $x = - t$ . So the singular support is the union of these two lines: the light cone of the origin in 1+1-d Minkowski spacetime.

Step 3. Identify the wave-front set in cotangent space. At a point $(t_{0}, t_{0})$ on the right-moving light ray $x = t$ , the singularity of $δ (x - t)$ is one-sided: it sees the direction $ξ = 1$ in $x$ paired against $τ = - 1$ in $t$ , so the wave-front-set fibre over $(t_{0}, t_{0})$ is ${(1, - 1) \cdot s : s > 0}$ , the positive ray in the $(τ, ξ) = (- 1, + 1)$ direction. Similarly, at a point $(t_{0}, - t_{0})$ on the left-moving ray, the wave-front-set fibre is the positive ray in the direction $(τ, ξ) = (- 1, - 1)$ .

Step 4. Compute the principal symbol of the wave operator. The d'Alembertian $□$ has principal symbol $p (t, x; τ, ξ) = - τ^{2} + ξ^{2}$ , the Lorentzian quadratic form on $T^{*} R^{1 + 1}$ . The characteristic variety ${p = 0}$ is ${τ = \pm ξ}$ , the two-sheeted light cone in cotangent space. The wave-front-set directions found in Step 3 — $(- 1, + 1)$ and $(- 1, - 1)$ — satisfy $τ = - ξ$ and $τ = ξ$ respectively, so they lie on the characteristic variety.

Step 5. Compute the Hamiltonian vector field. Differentiating $p = - τ^{2} + ξ^{2}$ with respect to the cotangent variables gives the coefficients of the Hamiltonian field: the spacetime component picks up $- 2 τ$ in the time direction and $+ 2 ξ$ in the space direction, and the cotangent components vanish because $p$ does not depend on $(t, x)$ in flat spacetime. On the right-moving sheet $τ = - ξ$ , the Hamiltonian field becomes $2 ξ$ in time and $2 ξ$ in space, which integrates to straight lines of slope $1$ in spacetime with the cotangent direction $(τ, ξ) = (- ξ, ξ)$ preserved along the flow.

Step 6. Verify Hörmander's theorem. The wave-front-set fibre at $(0, 0; - 1, + 1)$ flows under the Hamiltonian field to the fibre at $(2 s, 2 s; - 1, + 1)$ for every $s > 0$ , parametrising the right-moving light ray; similarly on the left. The wave-front set computed in Step 3 is the union of these flow lines, exactly the bicharacteristic flow-out of the initial singular point under the Hamiltonian flow.

What this tells us: the wave-front set is not just supported on the light cone — it is generated by the bicharacteristic flow of the principal symbol, starting from the singularities of the initial data. Hörmander's theorem packages this observation into a precise statement that holds for every operator of real principal type, on every manifold, with the geometric optics of waves on curved spacetimes as the master example.

Check your understanding Beginner

Exercise (easy, multiple choice).

For the 1+1-dimensional flat wave equation $u_{tt} - u_{xx} = 0$ with a singularity at the spacetime origin, along which curves do the singularities propagate?

A. Vertical lines $x = const$ B. Horizontal lines $t = const$ C. The two diagonal light rays $x = t$ and $x = - t$ D. A circle around the origin

Hint

The d'Alembert formula splits the initial pulse into a left-moving copy and a right-moving copy, each travelling at the wave speed (set to $1$ here).

Answer

C. The two diagonal light rays $x = t$ and $x = - t$ .

Feedback-correct: yes. The singularities of the flat 1+1-d wave equation propagate at the wave speed in both directions, giving the two diagonal light rays through the origin. Feedback-wrong: vertical lines correspond to standing waves (no spatial motion); horizontal lines correspond to instantaneous propagation (no time delay); circles correspond to elliptic-equation propagation. None of these matches the hyperbolic wave equation.

Exercise (easy, true-false).

True or false: on a Lorentzian manifold, the bicharacteristics of the wave operator $□_{g}$ are exactly the null geodesics of the metric.

Hint

The principal symbol of $□_{g}$ is $g^{μν} ξ_{μ} ξ_{ν}$ , and the Hamiltonian flow of this symbol on the characteristic variety ${g^{μν} ξ_{μ} ξ_{ν} = 0}$ is the geodesic flow restricted to lightlike covectors.

Answer

True. The principal symbol of $□_{g}$ on a Lorentzian manifold $(M, g)$ is $p (x, ξ) = g^{μν} (x) ξ_{μ} ξ_{ν}$ . The Hamilton equations $\overset{x}{˙}^{μ} = 2 g^{μν} ξ_{ν}$ together with the corresponding equation for the cotangent variables (involving derivatives of the metric in $x$ ) recover the geodesic equations after lowering with $g$ . Restricting to the characteristic set ${p = 0}$ selects exactly the null (lightlike) geodesics. So Hörmander's theorem says singularities of wave-equation solutions on a Lorentzian manifold propagate along null geodesics — the rigorous microlocal version of geometric optics.

Exercise (easy, multiple choice).

The characteristic variety of an elliptic operator like the Laplacian $- Δ$ on $R^{n}$ is what subset of $T^{*} R^{n} ∖ 0$ ?

A. The entire punctured cotangent bundle B. The conormal bundle of the boundary of the domain C. The empty set (no characteristic directions) D. A union of null geodesics

Hint

An operator is elliptic when its principal symbol does not vanish on the punctured cotangent bundle. The Laplacian has principal symbol $∣ ξ ∣^{2}$ , which is positive for $ξ \neq = 0$ .

Answer

C. The empty set (no characteristic directions).

Feedback-correct: yes. The principal symbol of $- Δ$ is $∣ ξ ∣^{2}$ , strictly positive on $T^{*} R^{n} ∖ 0$ , so the characteristic variety ${p = 0}$ is empty. Hörmander's theorem then says — by vacuous quantification over the empty characteristic set — that the wave-front set of a solution cannot propagate at all: it stays exactly where the wave-front set of $- Δ u$ is. This is the wave-front-set version of elliptic regularity. Wave operators are not elliptic — their characteristic variety is the light cone, which is non-empty and supports the bicharacteristic flow that carries singularities.

Formal definition Intermediate+

Let $X \subseteq R^{n}$ be open and let $P \in Ψ^{m} (X)$ be a properly supported classical pseudo-differential operator of order $m$ . Write $p_{m} \in S_{1, 0}^{m} / S_{1, 0}^{m - 1}$ for its principal symbol, regarded as a function on $T^{*} X ∖ 0$ that is positively homogeneous of degree $m$ in the fibre variable $ξ$ . Recall that $T^{*} X ∖ 0 = X \times (R^{n} ∖ 0)$ carries the canonical symplectic form $ω = \sum_{j} d ξ_{j} \land d x_{j}$ , and that the Hamiltonian vector field of a smooth function $f$ on $T^{*} X$ is the unique vector field $H_{f}$ satisfying $ω (H_{f}, \cdot) = - df$ , equivalently $$ H_f = \sum_{j=1}^n \Big( \frac{\partial f}{\partial \xi_j} \frac{\partial}{\partial x_j} - \frac{\partial f}{\partial x_j} \frac{\partial}{\partial \xi_j} \Big). $$

Definition (real principal type). The operator $P \in Ψ^{m} (X)$ is of real principal type if its principal symbol $p_{m}$ is real-valued and $d p_{m} (x, ξ) \neq = 0$ for every $(x, ξ)$ on the characteristic variety $Char (P) = {(x, ξ) \in T^{*} X ∖ 0 : p_{m} (x, ξ) = 0}$ . The non-vanishing of $d p_{m}$ on $Char (P)$ guarantees that the characteristic variety is a smooth conic hypersurface in $T^{*} X ∖ 0$ and that the Hamiltonian field $H_{p_{m}}$ is nowhere tangent to the zero section.

Definition (bicharacteristic). A bicharacteristic of $P$ is an integral curve of $H_{p_{m}}$ in $T^{*} X ∖ 0$ . A null bicharacteristic is a bicharacteristic that lies entirely in $Char (P)$ . Equivalently, a curve $γ : I \to T^{*} X ∖ 0$ is a null bicharacteristic if it satisfies the Hamilton equations $\overset{γ}{˙} (s) = H_{p_{m}} (γ (s))$ and the initial condition $p_{m} (γ (0)) = 0$ (the second condition is then preserved by the flow because $H_{p_{m}} p_{m} = 0$ for any $p_{m}$ ).

Definition (operator wave-front set $WF^{'}$ ). For $A \in Ψ^{m} (X)$ , the operator wave-front set $WF^{'} (A) \subseteq T^{*} X ∖ 0$ is the smallest closed conic subset outside which the symbol of $A$ is rapidly decreasing. Equivalently, $(x_{0}, ξ_{0}) \in / WF^{'} (A)$ iff there is a $Ψ$ DO $B \in Ψ^{0} (X)$ with $σ_{0} (B) (x_{0}, ξ_{0}) \neq = 0$ and $B A \in Ψ^{- \infty} (X)$ — the operator $A$ is microlocally smoothing at $(x_{0}, ξ_{0})$ . This is the operator-side analogue of the distribution wave-front set; the two are linked by $WF (A u) \subseteq WF^{'} (A) \cap WF (u)$ for every $u \in D^{'} (X)$ .

Theorem (Hörmander 1971, propagation of singularities). *Let $P \in Ψ^{m} (X)$ be properly supported of real principal type. For every $u \in D^{'} (X)$ , the set $WF (u) ∖ WF (P u)$ is contained in $Char (P) = p_{m}^{- 1} (0)$ and is invariant under the Hamiltonian flow $Φ_{t}^{H_{p_{m}}}$ of $p_{m}$ on $T^{*} X ∖ 0$ . Concretely, if $(x_{0}, ξ_{0}) \in WF (u) ∖ WF (P u)$ , then $p_{m} (x_{0}, ξ_{0}) = 0$ , and the entire integral curve of $H_{p_{m}}$ through $(x_{0}, ξ_{0})$ (extended until it meets $WF (P u)$ or leaves $T^{*} X ∖ 0$ ) is contained in $WF (u)$ .*

The first containment $WF (u) ∖ WF (P u) \subseteq Char (P)$ is the elliptic-regularity statement of unit 02.14.02 in negative form: on the non-characteristic set ${p_{m} \neq = 0}$ where $P$ is elliptic, $WF (u) = WF (P u)$ , so the difference is empty. The substantive content of the theorem is the flow-invariance on the characteristic set: singularities of $u$ that are not already singularities of $P u = f$ must lie on bicharacteristic curves, and the wave-front set is closed under translation along those curves until it meets a singularity of the source.

Notation and conventions

$T^{*} X ∖ 0$ : the punctured cotangent bundle, with the canonical symplectic form $ω = \sum_{j} d ξ_{j} \land d x_{j}$ .
$H_{f}$ : the Hamiltonian vector field of $f \in C^{\infty} (T^{*} X)$ , defined by $ω (H_{f}, \cdot) = - df$ . Sign convention: with this choice, $H_{f} g = {f, g}$ , the Poisson bracket; some references (e.g. Arnold) use the opposite sign.
$Char (P) = {(x, ξ) : p_{m} (x, ξ) = 0}$ : the characteristic variety of $P$ , a conic subset of $T^{*} X ∖ 0$ . For elliptic $P$ this is empty; for real-principal-type $P$ it is a smooth conic hypersurface.
$WF^{'} (A)$ : the operator wave-front set of $A \in Ψ^{m}$ ; $WF^{'} (A) = ess supp (σ (A))$ , the essential support of the symbol modulo $S^{- \infty}$ .
$Φ_{t}^{H_{p_{m}}}$ : the Hamiltonian flow of $p_{m}$ , a one-parameter family of conic diffeomorphisms of $T^{*} X ∖ 0$ that preserves $Char (P)$ .
Real-principal-type assumption is the standard 1971 setting; relaxations to complex principal type (Duistermaat-Hörmander 1972) and to subprincipal type (Hörmander 1985 Vol. IV §27) require extra hypotheses on lower-order symbols and modify the bicharacteristic geometry, but the prototypical setting is the real-principal-type case.

Counterexamples to common slips

The theorem fails without the real-principal-type assumption. If $p_{m}$ is complex-valued, the bicharacteristic flow is not real and the wave-front-set statement needs modification (Duistermaat-Hörmander 1972).
The theorem says singularities propagate, not that they are generated. The statement is an invariance: if $(x_{0}, ξ_{0})$ is in $WF (u) ∖ WF (P u)$ , the entire bicharacteristic through that point is in $WF (u)$ — not that every bicharacteristic point automatically becomes a singularity.
The theorem requires $d p_{m} \neq = 0$ on $Char (P)$ . At a critical point of $p_{m}$ on the characteristic set, the Hamiltonian field vanishes and the bicharacteristic degenerates to a point; the theorem says nothing there. For the wave operator $□$ this never happens away from $ξ = 0$ (the zero section is excluded by convention), but for more exotic operators the critical-point set is a real obstruction.
The wave-front set is not the singular support — propagation holds in cotangent space, with the direction $ξ$ playing a load-bearing role. Two singularities at the same base point with different cotangent directions propagate along different bicharacteristics.

Key theorem with proof Intermediate+

Theorem (Hörmander 1971, propagation of singularities; Hörmander Vol. III Theorem 26.1.1 ^{[Hörmander Vol. III §26]}). Let $P \in Ψ^{m} (X)$ be a properly supported classical pseudo-differential operator on an open $X \subseteq R^{n}$ of real principal type, with real principal symbol $p_{m} \in S_{1, 0}^{m}$ . For every distribution $u \in D^{'} (X)$ , $$ \mathrm{WF}(u) \setminus \mathrm{WF}(Pu) \subseteq p_m^{-1}(0) $$ *and $WF (u) ∖ WF (P u)$ is invariant under the Hamiltonian flow $Φ_{t}^{H_{p_{m}}}$ of $p_{m}$ on $T^{*} X ∖ 0$ .*

Proof. The argument runs in four steps: reduce to the case of a normal-form first-order operator $D_{t} + Q (t, x, D_{x})$ via a conic conjugation; construct the propagator $E (t)$ for the normal form as a Fourier integral operator; use Egorov's theorem on the propagator to transport the wave-front set along the Hamiltonian flow; and combine with an energy estimate to control the wave-front set modulo $WF (P u)$ .

Step 1: reduction to a normal-form first-order operator. The first containment $WF (u) ∖ WF (P u) \subseteq p_{m}^{- 1} (0)$ is elliptic regularity in negative form (unit 02.14.02 Theorem). For the flow-invariance, work near a fixed point $(x_{0}, ξ_{0}) \in Char (P)$ . Because $d p_{m} (x_{0}, ξ_{0}) \neq = 0$ by real principal type, Darboux's theorem in the conic-symplectic category provides a homogeneous canonical transformation $χ : U \to V$ between conic neighbourhoods of $(x_{0}, ξ_{0})$ in $T^{*} X$ and a conic neighbourhood of $(0, e_{n}^{*}) \in T^{*} R^{n}$ such that $p_{m} \circ χ^{- 1} (y, η) = η_{1}$ — the symbol is conjugated to the first dual coordinate. Lift $χ$ to a Fourier integral operator $F \in I^{0} (X \times R^{n}, χ)$ with $WF^{'} (F) \subseteq graph (χ)$ ; conjugation $P = F P F^{- 1}$ produces an operator with principal symbol $η_{1}$ , i.e. $P = D_{1} + R$ with $R \in Ψ^{0}$ . (Lower-order corrections from $F$ -conjugation absorb into $R$ .) Wave-front-set invariance under Hamilton flow of $p_{m}$ transforms to wave-front-set invariance under translation in $y_{1}$ , the Hamilton flow of $η_{1}$ .

Step 2: the propagator $E (t)$ for $D_{t} + R$ . With time variable $t = y_{1}$ , the normal-form operator is $D_{t} + R (t, y^{'}, D_{y^{'}})$ where $y^{'} = (y_{2}, \dots, y_{n})$ and $R \in Ψ^{0}$ smoothly depending on $t$ . The Cauchy problem $$ (D_t + R) u = 0, \qquad u(0) = u_0 \in \mathcal{S}'(\mathbb{R}^{n-1}) $$ is well-posed in tempered-distribution spaces: define $u (t) = E (t) u_{0}$ where the propagator $E (t) : S^{'} (R^{n - 1}) \to S^{'} (R^{n - 1})$ satisfies $\partial_{t} E (t) + i R E (t) = 0$ , $E (0) = I$ . Existence and uniqueness follow from Picard iteration in the symbol class because $R \in Ψ^{0}$ is bounded on Sobolev spaces (Calderón-Vaillancourt theorem, unit 02.14.02 Master tier). The propagator $E (t)$ is a unitary operator on $L^{2}$ if $R$ is formally self-adjoint, and a Fourier integral operator more generally.

Step 3: Egorov's theorem. The conjugation of a $Ψ$ DO $A \in Ψ^{k}$ by the propagator $E (t)$ is again a $Ψ$ DO of the same order with a controlled principal symbol: $$ E(t) A E(t)^{-1} = A(t) \in \Psi^k, \qquad \sigma_k(A(t)) = \sigma_k(A) \circ \Phi_t^{H_{\sigma_1(R)}}, $$ where $Φ_{t}^{H_{σ_{1} (R)}}$ is the Hamiltonian flow of the principal symbol $σ_{1} (R) = η_{1}$ of the normal form (i.e. translation $(t, y^{'}; τ, η^{'}) \mapsto (t + s, y^{'}; τ, η^{'})$ — actually, since $R \in Ψ^{0}$ here, the relevant Hamiltonian field on the symbol level comes from the full first-order $D_{t} + R$ ). This is Egorov's theorem (Hörmander Vol. III Theorem 25.1.2 ^{[Hörmander Vol. III §25.1]}). The proof is by differentiating $A (t)$ in $t$ and applying the composition law: $\partial_{t} A (t) = i [R, A (t)]$ , with principal symbol ${σ_{1} (R), σ_{k} (A (t))}$ by the leading-order commutator formula $[R, A]$ has symbol $\frac{1}{i} {r, a} + O (lower)$ . This is exactly the ODE for the Hamiltonian flow of $σ_{1} (R)$ acting on $σ_{k} (A)$ .

Step 4: wave-front-set transport. Pick $(x_{0}, ξ_{0}) \in WF (u) ∖ WF (P u)$ on the characteristic set, work in the normal-form coordinates of Step 1, and consider the bicharacteristic $γ (s) = Φ_{s}^{H_{p_{m}}} (x_{0}, ξ_{0})$ . Want: $γ (s) \in WF (u)$ for every $s$ in an open interval around $0$ on which $γ$ stays in the conic neighbourhood and avoids $WF (P u)$ . Suppose for contradiction that $γ (s_{0}) \in / WF (u)$ for some such $s_{0} > 0$ . Then there is a $Ψ^{0}$ microlocal cutoff $A_{0}$ with $σ_{0} (A_{0}) (γ (s_{0})) \neq = 0$ such that $A_{0} u \in C^{\infty}$ in a conic neighbourhood. Transport: define $A (s) = E (s) A_{0} E (s)^{- 1} \in Ψ^{0}$ for the propagator $E$ of the normal-form operator, so by Egorov $σ_{0} (A (s)) = σ_{0} (A_{0}) \circ Φ_{s}^{H_{p_{m}}}$ . At $s = 0$ , $A (0)$ has principal symbol non-vanishing at $(x_{0}, ξ_{0})$ . But $A (0) u = A_{0} \cdot u + (error of order WF (P u))$ via the propagator identity, and the error is microlocally smooth in a neighbourhood of $(x_{0}, ξ_{0})$ because $(x_{0}, ξ_{0}) \in / WF (P u)$ . So $A (0) u$ is microlocally smooth at $(x_{0}, ξ_{0})$ , contradicting $(x_{0}, ξ_{0}) \in WF (u)$ . The contradiction shows the bicharacteristic stays in $WF (u)$ , proving Hamiltonian-flow invariance.

The full argument requires care with the energy-estimate / commutator-positivity bookkeeping at each step; the canonical reference is Hörmander Vol. III §26.1, which presents the argument in essentially the form above. $□$

Bridge. Hörmander's theorem closes the microlocal calculus: pseudolocality (wave-front set non-increasing under $Ψ$ DO action), elliptic regularity (wave-front set invariant on the non-characteristic set), and propagation of singularities (wave-front set bicharacteristic-flow invariant on the characteristic set) together describe how every linear $Ψ$ DO acts on wave-front sets. The theorem reads, in physical terms: an operator of real principal type has a well-defined classical limit — its principal-symbol Hamiltonian flow — and the singularities of solutions follow that classical limit. This is the rigorous microlocal version of the WKB / geometric-optics intuition that high-frequency asymptotics of wave-equation solutions are governed by Hamilton-Jacobi theory of the symbol. The same pattern appears again in 13.09.03 (Hadamard states on a globally hyperbolic spacetime), where the wave-front-set bound on the two-point function of a quasi-free state is forced by propagation of singularities for the Klein-Gordon operator; in 05.02.06 (geodesic Hamiltonian flow), where the Lorentzian extension of the Riemannian geodesic-flow Hamiltonian is exactly the principal symbol of $□_{g}$ ; and in Atiyah-Bott Lefschetz fixed-point theory, where the contribution of a fixed point to an analytic Lefschetz number depends on the bicharacteristic structure near the fixed point. Putting these together produces every modern microlocal application — Hadamard states, parametrices for non-elliptic operators, semiclassical resolvent estimates with trapped sets, decay rates for the wave equation on black-hole spacetimes (Dyatlov-Zworski 2019 Mathematical Theory of Scattering Resonances ^{[source pending]}) — and is the load-bearing technical ingredient of microlocal analysis as a discipline.

Exercises Intermediate+

Exercise 1 (easy, symbolic).

Write down the principal symbol of the wave operator $□ = \partial_{t}^{2} - Δ_{x}$ on $R^{1 + n}$ , and identify the characteristic variety as a subset of $T^{*} R^{1 + n} ∖ 0$ .

Hint

Each $\partial_{t}$ becomes multiplication by $i τ$ on the Fourier side, and each $\partial_{x_{j}}$ becomes $i ξ_{j}$ . The operator $□$ has order $2$ , so the principal symbol is the quadratic polynomial in $(τ, ξ)$ obtained by squaring.

Answer

Principal symbol $p_{2} (t, x; τ, ξ) = - τ^{2} + ∣ ξ ∣^{2} = - τ^{2} + ξ_{1}^{2} + \dots + ξ_{n}^{2}$ , the Lorentzian quadratic form on $T^{*} R^{1 + n}$ . The characteristic variety is ${p_{2} = 0} = {(t, x; τ, ξ) : τ^{2} = ∣ ξ ∣^{2}, (τ, ξ) \neq = 0}$ , the punctured light cone in cotangent space, a smooth conic hypersurface in $T^{*} R^{1 + n} ∖ 0$ . It splits into two sheets $τ = + ∣ ξ ∣$ (future-directed) and $τ = - ∣ ξ ∣$ (past-directed) under any time-orientation, joined along the zero section that has been excluded. Rubric: full credit for the symbol formula plus identification of the two-sheeted light cone.

Exercise 2 (easy, numeric).

For the 1+1-d wave equation principal symbol $p (t, x; τ, ξ) = - τ^{2} + ξ^{2}$ , compute the Hamiltonian vector field $H_{p}$ at the point $(t, x; τ, ξ) = (0, 0; 1, 1)$ (which lies on the characteristic set $τ = ξ$ ).

Hint

$H_{p} = (\partial p / \partial τ) \partial_{t} + (\partial p / \partial ξ) \partial_{x} - (\partial p / \partial t) \partial_{τ} - (\partial p / \partial x) \partial_{ξ}$ . The symbol does not depend on $(t, x)$ in the flat case, so the $\partial_{τ}$ and $\partial_{ξ}$ components vanish.

Answer

$H_{p} = - 2 τ \partial_{t} + 2 ξ \partial_{x}$ . At $(0, 0; 1, 1)$ , $H_{p} = - 2 \partial_{t} + 2 \partial_{x}$ , a vector pointing in the $+ x, - t$ direction with both components of magnitude $2$ . After reparametrising the curve so it moves forward in time, this is the right-moving null direction $\partial_{t} + \partial_{x}$ (signs flip because $τ$ in $T^{*}$ is dual to $t$ ). The integral curve from $(0, 0; 1, 1)$ is $(t, x; τ, ξ) = (- 2 s, 2 s; 1, 1)$ , the right-moving null line through the spacetime origin (after the standard sign-flip $s \mapsto - s$ to align with future time). Rubric: full credit for the formula and the identification of the integral curve as a null geodesic.

Exercise 3 (medium, short-answer).

Verify that the bicharacteristics of the d'Alembertian on flat 1+1-d Minkowski spacetime are exactly the null geodesics (straight lines of slope $\pm 1$ ), and that the bicharacteristic preserves the cotangent direction.

Hint

Solve the Hamilton equations $\overset{x}{˙}^{μ} = \partial p / \partial ξ_{μ}$ , $\dot{ξ}_{μ} = - \partial p / \partial x^{μ}$ for $p = - τ^{2} + ξ^{2}$ . The cotangent direction is constant along the flow because $p$ does not depend on $(t, x)$ .

Answer

Hamilton equations for $p = - τ^{2} + ξ^{2}$ : $\dot{t} = \partial_{τ} p = - 2 τ$ , $\overset{x}{˙} = \partial_{ξ} p = 2 ξ$ , $\overset{τ}{˙} = - \partial_{t} p = 0$ , $\dot{ξ} = - \partial_{x} p = 0$ . So $τ$ and $ξ$ are constants of motion. Integrating: $t (s) = t_{0} - 2 τ_{0} s$ and $x (s) = x_{0} + 2 ξ_{0} s$ . The slope in spacetime is $d x / d t = - ξ_{0} / τ_{0}$ . On the characteristic sheet $τ_{0} = ξ_{0}$ , this slope is $- 1$ ; on the sheet $τ_{0} = - ξ_{0}$ , this slope is $+ 1$ . Both are null directions (slope $\pm 1$ is the speed of light in normalised units). So bicharacteristics are exactly straight null lines, with cotangent direction $(τ_{0}, ξ_{0})$ preserved along the line. After a sign flip to orient forward in time ( $s \to - s$ on the past-pointing sheet), these are the standard right- and left-moving null geodesics. Rubric: full credit for the explicit integration plus the identification of the slope as null.

Exercise 4 (medium, short-answer).

State Egorov's theorem precisely and use it to derive the wave-front-set transport law: if $E (t)$ is the propagator for $D_{t} + R$ with $R \in Ψ^{1}$ self-adjoint having real principal symbol $r$ , then for $u \in D^{'}$ , $WF (E (t) u) = Φ_{t}^{H_{r}} (WF (u))$ .

Hint

Egorov: $E (t) A E (t)^{- 1} \in Ψ^{k}$ for $A \in Ψ^{k}$ , with principal symbol $σ_{k} (A) \circ Φ_{t}^{H_{r}}$ (the symbol is transported by the Hamiltonian flow of $r$ ). Wave-front-set transport then follows by applying Egorov to a microlocal cutoff that detects $WF (u)$ .

Answer

Egorov's theorem (Hörmander Vol. III §25.1): Let $R \in Ψ^{1} (X)$ be properly supported, formally self-adjoint, with real principal symbol $r \in S_{1, 0}^{1}$ of real principal type. Let $E (t)$ be the unique unitary propagator solving $\partial_{t} E + i R E = 0$ , $E (0) = I$ . Then for every $A \in Ψ^{k} (X)$ , the conjugate $A (t) = E (t) A E (t)^{- 1}$ lies in $Ψ^{k} (X)$ with principal symbol $$ \sigma_k(A(t))(x, \xi) = \sigma_k(A)(\Phi_{-t}^{H_r}(x, \xi)), $$ i.e. the symbol of $A$ pulled back by the Hamiltonian flow of $r$ . (Sign conventions vary; the key point is that the principal symbol is transported by the Hamiltonian flow of the principal symbol of $R$ .)

Wave-front-set transport. Pick $(x_{0}, ξ_{0}) \in WF (u)$ and let $γ (t) = Φ_{t}^{H_{r}} (x_{0}, ξ_{0})$ . Suppose for contradiction that $γ (t_{0}) \in / WF (E (t_{0}) u)$ . Then there is a microlocal cutoff $A \in Ψ^{0}$ with $σ_{0} (A) (γ (t_{0})) \neq = 0$ and $A E (t_{0}) u \in C^{\infty}$ microlocally near $γ (t_{0})$ . By Egorov, $E (- t_{0}) A E (t_{0}) = A (- t_{0}) \in Ψ^{0}$ with $σ_{0} (A (- t_{0})) (x_{0}, ξ_{0}) = σ_{0} (A) (γ (t_{0})) \neq = 0$ . So $A (- t_{0}) u = E (- t_{0}) A E (t_{0}) u$ is microlocally smooth at $(x_{0}, ξ_{0})$ , because $E (t_{0})$ is unitary on $L^{2}$ (and on Sobolev spaces, with explicit wave-front-set continuity). But $A (- t_{0}) u$ smooth at $(x_{0}, ξ_{0})$ with $σ_{0} (A (- t_{0})) (x_{0}, ξ_{0}) \neq = 0$ contradicts $(x_{0}, ξ_{0}) \in WF (u)$ . Hence $γ (t_{0}) \in WF (E (t_{0}) u)$ , and by symmetry of the argument the wave-front sets are related by the Hamiltonian flow. Rubric: full credit for the precise Egorov statement plus the transport argument.

Exercise 5 (medium, symbolic).

Compute the principal symbol of the Klein-Gordon operator $□_{g} + m^{2}$ on a Lorentzian manifold $(M, g)$ with signature $(-, +, +, +)$ and verify that the characteristic variety is the null cone, independent of the mass parameter $m$ .

Hint

The mass term is order $0$ , so it does not contribute to the principal symbol. Only the second-order part $□_{g} = g^{μν} \nabla_{μ} \nabla_{ν}$ contributes, with principal symbol $g^{μν} ξ_{μ} ξ_{ν}$ .

Answer

The Klein-Gordon operator is $□_{g} + m^{2}$ on $(M, g)$ , with $□_{g} = g^{μν} \nabla_{μ} \nabla_{ν}$ the Lorentzian d'Alembertian. The mass term $m^{2}$ is a smooth function on $M$ , contributing a $Ψ^{0}$ multiplication operator with principal symbol equal to $m^{2}$ at order $0$ — order $0 < 2$ , so it does not enter the principal symbol of the order- $2$ operator. The principal symbol is therefore $$ \sigma_2(\Box_g + m^2)(x, \xi) = g^{\mu\nu}(x)\xi_\mu \xi_\nu, $$ the Lorentzian inverse-metric quadratic form on $T^{*} M$ . The characteristic variety ${g^{μν} ξ_{μ} ξ_{ν} = 0}$ is the null cone of $g$ in $T^{*} M$ , the set of lightlike (null) covectors at each point. This is independent of $m$ : massive and massless Klein-Gordon equations have the same characteristic variety and hence the same propagation-of-singularities geometry, with the mass entering only at lower order in the symbol expansion and the bicharacteristic flow. The associated Hamiltonian flow on the null cone is the null-geodesic flow of $(M, g)$ , and propagation of singularities reads: singularities of solutions of the Klein-Gordon equation on a Lorentzian manifold propagate along null geodesics, regardless of the mass. Rubric: full credit for the symbol formula plus the null-cone identification with the mass-independence observation.

Exercise 6 (medium, short-answer).

Show that the wave-front set of the retarded fundamental solution $E^{+}$ of the flat d'Alembertian on $R^{1 + n}$ is supported on pairs of points joined by future-directed null geodesics, with parallel cotangents.

Hint

$E^{+}$ satisfies $□ E^{+} = δ$ with $supp E^{+} \subset {(t, x) : t \geq 0, t \geq ∣ x ∣}$ (forward light cone). Hörmander's theorem applied to $E^{+}$ (as a distribution on $R^{1 + n}$ ) gives the wave-front set as the bicharacteristic flow-out of $WF (δ) = {(0; τ, ξ) : (τ, ξ) \neq = 0}$ on the characteristic variety.

Answer

The retarded fundamental solution $E^{+}$ on $R^{1 + n}$ satisfies $□ E^{+} = δ_{(0, 0)}$ (Dirac mass at the spacetime origin) with forward-cone support. By Hörmander's theorem, $WF (E^{+}) ∖ WF (δ_{(0, 0)})$ lies on the characteristic variety ${- τ^{2} + ∣ ξ ∣^{2} = 0}$ — the light cone in cotangent space — and is invariant under the bicharacteristic flow. The wave-front set of $δ_{(0, 0)}$ on $R^{1 + n}$ is the entire conic fibre over the origin: ${(0, 0; τ, ξ) : (τ, ξ) \neq = 0}$ . Intersecting with the characteristic variety picks out ${(0, 0; τ, ξ) : τ^{2} = ∣ ξ ∣^{2}, (τ, ξ) \neq = 0}$ , the punctured light cone over the origin. The bicharacteristic flow-out is straight-line transport in spacetime along null directions with cotangents preserved, giving $$ \mathrm{WF}(E^+) = {(t, x; \tau, \xi) : t \geq 0,, (t, x) \in \partial J^+(0),, \tau^2 = |\xi|^2,, (\tau, \xi) \text{ parallel-transported along the null geodesic from } 0 \text{ to } (t, x)}, $$ where $\partial J^{+} (0)$ is the boundary of the future of the origin (i.e. the future light cone $t = ∣ x ∣ > 0$ ). On flat spacetime "parallel-transported" reduces to "constant", so the wave-front-set fibre over $(t, x) = (∣ x ∣, x)$ is the one-dimensional ray ${(τ, ξ) \propto (- 1, x /∣ x ∣)}$ , the unique null direction pointing back to the source. This wave-front-set pattern is what Radzikowski 1996 ^{[source pending]} turns into the curved-spacetime Hadamard-state criterion. Rubric: full credit for the bicharacteristic flow-out plus the parallel-transport identification.

Exercise 7 (hard, short-answer).

Construct a microlocal parametrix at a point $(x_{0}, ξ_{0}) \in Char (P)$ for a real-principal-type operator $P \in Ψ^{m} (X)$ , using the propagation-of-singularities theorem to invert $P$ modulo smoothing operators on a conic neighbourhood transverse to the bicharacteristic flow.

Hint

Cannot invert $P$ pointwise at characteristic points (the principal symbol vanishes there). The parametrix is constructed by integrating along the bicharacteristic from a transversal Cauchy hypersurface; the construction lifts to the symbol calculus via a Fourier integral operator with phase function generating the bicharacteristic flow.

Answer

By Step 1 of the Key-theorem proof, conic-conjugate $P$ to a normal form $P = D_{t} + R$ with $R \in Ψ^{0}$ in a conic neighbourhood of $(x_{0}, ξ_{0})$ . The Cauchy problem for $P$ , $P u = f$ , $u ∣_{t = 0} = u_{0}$ , has the explicit solution $u (t) = E (t) u_{0} + \int_{0}^{t} E (t - s) f (s) d s$ where $E (t)$ is the propagator constructed in Step 2. Encode this as an operator-valued kernel: define $$ \widetilde{B} : f \mapsto \int_0^t E(t - s) f(s), ds, $$ the right inverse of $P$ on the half-space $t \geq 0$ with zero Cauchy data. This $B$ is a Fourier integral operator of order $- 1$ (one degree below $P$ 's order $+ 1$ ) with phase function generating the bicharacteristic flow. The composition $P B = I$ on the half-space modulo a Cauchy-data term, and $B P - I$ is microlocally smoothing in a conic neighbourhood transverse to the bicharacteristic through $(x_{0}, ξ_{0})$ . Conjugating back by the FIO $F$ produces the microlocal parametrix $B = F^{- 1} B F \in Ψ_{(FIO-extended)}^{- m}$ at $(x_{0}, ξ_{0})$ . The construction is exactly the propagation parametrix of Hörmander Vol. IV §29.1 ^{[Hörmander Vol. III §26]}; the parametrix lives in the Fourier-integral-operator class rather than the pseudo-differential class because the bicharacteristic transport is not encoded by a pseudo-differential symbol. Rubric: full credit for the normal-form reduction, the propagator-integral construction, and the FIO conjugation back to the original coordinates.

Exercise 8 (hard, short-answer).

State Duistermaat-Hörmander's 1972 extension to operators of complex principal type, and explain in qualitative terms how the bicharacteristic flow is replaced by a real-symplectic foliation in the complex case.

Hint

For complex-valued $p_{m}$ , $H_{p_{m}}$ has both real and imaginary parts; the bicharacteristic flow is no longer well-defined as a single real curve, but the two real one-parameter families generated by $Re H_{p_{m}}$ and $Im H_{p_{m}}$ commute and foliate the characteristic set by 2-dimensional symplectic leaves.

Answer

Duistermaat-Hörmander 1972 Acta Math. 128 §6 ^{[Duistermaat-Hörmander 1972]}: Let $P \in Ψ^{m} (X)$ have complex principal symbol $p_{m} = a + ib$ with $a, b \in S_{1, 0}^{m}$ real and the joint vanishing ${a = b = 0}$ a smooth conic submanifold of $T^{*} X ∖ 0$ on which ${a, b} = 0$ (Poisson bracket vanishes, equivalently $H_{a}$ and $H_{b}$ commute) and the differentials $d a, d b$ are linearly independent. Then $WF (u) ∖ WF (P u)$ is contained in ${a = b = 0}$ and is invariant under the commuting two-dimensional foliation generated by $H_{a}$ and $H_{b}$ on this characteristic submanifold.

Qualitatively, the difference is dimensional. In the real-principal-type case, $p_{m}$ is a single real function whose Hamiltonian field $H_{p_{m}}$ generates a one-dimensional flow on $T^{*} X$ ; the bicharacteristic is a curve, and the wave-front set is invariant under translation along curves. In the complex-principal-type case, $p_{m}$ has two real components $a$ and $b$ , each generating its own Hamiltonian flow, and the commutation condition ${a, b} = 0$ ensures the two flows combine into a single two-dimensional symplectic foliation of the joint characteristic set ${a = b = 0}$ . Wave-front-set invariance now means invariance under translation in both directions on the leaves, not just along a single curve.

Example: the $\overset{ˉ}{\partial}$ -operator on $C^{n}$ has complex principal symbol $ζ_{j}$ (the dual coordinate to $z_{j}$ ), real part $ξ_{j}$ , imaginary part $η_{j}$ (writing $ζ = ξ + i η$ ). The two Hamiltonian fields are translations in opposite-axis directions, and the leaves are real two-dimensional planes. Propagation of singularities for $\overset{ˉ}{\partial}$ then says wave-front sets are invariant on these 2-dimensional leaves; the canonical application is to the $\overset{ˉ}{\partial}$ -Neumann problem and Kohn-Nirenberg subellipticity on strongly pseudoconvex boundaries (Kohn-Nirenberg 1965 CPAM 18; Hörmander 1965 CPAM 18). Rubric: full credit for the precise Duistermaat-Hörmander statement plus the dimensional / foliation interpretation.

Advanced results Master

Theorem (Hörmander's theorem in operator wave-front-set form; Hörmander Vol. III Theorem 26.1.4 ^{[Hörmander Vol. III §26.1]}). Let $P \in Ψ^{m} (X)$ be properly supported of real principal type with real principal symbol $p_{m}$ . The operator wave-front set of the "forward parametrix" $B$ constructed by integrating along the bicharacteristic flow satisfies $$ \mathrm{WF}'(B) \subseteq \mathrm{diag}(T^*X) \cup C^+_P, $$ *where $C_{P}^{+} = {(Φ_{t}^{H_{p_{m}}} (x, ξ); x, ξ) : (x, ξ) \in Char (P), t \geq 0}$ is the forward bicharacteristic relation on $T^{*} X \times T^{*} X$ . The parametrix $B$ is a Fourier integral operator associated to the canonical relation $C_{P}^{+}$ , and $P B - I, B P - I$ are properly supported smoothing operators on a conic neighbourhood of the diagonal.*

The operator-side statement packages propagation as a Fourier-integral-operator construction: the parametrix for $P$ on the characteristic set lives in the FIO class, not the $Ψ$ DO class, because the bicharacteristic transport is not described by a pseudo-differential symbol. Sjöstrand's Singularités analytiques microlocales (Astérisque 1982) extends this to the analytic-wave-front-set setting via FBI / Bargmann-type transforms; Lebeau's Geometric optics for the elastic wave equation (1996) extends to systems with multiple characteristic sheets.

Theorem (Duistermaat-Hörmander 1972, complex principal type; Duistermaat-Hörmander 1972 §6 ^{[Duistermaat-Hörmander 1972]}). Let $P \in Ψ^{m} (X)$ have complex principal symbol $p_{m} = a + ib$ with ${a, b} = 0$ on ${a = b = 0}$ and $d a, d b$ linearly independent there. Then $WF (u) ∖ WF (P u) \subseteq {a = b = 0}$ and is invariant under the (real) two-dimensional symplectic foliation generated by $H_{a}$ and $H_{b}$ .

The complex-principal-type extension subsumes the $\overset{ˉ}{\partial}$ -Neumann problem on strongly pseudoconvex domains, Kohn-Nirenberg subellipticity, and the propagation of analytic singularities for the heat equation (with $p_{m} = τ + i ∣ ξ ∣^{2}$ , complex principal symbol). The flow-invariance generalises to higher-codimension symplectic leaves for $k$ -fold complex principal-type symbols, with leaves of dimension $2 k$ on the joint characteristic set.

Theorem (microlocal Hadamard condition; Radzikowski 1996 ^{[Radzikowski 1996]}). Let $(M, g)$ be a globally hyperbolic Lorentzian manifold and let $ω$ be a quasi-free state of the free Klein-Gordon field $(□_{g} + m^{2}) ϕ = 0$ . Then $ω$ is Hadamard iff its two-point function $ω_{2} \in D^{'} (M \times M)$ satisfies $$ \mathrm{WF}(\omega_2) = \big{ ((x_1, \xi_1), (x_2, -\xi_2)) \in T^(M \times M) \setminus 0 :\ (x_1, \xi_1) \sim_g (x_2, \xi_2),\ \xi_1 \in \overline{V^+{x_1}} \big}, $$ *where $\sim_{g}$ denotes the null-bicharacteristic equivalence (points connected by a null geodesic with $ξ$ the parallel transport of the cotangent) and $\overline{V^+{x_1}} $i s t h ec l ose df u t u r e l i g h t co n e in$ T^*_{x_1} M$.

The Radzikowski criterion is a direct consequence of propagation of singularities for the Klein-Gordon operator: the two-point function $ω_{2}$ satisfies the equation $(□_{g} + m^{2})_{x} ω_{2} = 0$ in the distribution sense, and Hörmander's theorem then constrains $WF (ω_{2})$ to the joint characteristic set of $(□_{g} + m^{2})_{x_{1}}$ and $(□_{g} + m^{2})_{x_{2}}$ , with bicharacteristic-flow invariance. The Hadamard condition adds the positivity asymmetry $ξ_{1} \in \overline{V_{x_{1}}^{+}}$ , selecting the "positive-frequency" part of the wave-front set — exactly the curved-spacetime analogue of the Minkowski spectrum condition for Wightman fields (Streater-Wightman 1964). Existence of Hadamard states (Fulling-Narcowich-Wald 1981, Junker-Olbermann 2001, Gérard-Wrochna 2014 ^{[source pending]}) verifies the wave-front-set condition by a deformation argument or by direct pseudo-differential construction.

Theorem (semiclassical propagation of singularities; Dimassi-Sjöstrand Ch. 9 ^{[Dimassi-Sjöstrand]}). *Let $P_{h} = p^{w} (x, h D; h) \in Ψ_{h}^{m, k}$ be a semiclassical $Ψ$ DO with principal symbol $p_{0} (x, ξ)$ real of real principal type. The semiclassical wave-front set $WF_{h} (u_{h}) \subseteq T^{*} X$ of a family of distributions $u_{h} \in D^{'} (X)$ satisfies $WF_{h} (u_{h}) ∖ WF_{h} (P_{h} u_{h}) \subseteq p_{0}^{- 1} (0)$ and is invariant under the Hamiltonian flow of $p_{0}$ , with explicit $O (h)$ control on the leading-order transport coefficient (the subprincipal symbol of $P_{h}$ ).*

The semiclassical theorem refines Hörmander 1971 to control of the leading $h$ -power of the symbol along the bicharacteristic flow, not just the closure properties of the wave-front set. It is the foundational tool for semiclassical resolvent estimates with trapped sets (Sjöstrand 1990, Dyatlov-Zworski 2019 ^{[source pending]}) and for the analysis of quantum chaos in Anosov billiard / quantum-ergodicity questions.

Theorem (Duistermaat's solvability theorem for real-principal-type operators; Duistermaat 1996 Fourier Integral Operators Ch. 6 ^{[source pending]}). *Let $P \in Ψ^{m} (X)$ be of real principal type with no trapped bicharacteristics (every bicharacteristic exits every compact subset of $T^{*} X ∖ 0$ in finite time). Then $P : C^{\infty} (X) \to C^{\infty} (X)$ is surjective, and there exists a properly supported parametrix $B \in Ψ_{FIO-extended}^{- m}$ with $P B - I, B P - I$ smoothing.*

Duistermaat's solvability theorem is the existence theorem corresponding to propagation of singularities: the FIO parametrix constructed by integrating the bicharacteristic flow exists globally whenever the flow has no trapping, and provides a right inverse to $P$ modulo smoothing operators. The trapping obstruction is genuine: an operator with trapped bicharacteristics (e.g. a wave operator on a spacetime with closed null geodesics) generically fails the solvability theorem at the level of the Sobolev orders that probe the trapped set.

Synthesis. Propagation of singularities is the load-bearing technical theorem of microlocal analysis: it asserts that the wave-front set of a distribution annihilated by an operator of real principal type is closed under the Hamiltonian flow of the principal symbol, so the singular structure of the solution is forced to follow the classical-mechanical bicharacteristic geometry of the operator. The central insight is that the principal-symbol calculus (unit 02.14.02) is exactly the symbol-side reflection of classical Hamiltonian mechanics on the cotangent bundle, and the wave-front-set calculus (unit 02.14.01) is the singularity-side recording of that mechanics. Hörmander's theorem unifies the two: the singular geometry of solutions is the integrated form of the symbol-side Hamiltonian dynamics. This is exactly the same pattern as in 05.02.06 (geodesic Hamiltonian flow), where the symbol-side Hamiltonian for the Laplace-Beltrami operator generates the Riemannian geodesic flow, and in 13.09.02 (Klein-Gordon Cauchy problem on a globally hyperbolic Lorentzian manifold), where the wave operator's Hamiltonian generates the null-geodesic flow. Putting these together produces every microlocal application — the Hadamard-state programme of algebraic QFT on curved spacetimes (Radzikowski 1996; Gérard 2019), the geometric-optics derivation of WKB asymptotics, semiclassical resolvent estimates and quantum-ergodicity bounds, and the analysis of scattering resonances on black-hole and other trapping spacetimes (Dyatlov-Zworski 2019). The bridge is the recognition that linear PDE of real principal type is the singularity-side projection of classical Hamiltonian mechanics, and the wave-front set is the singularity-side phase-space portrait of the solution.

Propagation of singularities identifies several apparently distinct geometric notions. On a Riemannian manifold, the bicharacteristics of the Laplace-Beltrami square root $- Δ_{g}$ are the unit-speed geodesics; the singularities of $e^{i t - Δ_{g}}$ acting on a delta function propagate along these geodesics, giving the wave-trace formula and the link to the length spectrum (Duistermaat-Guillemin 1975, Colin de Verdière 1973). On a Lorentzian manifold, the bicharacteristics of $□_{g}$ are the null geodesics, and propagation of singularities forces the wave-front-set bound on the Klein-Gordon causal propagator and the Hadamard criterion. In semiclassical analysis, the bicharacteristics of $- h^{2} Δ + V (x)$ are the classical orbits of a particle in the potential $V$ , and propagation of singularities for the semiclassical operator (Dimassi-Sjöstrand Ch. 9) underwrites the Bohr-Sommerfeld quantisation condition and the WKB approximation. Each of these specialisations is controlled by the same Hörmander 1971 statement applied to the appropriate principal symbol, with the Hamiltonian field $H_{p_{m}}$ encoding the relevant classical-mechanical flow. The unification is structural: classical Hamiltonian mechanics on the cotangent bundle is the leading-order classical limit of every linear $Ψ$ DO of real principal type, and singularities of solutions follow this limit exactly.

Full proof set Master

Theorem (propagation of singularities, Hörmander 1971), proof. Given in the Intermediate-tier section: reduction to the normal-form first-order operator $D_{t} + R$ via Darboux-type conic canonical transformation lifted to a Fourier integral operator (Step 1); construction of the propagator $E (t)$ for the normal form via Picard iteration in the symbol class with $L^{2}$ -continuity from Calderón-Vaillancourt (Step 2); Egorov's theorem on conjugation $E (t) A E (t)^{- 1} \in Ψ^{k}$ with principal-symbol transport along the Hamiltonian flow (Step 3); and the microlocal-cutoff contradiction argument that transports the wave-front set along the bicharacteristic (Step 4). The full canonical exposition is Hörmander Vol. III §26.1 Theorem 26.1.1 ^{[Hörmander Vol. III §26]}, with the FIO machinery in Vol. IV §25. $□$

Theorem (Egorov's theorem, Hörmander Vol. III §25.1). *Let $R \in Ψ^{1} (X)$ be properly supported, formally self-adjoint, with real principal symbol $r \in S_{1, 0}^{1}$ of real principal type. Let $E (t) : L^{2} (X) \to L^{2} (X)$ be the unique unitary propagator with $\partial_{t} E + i R E = 0$ , $E (0) = I$ . Then for every $A \in Ψ^{k} (X)$ , the conjugate $A (t) = E (t) A E (t)^{- 1} \in Ψ^{k} (X)$ with principal symbol $σ_{k} (A (t)) = σ_{k} (A) \circ Φ_{- t}^{H_{r}}$ , where $Φ_{t}^{H_{r}}$ is the Hamiltonian flow of $r$ on $T^{*} X ∖ 0$ .*

Proof. Differentiate $A (t)$ in $t$ using the propagator equation: $$ \partial_t A(t) = \partial_t E(t) \cdot A E(t)^{-1} + E(t) A \cdot \partial_t E(t)^{-1} = -iR E(t) A E(t)^{-1} + i E(t) A E(t)^{-1} R = i[A(t), R]. $$ This is an operator ODE in $Ψ^{k}$ , with initial condition $A (0) = A$ . The commutator $[A (t), R]$ has principal symbol ${σ_{k} (A (t)), σ_{1} (R)}$ by the leading-order commutator formula in the symbol calculus (unit 02.14.02 Composition law: $[A, R]$ has symbol $\frac{1}{i} {a, r} + O (lower order)$ , with the factor of $\frac{1}{i}$ accounted for by the $i$ in the operator ODE). So at the principal-symbol level, $$ \partial_t \sigma_k(A(t)) = {\sigma_k(A(t)), r}. $$ This is exactly the equation of motion for $σ_{k} (A (t))$ pushed forward along the Hamiltonian flow of $- r$ , equivalently pulled back along the flow of $+ r$ at parameter $- t$ : $$ \sigma_k(A(t)) = \sigma_k(A) \circ \Phi_{-t}^{H_r}, $$ which is the claimed transport law. The remainder estimate at lower orders follows by iterating the same argument in the symbol-asymptotic expansion of the commutator. $□$

Proposition (commutator estimate at the principal level). For $A \in Ψ^{k}$ and $R \in Ψ^{1}$ with respective principal symbols $a$ and $r$ , the commutator $[A, R] \in Ψ^{k}$ with principal symbol $σ_{k} ([A, R]) = \frac{1}{i} {r, a}$ .

Proof. The composition law (unit 02.14.02 Theorem) gives $A R \in Ψ^{k + 1}$ with symbol asymptotic expansion $σ (A R) \sim a r + \frac{1}{i} \partial_{ξ} a \cdot \partial_{x} r + O (lower)$ and similarly $R A \sim r a + \frac{1}{i} \partial_{ξ} r \cdot \partial_{x} a + O (lower)$ . Subtract: the $a r = r a$ leading terms cancel, and the order- $k$ terms (one less than $k + 1$ ) give $$ \sigma_k([A, R]) = \frac{1}{i}(\partial_\xi a \cdot \partial_x r - \partial_\xi r \cdot \partial_x a) = \frac{1}{i}{a, r} = -\frac{1}{i}{r, a}. $$ The sign convention ${a, r} = \sum (\partial_{ξ} a \cdot \partial_{x} r - \partial_{x} a \cdot \partial_{ξ} r)$ gives $σ_{k} ([A, R]) = \frac{1}{i} {a, r}$ . $□$

Proposition (the wave-front set of $E^{+}$ for the flat d'Alembertian on $R^{1 + n}$ ). $WF (E^{+})$ is the bicharacteristic flow-out, restricted to forward time, of the wave-front set of $δ_{(0, 0)}$ over the spacetime origin intersected with the characteristic variety.

Proof. $E^{+}$ satisfies $□ E^{+} = δ_{(0, 0)}$ . By Hörmander's theorem applied to $u = E^{+}$ , $f = δ_{(0, 0)}$ , $P = □$ : $$ \mathrm{WF}(E^+) \setminus \mathrm{WF}(\delta_{(0,0)}) \subseteq \mathrm{Char}(\Box) = {(t, x; \tau, \xi) : \tau^2 = |\xi|^2, (\tau, \xi) \neq 0} $$ and is invariant under the Hamiltonian flow $H_{p} = - 2 τ \partial_{t} + 2 ξ \cdot \nabla_{x}$ of $p = - τ^{2} + ∣ ξ ∣^{2}$ .

$WF (δ_{(0, 0)}) = {(0, 0; τ, ξ) : (τ, ξ) \neq = 0}$ (the entire punctured fibre over the origin in $T^{*} R^{1 + n}$ ). Intersect with the characteristic variety: $WF (δ_{(0, 0)}) \cap Char (□) = {(0, 0; τ, ξ) : τ^{2} = ∣ ξ ∣^{2}, (τ, ξ) \neq = 0}$ , the punctured light cone over the origin.

Bicharacteristic flow-out: the integral curves of $H_{p}$ from $(0, 0; τ_{0}, ξ_{0})$ with $τ_{0}^{2} = ∣ ξ_{0} ∣^{2}$ are straight lines $(t (s), x (s); τ_{0}, ξ_{0})$ with $\dot{t} = - 2 τ_{0}$ , $\overset{x}{˙} = 2 ξ_{0}$ . Restricting to $s \geq 0$ for the retarded propagator (forward-time support, $t \geq 0$ ), and noting that the support constraint $supp E^{+} \subseteq {t \geq ∣ x ∣}$ forces the orientation $τ_{0} < 0$ (so $\dot{t} = - 2 τ_{0} > 0$ ), the flow-out covers exactly the forward light cone in cotangent space: ${(t, x; τ, ξ) : t = ∣ x ∣, τ^{2} = ∣ ξ ∣^{2}, τ < 0, ξ ∥ x}$ . This is the standard wave-front set of the retarded fundamental solution (Hörmander Vol. I §8.2 Example 8.2.4 ^{[Hörmander Vol. I]}). $□$

Proposition (microlocal Hadamard condition follows from propagation of singularities). Let $(M, g)$ be a globally hyperbolic Lorentzian manifold and let $ω_{2} \in D^{'} (M \times M)$ be the two-point function of a quasi-free state of the Klein-Gordon field $(□_{g} + m^{2}) ϕ = 0$ with $ω_{2} (x_{1}, x_{2}) = ω (ϕ (x_{1}) ϕ (x_{2}))$ . Then $ω_{2}$ satisfies $(□_{g, x_{1}} + m^{2}) ω_{2} = 0$ and $(□_{g, x_{2}} + m^{2}) ω_{2} = 0$ , and $WF (ω_{2})$ is contained in the joint characteristic set of $(□_{g} + m^{2})_{x_{1}}$ and $(□_{g} + m^{2})_{x_{2}}$ on $T^(M \times M) \setminus 0$, with bicharacteristic-flow invariance in each factor.*

Proof. The two-point function $ω_{2} = ω (ϕ \otimes ϕ)$ inherits the equation of motion from the field: $ω (ϕ (□ + m^{2}) ϕ \cdot ϕ) = 0$ and $ω (ϕ \cdot ϕ (□ + m^{2}) ϕ) = 0$ in the distribution sense, by the canonical commutation relations and the assumption that $ϕ$ solves the Klein-Gordon equation as an operator-valued distribution. So $(□_{g, x_{1}} + m^{2}) ω_{2} = 0$ and $(□_{g, x_{2}} + m^{2}) ω_{2} = 0$ on $M \times M$ . Applying Hörmander's theorem to each factor separately: $WF (ω_{2})$ in the $x_{1}$ -factor lies on $Char ((□ + m^{2})_{x_{1}}) = {(x_{1}, ξ_{1}) : g^{μν} ξ_{1 μ} ξ_{1 ν} = 0}$ , the null cone in $T_{x_{1}}^{*} M$ , with bicharacteristic-flow invariance; similarly in the $x_{2}$ -factor. The joint characteristic set is the pair of null covectors, one at each base point; the joint bicharacteristic-flow invariance is the parallel transport along null geodesics in each factor. Adding the positivity asymmetry $ξ_{1} \in \overline{V_{x_{1}}^{+}}$ from Hadamard / spectrum-condition arguments (which require more than propagation of singularities — they require positivity of the state) gives the Radzikowski formula. $□$

The remaining advanced theorems (Duistermaat-Hörmander 1972 complex-principal-type extension; Duistermaat's solvability theorem; semiclassical propagation) are stated without proof here; see the respective references for the technical arguments, all of which build on Egorov-style symbol transport along the relevant Hamiltonian flow as the load-bearing step.

Connections Master

Wave-front set of a distribution 02.14.01. Hörmander's theorem is the operator-side propagation statement for the wave-front-set calculus of 02.14.01. The pullback, product, and pseudolocality results of that unit set up the wave-front-set framework; propagation of singularities completes the picture by saying how the wave-front set transports under solution of a linear PDE of real principal type. Together with elliptic regularity, propagation of singularities exhausts the wave-front-set behaviour of $Ψ$ DO action: on the non-characteristic set ellipticity holds; on the characteristic set the Hamiltonian flow controls things.
Pseudo-differential operators on a manifold 02.14.02. The proof of Hörmander's theorem runs entirely in the symbol calculus of 02.14.02: the normal-form reduction is a conic canonical transformation lifted to a Fourier integral operator; the propagator $E (t)$ for the normal-form first-order operator is constructed via Picard iteration in the symbol class with Calderón-Vaillancourt $L^{2}$ -continuity; Egorov's theorem is a direct consequence of the composition law plus the principal-level commutator estimate. So 02.14.02 is the operator-theoretic ground on which 02.14.03 builds the wave-front-set transport.
Smooth manifold 03.02.01. The intrinsic statement of Hörmander's theorem on a smooth manifold uses the chart-independent definition of the wave-front set on $T^{*} M ∖ 0$ (unit 02.14.01) and of the principal symbol $p_{m}$ as a function on $T^{*} M ∖ 0$ (unit 02.14.02). The Hamiltonian flow $Φ_{t}^{H_{p_{m}}}$ is then a one-parameter family of conic diffeomorphisms of $T^{*} M ∖ 0$ , intrinsic to $M$ and the operator $P$ . The whole microlocal apparatus thus lives on the cotangent bundle, not on any particular chart.
(Forward link) Hadamard states via the wave-front-set criterion 13.09.03. The Radzikowski 1996 ^{[source pending]} microlocal characterisation of Hadamard states on a globally hyperbolic Lorentzian manifold is a direct consequence of propagation of singularities applied to the Klein-Gordon two-point function. The wave-front-set bound on the causal propagator $E = E^{+} - E^{-}$ and on quasi-free two-point functions is itself the propagation-of-singularities statement; Hadamard-ness adds the positivity asymmetry $ξ_{1} \in \overline{V^{+}}$ from spectrum-condition / positivity arguments. Hörmander's theorem is the microlocal-analytic prerequisite for the entire Hadamard-state programme of algebraic QFT on curved spacetimes (Gérard 2019).
(Forward link) Geodesic Hamiltonian flow 05.02.06. The geodesic flow on the cotangent bundle of a Riemannian manifold is the Hamiltonian flow of the principal symbol of the Laplace-Beltrami operator, in exactly the sense used by Hörmander's theorem. On a Lorentzian manifold the same construction with the wave operator $□_{g}$ gives the null-geodesic flow on the null cone in $T^{*} M$ . Propagation of singularities then unifies the Riemannian and Lorentzian pictures: in both cases, singularities of solutions to the relevant linear PDE propagate along the symbol-side geodesic flow. The Riemannian wave-trace formula (Duistermaat-Guillemin 1975) and the curved-spacetime Hadamard programme are two specialisations of this unified picture.

Historical & philosophical context Master

Hörmander introduced the wave-front set and propagation of singularities in the foundational paper "Fourier integral operators I," Acta Math. 127 (1971) 79–183 ^{[Hörmander 1971]}. The wave-front set was defined in §2 (with credit to Sato's hyperfunction-side singular spectrum of 1969); the bicharacteristic flow and propagation theorem were proved in §3 and §6.1, building on earlier work of Lax (1957 Duke Math. 24) and Ludwig (1960 CPAM 13) on geometric-optics asymptotic expansions of wave-equation solutions. The Fourier-integral-operator framework — operators with phase-function and amplitude data, with the Lagrangian-submanifold geometry encoded in the canonical relation $C_{P}^{+}$ above — was developed in the same paper as a parametrix construction for operators of real principal type, extending the Maslov index calculation (Maslov 1965) to a global microlocal-analytic theory. Duistermaat and Hörmander then extended the framework to operators of complex principal type and to Fourier integral operators with the full canonical-relation structure in "Fourier integral operators II," Acta Math. 128 (1972) 183–269 ^{[Duistermaat-Hörmander 1972]}. The Hörmander four-volume Analysis of Linear Partial Differential Operators, Springer 1983–1985, definitively consolidated the theory; Vol. III §26 is the canonical textbook treatment of propagation of singularities for real-principal-type operators. The book Vol. IV adds the FIO calculus.

The applications to physics matured in two waves. The first, geometric-optics derivation of WKB asymptotics for wave equations, was already implicit in Hörmander's 1971 paper and was developed by Duistermaat in Fourier Integral Operators, Birkhäuser 1996 ^{[Duistermaat 1996]}, and by Taylor in Pseudodifferential Operators, Princeton 1981 (Ch. 6) ^{[Taylor 1981]}. The semiclassical refinement, with explicit $h$ -power control on the subprincipal symbol, was developed by Helffer-Robert (1980s), Sjöstrand (Astérisque 1982 and many subsequent papers), and consolidated in Dimassi-Sjöstrand's Spectral Asymptotics in the Semi-Classical Limit, Cambridge LMS Lecture Note Series 268, 1999 ^{[Dimassi-Sjöstrand]}.

The second wave — the application to algebraic quantum field theory on curved spacetimes — was opened by Marek Radzikowski's "Micro-local approach to the Hadamard condition in quantum field theory on curved space-time," Comm. Math. Phys. 179 (1996) 529–553 ^{[Radzikowski 1996]}, which characterised Hadamard states on a globally hyperbolic Lorentzian manifold by a wave-front-set condition on the two-point function. The existence of Hadamard states had been proven by Fulling, Narcowich, and Wald in "Singularity structure of the two-point function in quantum field theory in curved spacetime, II," Ann. Phys. 136 (1981) 243 via a deformation argument; Radzikowski showed the microlocal characterisation is equivalent to the older Kay-Wald 1991 Hadamard-form condition (Kay-Wald, "Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon," Phys. Rep. 207 (1991) 49). Gérard's Microlocal Analysis of Quantum Fields on Curved Spacetimes, EMS 2019 ^{[Gérard 2019]}, is the canonical modern textbook consolidation of the entire programme, with Ch. 4 developing the symplectic and Hamiltonian-flow apparatus and Ch. 7 stating the Radzikowski criterion as the central definition. Brunetti, Fredenhagen, and Köhler then used the wave-front-set criterion to construct renormalised Wick polynomials and time-ordered products on a curved background ("The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes," Comm. Math. Phys. 180 (1996) 633), opening the perturbative algebraic QFT framework on globally hyperbolic spacetimes that has driven the field since.

Bibliography Master

@article{Hormander1971FIO,
  author  = {H{\"o}rmander, Lars},
  title   = {Fourier integral operators. {I}},
  journal = {Acta Math.},
  volume  = {127},
  year    = {1971},
  pages   = {79--183}
}

@article{DuistermaatHormander1972,
  author  = {Duistermaat, J. J. and H{\"o}rmander, Lars},
  title   = {Fourier integral operators. {II}},
  journal = {Acta Math.},
  volume  = {128},
  year    = {1972},
  pages   = {183--269}
}

@book{HormanderVolIII,
  author    = {H{\"o}rmander, Lars},
  title     = {The Analysis of Linear Partial Differential Operators, Vol. {III}: Pseudo-Differential Operators},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {274},
  year      = {1985}
}

@book{HormanderVolIV,
  author    = {H{\"o}rmander, Lars},
  title     = {The Analysis of Linear Partial Differential Operators, Vol. {IV}: Fourier Integral Operators},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {275},
  year      = {1985}
}

@book{Duistermaat1996FIO,
  author    = {Duistermaat, J. J.},
  title     = {Fourier Integral Operators},
  publisher = {Birkh{\"a}user},
  year      = {1996}
}

@book{Taylor1991Tools,
  author    = {Taylor, Michael E.},
  title     = {Pseudodifferential Operators and Nonlinear {PDE}},
  publisher = {Birkh{\"a}user},
  series    = {Progress in Mathematics},
  volume    = {100},
  year      = {1991}
}

@book{DimassiSjostrand1999,
  author    = {Dimassi, Mouez and Sj{\"o}strand, Johannes},
  title     = {Spectral Asymptotics in the Semi-Classical Limit},
  publisher = {Cambridge University Press},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {268},
  year      = {1999}
}

@article{Radzikowski1996,
  author  = {Radzikowski, Marek J.},
  title   = {Micro-local approach to the {H}adamard condition in quantum field theory on curved space-time},
  journal = {Comm. Math. Phys.},
  volume  = {179},
  year    = {1996},
  pages   = {529--553}
}

@article{FullingNarcowichWald1981,
  author  = {Fulling, S. A. and Narcowich, F. J. and Wald, R. M.},
  title   = {Singularity structure of the two-point function in quantum field theory in curved spacetime, {II}},
  journal = {Ann. Phys.},
  volume  = {136},
  year    = {1981},
  pages   = {243--272}
}

@article{KayWald1991,
  author  = {Kay, B. S. and Wald, R. M.},
  title   = {Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate {K}illing horizon},
  journal = {Phys. Rep.},
  volume  = {207},
  year    = {1991},
  pages   = {49--136}
}

@book{Gerard2019Microlocal,
  author    = {G{\'e}rard, Christian},
  title     = {Microlocal Analysis of Quantum Fields on Curved Spacetimes},
  publisher = {European Mathematical Society},
  series    = {ESI Lectures in Mathematics and Physics},
  year      = {2019}
}

@article{BrunettiFredenhagenKohler1996,
  author  = {Brunetti, R. and Fredenhagen, K. and K{\"o}hler, M.},
  title   = {The microlocal spectrum condition and {W}ick polynomials of free fields on curved spacetimes},
  journal = {Comm. Math. Phys.},
  volume  = {180},
  year    = {1996},
  pages   = {633--652}
}

@book{BarGinouxPfaffle2007,
  author    = {B{\"a}r, Christian and Ginoux, Nicolas and Pf{\"a}ffle, Frank},
  title     = {Wave Equations on {L}orentzian Manifolds and Quantization},
  publisher = {European Mathematical Society},
  series    = {ESI Lectures in Mathematics and Physics},
  year      = {2007}
}

@book{DyatlovZworski2019,
  author    = {Dyatlov, Semyon and Zworski, Maciej},
  title     = {Mathematical Theory of Scattering Resonances},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {200},
  year      = {2019}
}

Prerequisites

02.14.01
02.14.02
03.02.01

Tier anchors

beginner: the rigorous microlocal statement that singularities of wave-equation solutions ride along light rays; for a Lorentzian wave operator, along null geodesics
intermediate: Hörmander *The Analysis of Linear Partial Differential Operators* Vol. III §26; Taylor *Tools for PDE* Ch. 4; Dimassi-Sjöstrand *Spectral Asymptotics in the Semi-Classical Limit* Ch. 9
master: Hörmander 1971 *Acta Math.* 127 §3; Duistermaat-Hörmander 1972 *Acta Math.* 128 §6; Hörmander Vol. III §26; Gérard *Microlocal Analysis of Quantum Fields on Curved Spacetimes* Ch. 4; Sjöstrand-Zworski *Lecture Notes on Resolvents and Trapping*

References

Hörmander — Fourier integral operators I · *Acta Math.* 127 (1971) 79–183; §3 (Hamiltonian flow and bicharacteristics); §6 Theorem 6.1.1 (propagation of singularities for operators of real principal type)
Duistermaat-Hörmander — Fourier integral operators II · *Acta Math.* 128 (1972) 183–269; §6 (propagation of singularities for operators of real principal type, sharpened to a sharper-than-WF statement on the principal-type characteristic set)
Hörmander — The Analysis of Linear Partial Differential Operators, Vol. III · §26 (Propagation of singularities); §26.1 Theorem 26.1.1 (the modern textbook statement); §26.4 (geometric optics applications)
Gérard — Microlocal Analysis of Quantum Fields on Curved Spacetimes · Ch. 4 (Symplectic geometry of $T^*M$, Hamiltonian flow of the principal symbol on a Lorentzian background); EMS Lectures, Zürich 2019
Taylor — Tools for PDE · AMS Mathematical Surveys and Monographs 81, Ch. 4 (Propagation of singularities and parametrices for operators of real principal type)
Dimassi-Sjöstrand — Spectral Asymptotics in the Semi-Classical Limit · London Math. Soc. Lecture Note Series 268, Cambridge 1999, Ch. 9 (Propagation of singularities in the semiclassical setting via Egorov's theorem)
Radzikowski — Micro-local approach to the Hadamard condition in quantum field theory on curved space-time · *Comm. Math. Phys.* 179 (1996) 529–553; the Hadamard-state criterion as a wave-front-set condition on the two-point function, downstream of propagation of singularities for the Klein-Gordon operator
Bär-Ginoux-Pfäffle — Wave Equations on Lorentzian Manifolds and Quantization · ESI Lectures in Mathematics and Physics (EMS 2007), Ch. 3; free arXiv preprint at 0806.1036; classical-PDE companion treating the wave equation on a globally hyperbolic spacetime

Estimated time

beginner: 18m
intermediate: 42m
master: 80m