Contents

Idea

In microlocal analysis the propagation of singularities theorem (Duistermaat-Hörmander 72, section 6.1) characterizes the wave front set of a distributional solution to a partial differential equation in terms of the principal symbol of the corresponding differential operator: It says that the singularity wave fronts of the solution propagate along the Hamiltonian flow of the principal symbol of the differential operator (its bicharacteristic flow).

Hence this theorem makes fully manifest how the concept of wave front set enhances that of singular support by adding directional information about the singularities.

One implication is the proof of existence of solutions to suitable (possibly inhomogeneous) differential equations (Duistermaat-Hörmander 72, section 6.3).

For example the proof of existence the Feynman propagator on a Lorentzian manifold is obtained this way (Duistermaat-Hörmander 72, theorem 6.5.3 and beginning of section 6.6, Radzikowski 96, section 4).

Definitions

Definition

(properly supported peudo-differential operator)

A pseudo-differential operator $Q$ on a manifold $X$ is called properly supported if for each compact subset $K \subset X$ there exists a compact subset $K' \subset X$ such that for $u$ a distribution with support in $K$ it follows that the derivative of distributions $Q u$ has support in $K'$ and such that $u\vert_{K'} = 0$ implies $(Q u)\vert_{K} = 0$.

(Hörmander 85 (18.1.21) recalled e.g. in Radzikowski 96. p. 8,9)

Example

(differential operators are properly supported pseudo-differential operators)

Every ordinary differential operator, regarded as a pseudo-differential operator, is properly supported (def. ), since differential operators do not increase support.

Definition

(symbol order)

A smooth function $q$ on a cotangent bundle (e.g. symbol of a differential operator) is of order $m$ (and type $1,0$, denoted $q \in S^m = S^m_{1,0}$), for $m \in \mathbb{N}$, if on each coordinate chart $((x^i), (k_i))$ we have that for every compact subset $K$ of the base space and all multi-indices $\alpha$ and $\beta$, there is a real number $C_{\alpha, \beta,K } \in \mathbb{R}$ such that the absolute value of the partial derivatives of $q$ is bounded by

$\left\vert \frac{\partial^\alpha}{\partial k_\alpha} \frac{\partial^\beta}{\partial x^\beta} q(x,k) \right\vert \;\leq\; C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}$

for all $x \in K$ and all cotangent vectors $k$ to $x$.

A Fourier integral operator $Q$ is of symbol class $L^m = L^m_{1,0}$ if it is of the form

$Q f (x) \;=\; \int \int e^{i k \cdot (x - y)} q(x,y,k) f(y) \, d y d k$

with symbol $q$ of order $m$, in the above sense.

Example

The wave operator/Klein-Gordon operator on Minkowski spacetime is of class $L^2$, according to def. .

Statement

Proposition

(propagation of singularities theorem)

Let $Q$ be a pseudo-differential operator on some smooth manifold $X$ which is properly supported (def. ) and of symbol class $L^m$ (def. ) with real principal symbol $q$ that is homogeneous of degree $m$.

For $u \in \mathcal{D}'(X)$ a distribution with $Q u = f$, then the complement of the wave front set of $u$ by that of $f$ is contained in the set of covectors on which the principal symbol $q$ vanishes:

$WF(u) \setminus WF(f) \;\subset\; q^{-1}(0) \,.$

Moreover, $WF(u)$ is invariant under the bicharacteristic flow induced by the Hamiltonian vector field of $q$ with respect to the canonical symplectic manifold structure on the cotangent bundle (here).

(Duistermaat-Hörmander 72, theorem 6.1.1, recalled e.g. as Radzikowski 96, theorem 4.6

Examples

Example

For $(X,e)$ a globally hyperbolic spacetime and $P$ a hyperbolic differential operator such as the wave operator/Klein-Gordon operator, then the wave front set of any solution $f$ to $P f = 0$ is a union of cotangent vectors along lightlike geodesics .

Proof

This follows by prop. by the fact that the bicharacteristic strips of the wave operator/Klein-Gordon operator are cotangent vectors of lightlike geodesics, by this example.

References

The theorem is due to

discussed in

• Lars Hörmander, The analysis of partial differential operators III, Springer 1985

Review in the context of the free scalar field on globally hyperbolic spacetimes (with $Q$ the wave operator/Klein-Gordon operator) is in

• Marek Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553 (Euclid)

which otherwise discusses the corresponding Hadamard distributions.