nLab bicharacteristic flow

Definition

Let $X$ be a smooth manifold and let $D$ be a differential operator on (smooth sections of) the trivial line bundle over $X$ (or more generally a properly supported pseudo-differential operator). Then the principal symbol $q(D)$ of $D$ is equivalently a smooth function on the cotangent bundle $T^\ast X$ (by this example). With the cotangent bundle canonically regarded as a symplectic manifold, let

$$v_{q(D)} \in \Gamma\left(T\left(T^\ast X\right) \right)$$

be the corresponding Hamiltonian vector field.

Definition

The bicharacteristic flow of $D$ is the Hamiltonian flow of the Hamiltonian vector field $v_{q(D)}$ inside the submanifold defined by $q = 0$. Moreover:

1. A single flow line in $T^\ast X$ is called a bicharacteristic strip of $D$,

2. the projection of such to a curve in $X$ is called a bicharacteristic curve.

3. The relation $C$ on $T^\ast X$ given by

   $$\left((x_1,k_1) \sim (x_2, k_2)\right) \;\coloneqq\; \left( q(x_i,k_i) = 0 \;\;\text{and}\;\; (x_1,k_1) \,\text{is connected to}\, (x_2,k_2) \,\text{by a bicharacteristic strip} \right)$$

   is called the bicharacteristic relation.

Examples

Of the Klein-Gordon operator

Example

(bicharacteristic curves of wave operator/Klein-Gordon operators are the lightlike geodesics)

Let $(X,g)$ be a Lorentzian manifold and let $D \coloneqq \Box_g - m^2$ be its wave operator/Klein-Gordon operator.

Then the bicharacteristic curves of $D$ (def. ) are precisely the lightlike geodesics of $(X,e)$, and the bicharacteristic strips are precisely these geodesices with their cotangent vectors.

Accordingly two cotangent vectors are bicharacteristically related $(x_1,k_1) \sim (x_2,k_2)$ precisely if there is a lightlike geodesic connecting the points, with $k_1$ and $k_2$ the corresponding cotangents, hence one the result of parallel transport of the other along the geodesic.

(Radzikowski 96, prop. 4.2 and below (6))

Specifically on Minkowski spacetime:

Example

(bicharacteristic flow of Klein-Gordon operator on Minkowski spacetime)

Let $\mathbb{R}^{p,1}$ be Minkowski spacetime of dimension $p+1$ consider the Klein-Gordon operator

$$D \;=\; \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} - \left(\tfrac{m c}{\hbar}\right)^2 \,.$$

Its principal symbol is the function

$$\array{ T^\ast \mathbb{R}^{p,1} &\overset{q}{\longrightarrow}& \mathbb{R} \\ (x,k) &\mapsto& \eta^{\mu \nu} k_\mu k_\nu }$$

Hence $q(k) = 0$ is the condition that the wave vector $k$ be lightlike.

The Hamiltonian vector field corresponding to $q$ is

$$\begin{aligned} v_q & = -\tfrac{1}{2} \eta^{\mu \nu} k_\mu \partial_{x^\nu} \\ & = -\tfrac{1}{2} k^\mu \partial_{x^\mu} \end{aligned}$$

in that

$$\begin{aligned} \iota_{v_q} d k_\mu \wedge d x^\mu &= \tfrac{1}{2} \eta^{\mu \nu} k_\mu d k_\mu \\ & = d q(k) \end{aligned}$$

It follows that the bicharacteristic curves are precisely the lightlike curves

$$\array{ \mathbb{R} &\overset{\gamma_k}{\longrightarrow}& \mathbb{R}^{p,1} \\ \tau &\mapsto& (\gamma^\mu(0) + \tau k^\mu) }$$

and the corresponding bicharacteristic strips are these with their lightlike contangent vector constantly carried along

$$\array{ \mathbb{R} &\overset{\gamma_k}{\longrightarrow}& T^\ast\mathbb{R}^{p,1} \\ \tau &\mapsto& \left((\gamma^\mu(0) + \tau k^\mu),(k_\mu)\right) }$$

Properties

Propagation of singularities

The propagation of singularities theorem says that the wave front set of a distributional solution to the differential equation of a sufficiently nice differential operator (or generally of a properly supported pseudo-differential operator) is preserved by the bicharacteristic flow.

References

• Johann Duistermaat, Lars Hörmander, Fourier integral operators II, Acta Mathematica 128, 183-269, 1972 (Euclid)

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)