# nLab bicharacteristic flow

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

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.

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)