nLab
bicharacteristic flow

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 XX be a smooth manifold and let DD be a differential operator on (smooth sections of) the trivial line bundle over XX (or more generally a properly supported pseudo-differential operator). Then the principal symbol q(D)q(D) of DD is equivalently a smooth function on the cotangent bundle T *XT^\ast X (by this example). With the cotangent bundle canonically regarded as a symplectic manifold, let

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

be the corresponding Hamiltonian vector field.

Definition

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

  1. A single flow line in T *XT^\ast X is called a bicharacteristic strip of DD,

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

  3. The relation CC on T *XT^\ast X given by

    ((x 1,k 1)(x 2,k 2))(q(x i,k i)=0and(x 1,k 1)is connected to(x 2,k 2)by a bicharacteristic strip) \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)(X,g) be a Lorentzian manifold and let D gm 2D \coloneqq \Box_g - m^2 be its wave operator/Klein-Gordon operator.

Then the bicharacteristic curves of DD (def. ) are precisely the lightlike geodesics of (X,e)(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)(x 2,k 2)(x_1,k_1) \sim (x_2,k_2) precisely if there is a lightlike geodesic connecting the points, with k 1k_1 and k 2k_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 p,1\mathbb{R}^{p,1} be Minkowski spacetime of dimension p+1p+1 consider the Klein-Gordon operator

D=η μνx μx ν(mc) 2. 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

T * p,1 q (x,k) η μνk μk ν \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)=0q(k) = 0 is the condition that the wave vector kk be lightlike.

The Hamiltonian vector field corresponding to qq is

v q =12η μνk μ x ν =12k μ x μ \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

ι v qdk μdx μ =12η μνk μdk μ =dq(k) \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

γ k p,1 τ (γ μ(0)+τk μ) \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

γ k T * p,1 τ ((γ μ(0)+τk μ),(k μ)) \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

Review in the context of the free scalar field on globally hyperbolic spacetimes (with QQ 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)

Last revised on August 27, 2018 at 14:58:09. See the history of this page for a list of all contributions to it.