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\begin{document}

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\section*{bicharacteristic flow}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{OfTheKleinGordonOperator}{Of the Klein-Gordon operator}\dotfill \pageref*{OfTheKleinGordonOperator} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{propagation_of_singularities}{Propagation of singularities}\dotfill \pageref*{propagation_of_singularities} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{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 \href{symbol+of+a+differential+operator#BicharacteristicFlow}{this example}). With the [[cotangent bundle]] canonically regarded as a [[symplectic manifold]], let

\begin{displaymath}
v_{q(D)} \in \Gamma\left(T\left(T^\ast X\right) \right)
\end{displaymath}
be the corresponding [[Hamiltonian vector field]].

\begin{defn}
\label{BicharacteristicFlow}\hypertarget{BicharacteristicFlow}{}
The \emph{bicharacteristic flow} of $D$ is the [[Hamiltonian flow|Hamiltonian]] [[flow of a vector field|flow]] of the [[Hamiltonian vector field]] $v_{q(D)}$ inside the submanifold defined by $q = 0$. Moreover:

\begin{enumerate}%
\item A single [[flow line]] in $T^\ast X$ is called a \emph{bicharacteristic strip} of $D$,


\item the [[projection]] of such to a [[curve]] in $X$ is called a \emph{bicharacteristic curve}.


\item The [[relation]] $C$ on $T^\ast X$ given by

\begin{displaymath}
\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)
\end{displaymath}
is called the \emph{bicharacteristic relation}.



\end{enumerate}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{OfTheKleinGordonOperator}{}\subsubsection*{{Of the Klein-Gordon operator}}\label{OfTheKleinGordonOperator}

\begin{example}
\label{BicharachteristicFlowOfKleinGordonOperator}\hypertarget{BicharachteristicFlowOfKleinGordonOperator}{}
\textbf{(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. \ref{BicharacteristicFlow}) 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.

\end{example}
(\hyperlink{Radzikowski96}{Radzikowski 96, prop. 4.2 and below (6)})

Specifically on [[Minkowski spacetime]]:

\begin{example}
\label{}\hypertarget{}{}
\textbf{([[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]]

\begin{displaymath}
D
  \;=\;
  \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu}
  -
  \left(\tfrac{m c}{\hbar}\right)^2
  \,.
\end{displaymath}
Its [[principal symbol]] is the function

\begin{displaymath}
\itexarray{
    T^\ast \mathbb{R}^{p,1}
      &\overset{q}{\longrightarrow}&
    \mathbb{R}
    \\
    (x,k) &\mapsto& \eta^{\mu \nu} k_\mu k_\nu
  }
\end{displaymath}
Hence $q(k) = 0$ is the condition that the [[wave vector]] $k$ be [[lightlike]].

The [[Hamiltonian vector field]] corresponding to $q$ is

\begin{displaymath}
\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}
\end{displaymath}
in that

\begin{displaymath}
\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}
\end{displaymath}
It follows that the [[bicharacteristic curves]] are precisely the [[lightlike]] [[curves]]

\begin{displaymath}
\itexarray{
    \mathbb{R} &\overset{\gamma_k}{\longrightarrow}& \mathbb{R}^{p,1}
    \\
    \tau &\mapsto& (\gamma^\mu(0) +  \tau k^\mu)
  }
\end{displaymath}
and the corresponding [[bicharacteristic strips]] are these with their lightlike contangent vector constantly carried along

\begin{displaymath}
\itexarray{
    \mathbb{R} &\overset{\gamma_k}{\longrightarrow}& T^\ast\mathbb{R}^{p,1}
    \\
    \tau &\mapsto& \left((\gamma^\mu(0) +  \tau k^\mu),(k_\mu)\right)
  }
\end{displaymath}
\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{propagation_of_singularities}{}\subsubsection*{{Propagation of singularities}}\label{propagation_of_singularities}

The \emph{[[propagation of singularities theorem]]} says that the [[wave front set]] of a [[distribution|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.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Johann Duistermaat]], [[Lars Hörmander]], \emph{Fourier integral operators II}, Acta Mathematica 128, 183-269, 1972 (\href{https://projecteuclid.org/euclid.acta/1485889724}{Euclid})

\end{itemize}
Review in the context of the [[free field|free]] [[scalar field]] on [[globally hyperbolic spacetimes]] (with $Q$ the [[wave operator]]/[[Klein-Gordon operator]]) is in

\begin{itemize}%
\item [[Marek Radzikowski]], \emph{Micro-local approach to the Hadamard condition in quantum field theory on curved space-time}, Commun. Math. Phys. 179 (1996), 529--553 (\href{http://projecteuclid.org/euclid.cmp/1104287114}{Euclid})

\end{itemize}
[[!redirects bicharacteristic flows]]

[[!redirects bicharacteristic strip]] [[!redirects bicharacteristic strips]]

[[!redirects bicharacteristic curve]] [[!redirects bicharacteristic curves]]



\end{document}