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\begin{document}

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\section*{formal adjoint differential operator}

\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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Over a [[smooth manifold]] $\Sigma$ of [[dimension]] $p+1$, let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] and $\tilde E^\ast \coloneqq E^\ast \otimes_{\Sigma} \wedge_\Sigma^{p+1} T^\ast \Sigma$ the [[tensor product of vector bundles]] of the [[dual vector bundle]] with the [[differential n-form|differential (p+1)-form bundle]].

\begin{defn}
\label{FormallyAdjointDifferentialOperators}\hypertarget{FormallyAdjointDifferentialOperators}{}
\textbf{(formally adjoint differential operators)}

Two [[differential operators]]

\begin{displaymath}
P, P^\ast
   \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow
  \Gamma_\Sigma(\tilde E^\ast)
\end{displaymath}
are called \emph{formally adjoint} if there exists a [[bilinear map|bilinear]] [[differential operator]]

\begin{equation}
K \;\colon\;
  \Gamma_\Sigma(E) \otimes \Gamma_\Sigma(E)
    \longrightarrow
   \Gamma_\Sigma(\wedge^{p} T^\ast \Sigma)
\label{FormallyAdjointDifferentialOperatorWitness}\end{equation}
such that for all $\Phi_1, \Phi_2 \in \Gamma_\Sigma(E)$ we have

\begin{displaymath}
P(\Phi_1) \cdot \Phi_2
  -
  \Phi_1 \cdot P^\ast(\Phi_2)
   \;=\;
  d K(\Phi_1, \Phi_2)
\end{displaymath}
This implies by [[Stokes' theorem]], in the case of [[compact support]], that under an [[integral]] $P$ and $P^\ast$ are related via [[integration by parts]].

\end{defn}
(\hyperlink{Khavkine14}{Khavkine 14, def. 2.4})

See also (\hyperlink{VinogradovKrasilshchik99}{Vinogradov-Krasilshchik 99, chapter 5, \S{}2.3})

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{FormallySelfAdjointKleinGordonOperator}\hypertarget{FormallySelfAdjointKleinGordonOperator}{}
\textbf{([[Klein-Gordon operator]] is [[formally self-adjoint differential operator]])}

Let $\Sigma = \mathbb{R}^{p,1}$ be [[Minkowski spacetime]] with [[Minkowski metric]] $\eta$ and let $E \coloneqq \Sigma \times \mathbb{R}$ be the [[trivial line bundle]]. The canonical [[volume form]] $dvol_\Sigma$ induces an [[isomorphism]] $\tilde E^\ast \simeq E$.

Consider then the [[Klein-Gordon operator]]

\begin{displaymath}
(\Box - m^2)
    \;\colon\; 
  \Gamma_\Sigma(\Sigma \times \mathbb{R})
     \longrightarrow
  \Gamma_\Sigma(\Sigma \times \mathbb{R}) \otimes \langle dvol_\Sigma\rangle
  \,.
\end{displaymath}
This is its own formal adjoint (def. \ref{FormallyAdjointDifferentialOperators}) witnessed by the bilinear differential operator \eqref{FormallyAdjointDifferentialOperatorWitness} given by

\begin{displaymath}
K(\Phi_1, \Phi_2)
  \;\coloneqq\;
  \left(
    \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
    -
    \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
  \right)
  \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
  \,.
\end{displaymath}
\end{example}
\begin{proof}
\begin{displaymath}
\begin{aligned}
    d K(\Phi_1, \Phi_2)
    & =
    d
    \left(
      \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
      -
      \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
    \right)
    \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
    \\
    &=
    \left(
    \left(
      \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
      + 
      \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu}
    \right)
    -
    \left(
      \eta^{\mu \nu}
      \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu}  
      + 
      \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
    \right)
    \right)
    dvol_\Sigma
    \\
    & =
    \left(
      \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
      -
      \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
    \right)
    dvol_\Sigma    
    \\
    & =
    \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2)
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{example}
\label{DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}\hypertarget{DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}{}
\textbf{([[Dirac operator]] on [[Dirac spinors]] is [[formally self-adjoint differential operator]])}

The [[Dirac operator]] on [[Dirac spinors]] is a [[formally self-adjoint differential operator|formally anti-self adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}):

\begin{displaymath}
D^\ast = - D
  \,.
\end{displaymath}
\end{example}
\begin{proof}
We spell out the proof over [[Minkowski spacetime]]. Then general case is a straightforward generalization

Regard the Dirac operator as taking values in the [[dual vector bundle|dual]] [[spin bundle]] by using the [[Dirac conjugate]] $\overline{(-)}$:

\begin{displaymath}
\itexarray{
    \Gamma_\Sigma(\Sigma \times S)
      &\overset{D}{\longrightarrow}&  
    \Gamma_\Sigma(\Sigma \times S^\ast)
    \\
   \Psi &\mapsto&
   \overline{(-)} \gamma^\mu \partial_\mu  \Psi
  }
\end{displaymath}
Then we need to show that there is $K(-,-)$ such that for all [[pairs]] of [[spinor]] [[sections]] $\Psi_1, \Psi_2$ we have

\begin{displaymath}
\overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1)
  - 
  \overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2)
  \;=\;
  d K(\psi_1, \psi_2)
  \,.
\end{displaymath}
But the spinor-to-vector pairing is symmetric (see at \emph{[[spin representation]]}), hence this is equivalent to

\begin{displaymath}
\overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2
  +
  \overline{\Psi_1}\gamma^\mu (\partial_\mu \Psi_2)
  \;=\;
  d K(\psi_1, \psi_2)
  \,.
\end{displaymath}
By the [[product law]] of [[differentiation]], this is solved, for all $\Psi_1, \Psi_2$, by

\begin{displaymath}
K(\Psi_1, \Psi_2) 
    \;\coloneqq\;
  \left( \overline{\Psi_1} \gamma^\mu \Psi_2\right)
  \, 
  \iota_{\partial_\mu} dvol
  \,.
\end{displaymath}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[generalized solution of a differential equation]]


\item [[Green hyperbolic differential operator]]



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

\begin{itemize}%
\item [[Peter Olver]], chapter 5.3, around p. 328-330 of \emph{Applications of Lie groups to differential equations}, Springer; \emph{Equivalence, invariants, and symmetry}, Cambridge Univ. Press 1995.


\item [[Alexandre Vinogradov]], I. S. Krasilshchik (eds.) \emph{Symmetries and Conservation Laws for Differential Equations of Mathematical Physics}, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. (\href{ftp://softbank.iust.ac.ir/MathBooks/V/Vinogradov%20-%20Symmetries%20and%20Conservation%20Laws%20for%20Differential%20equations%20of%20mathematical%20physics.pdf}{pdf})


\item [[Igor Khavkine]], \emph{Covariant phase space, constraints, gauge and the Peierls formula}, Int. J. Mod. Phys. A, 29, 1430009 (2014) (\href{https://arxiv.org/abs/1402.1282}{arXiv:1402.1282})



\end{itemize}
[[!redirects formal adjoint differential operators]]

[[!redirects formally adjoint differential operator]] [[!redirects formally adjoint differential operators]]

[[!redirects formally self-adjoint differential operator]] [[!redirects formally self-adjoint differential operators]]



\end{document}