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\section*{Klein-Gordon equation}

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

\hypertarget{variational_calculus}{}\paragraph*{{Variational calculus}}\label{variational_calculus}

[[!include variational calculus - contents]]

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{formal_selfadjointness}{Formal self-adjointness}\dotfill \pageref*{formal_selfadjointness} \linebreak
\noindent\hyperlink{bicharacteristic_flow_and_propagation_of_singularities}{Bicharacteristic flow and propagation of singularities}\dotfill \pageref*{bicharacteristic_flow_and_propagation_of_singularities} \linebreak
\noindent\hyperlink{FundamentalSolutions}{Fundamental solutions}\dotfill \pageref*{FundamentalSolutions} \linebreak
\noindent\hyperlink{relation_to_schrdinger_equation}{Relation to Schr\"o{}dinger equation}\dotfill \pageref*{relation_to_schrdinger_equation} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Klein-Gordon equation} is the [[linear differential equation|linear]] [[partial differential equation]] which is the [[equation of motion]] of a [[free field|free]] [[scalar field|scalar]] [[field (physics)|field]] of possibly non-vanishing [[mass]] $m$ on some (possibly [[curved spacetime|curved]]) [[spacetime]] ([[Lorentzian manifold]]): it is the relativistic [[wave equation]] with inhomogenety the mass $m^2$.

The structure of the Klein-Gordon equation appears also in the [[equations of motion]] of richer [[field (physics)|fields]] than just [[scalar fields]], where now the underlying [[field bundle]] may more generally be some [[vector bundle]]. Therefore the [[fundamental solutions]] of the Klein-Gordon equation, called the \emph{[[propagators]]} (see \hyperlink{FundamentalSolutions}{below}) pervades all of relativistic [[perturbative quantum field theory]].

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

Given a [[spacetime]] $(X,g)$ (a [[pseudo-Riemannian manifold]]) and a [[real number]] $m \in \mathbb{R}_{\geq 0}$, then the \emph{Klein-Gordon equation} is the [[differential equation]] on [[smooth functions]] $\phi \colon X \to \mathbb{R}$ given by

\begin{displaymath}
\left(
    \Box_g 
      - 
    \left(
      \tfrac{m c}{\hbar}
    \right)^2
  \right) \phi 
   \;=\; 0
  \,,
\end{displaymath}
where $\Box_g$ denotes the \emph{[[wave operator]]} on $(X,g)$ (the analog of the [[Laplace operator]] in [[Lorentzian geometry]]) and where $\tfrac{m c }{\hbar}$ is for the purpose of pure [[PDE]] theory just a [[real number]], while regarded as equipped with [[physical units]] it is the inverse \emph{[[Compton wavelength]]} for [[mass]] $m$.

This is the [[equation of motion]] of the [[free field|free]] [[scalar field]] on $X$, of [[mass]] $m$ and subject to a [[background field]] of [[gravity]] as encoded in the metric $g$.

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

If $(X,g) = \mathbb{R}^{p,1}$ is [[Minkowski spacetime]] equipped with its canonical [[coordinate functions]] $x^0 = c t$ and $\{x^i\}_{i = 1}^p$, then the Klein-Gordon equation reads as follows (using [[Einstein summation convention]])

\begin{displaymath}
\left(
    \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu}
      - 
    \left(
      \tfrac{m c}{\hbar}
    \right)^2
  \right) \phi 
   \;=\; 
  0
\end{displaymath}
hence

\begin{displaymath}
\left(
    -\tfrac{1}{c^2} \frac{\partial^2}{\partial t^2}
    +
    \underoverset{i = 1}{p}{\sum}\frac{\partial}{\partial x^i} \frac{\partial}{\partial x^i}
      - 
    \left(
      \tfrac{m c}{\hbar}
    \right)^2
  \right) \phi 
   \;=\; 
  0
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{formal_selfadjointness}{}\subsubsection*{{Formal self-adjointness}}\label{formal_selfadjointness}

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

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 differential operator|formal adjoint]] witnessed by the bilinear differential operator 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}
\hypertarget{bicharacteristic_flow_and_propagation_of_singularities}{}\subsubsection*{{Bicharacteristic flow and propagation of singularities}}\label{bicharacteristic_flow_and_propagation_of_singularities}

The [[bicharacteristic strips]] of the Klein-Gordon operator are [[cotangent vectors]] along [[lightlike]] [[geodesics]] (\href{bicharacteristic+flow#BicharachteristicFlowOfKleinGordonOperator}{this example}).

\hypertarget{FundamentalSolutions}{}\subsubsection*{{Fundamental solutions}}\label{FundamentalSolutions}

On a [[globally hyperbolic spacetime]] $M$ the Klein-Gordon equation has unique advanced and retarded [[Green functions]], $\Delta_R \in \mathcal{D}'(M\times M)$ and $\Delta_A \in \mathcal{D}'(M\times M)$ respectively.

The [[advanced and Green functions]] are uniquely distinguished by their [[support of a distribution|support]] properties. Namely, $(x,y) \in \operatorname{supp} \Delta_R$ only if $x$ is in the [[causal future]] of $y$, while $(x,y) \in \operatorname{supp} \Delta_A$ only if $x$ is in the [[causal past]] of $y$.

Their difference $\Delta_S = \Delta_R - \Delta_A$ is a bisolution known as the [[causal propagator]], which is the [[Peierls bracket]] which gives the [[Poisson bracket]] on the [[covariant phase space]] of the [[free field|free]] [[scalar field]]. This in turn defines the [[Wick algebra]] of the free scalar field, which yields the [[quantization]] of the free scalar field to a [[quantum field theory]].

Other important Green functions or bisolutions include any (anti-)[[Feynman propagator]] ($\Delta_{\bar{F}}$) $\Delta_F$ and [[Hadamard propagator]]. Unfortunately, it is not possible to identify them by a simple support condition. On Minkowski space, they are identified by the support of their Fourier transform. On curved spacetimes, there are two possibilities. One specifies the asymptotic expansion $\Delta(x,y)$ in a geodesically convex neighborhood of the diagonal $x=y$ to be of a special \emph{Hadamard form}. The other specifies constraints on the [[wavefront set]] $WF(\Delta)$. The possibilities were proven to be equivalent in \hyperlink{Radzikowski96}{Radzikowski 96}, which made essential use of the relevant notions of microlocal analysis and of \emph{distinguished parametrices} introduced in \hyperlink{DuistermaatHoermander72}{DuistermaatH\"o{}rmander 72}.

According to \hyperlink{Radzikowski96}{Radzikowski 96}, the constraints on the wavefront sets of important Green functions and bisolutions can be diagrammatically illustrated as follows below. Note that the primed [[wavefront set]] of a distribution on $M\times M$ is defined as $WF'(\Delta) = \{ (x,y;p,-q) \in T^*(M\times M) \mid (x,y;p,q) \in WF(\Delta) \}$. The diagrams illustrate tuples $(x,y; p,q) \in T^*(M\times M)$, where the vertex of a cone corresponds to $y$ and the cone illustrates all points $x$ linked to $y$ by null geodesics; the arrows illustrate the allowed directions of $p$, with $p$ and $q$ linked by parallel transport and both tangent to null geodesic linking $x$ and $y$.

[[!include propagators - table]]

These propagators govern the construction of the [[Wick algebra]] of [[quantum observables]] of the [[free field|free]] [[scalar field]] on the given [[globally hyperbolic spacetime]], as well as the further [[deformation quantization]] to [[interaction|interacting]] [[perturbative quantum field theory]] [[quantum field theory on curved spacetimes|on curved spacetimes]] via [[causal perturbation theory]]. See at \emph{[[locally covariant perturbative quantum field theory]]} for more on this.

\hypertarget{relation_to_schrdinger_equation}{}\subsubsection*{{Relation to Schr\"o{}dinger equation}}\label{relation_to_schrdinger_equation}

Sometimes the Klein-Gordon equation is thought of as a [[general relativity|relativistic]] refinement of the [[Schrödinger equation]] (as one passes from the non-relativistic to the [[relativistic particle]]). But this requires some care. A priori the Klein-Gordon equation takes as arguments a [[field (physics)|field]] on [[spacetime]], whereas the Schr\"o{}dinger equation takes as argument a [[wave function]] on [[phase space]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[wave equation]], [[wave]]


\item [[plane wave]]

[[wave vector]], [[wavelength]], [[frequency]]


\item [[Fourier analysis]]


\item [[heat equation]]


\item [[wave equation]]


\item [[Schrödinger equation]]


\item [[Dirac equation]]



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

The Klein-Gordon equation is named after [[Oskar Klein]] and [[Walter Gordon]].

An overview over the KG [[propagators]] on [[Minkowski spacetime]] is given in

\begin{itemize}%
\item [[Mikica Kocic]], \emph{Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions}, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]])

\end{itemize}
The [[Hadamard propagator]] for the Klein-Gordon equation on general [[globally hyperbolic spacetimes]] was found 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}
The original reference on the relevant notions of microlocal analysis and distinguished parametrices of the Klein-Gordon equation is

\begin{itemize}%
\item [[Johann Duistermaat]], [[Lars Hörmander]], \emph{Fourier integral operators. II}, Acta Mathematica 128 (1972), 183-269 (\href{http://dx.doi.org/10.1007/bf02392165}{doi})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], \emph{Wave Equations on Lorentzian Manifolds and Quantization}, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (\href{https://arxiv.org/abs/0806.1036}{arXiv:0806.1036})


\item [[Nicolas Ginoux]], \emph{Linear wave equations}, Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], \emph{Quantum Field Theory on Curved Spacetimes: Concepts and Methods}, Lecture Notes in Physics, Vol. 786, Springer, 2009



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation}{Klein-Gordon equation}}

\end{itemize}
[[!redirects Klein-Gordon equations]]

[[!redirects Klein-Gordon operator]] [[!redirects Klein-Gordon operators]]



\end{document}