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\begin{document}

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\section*{Tomonaga-Schwinger equation}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \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 generalization of the [[Schrödinger equation]] from [[quantum mechanics]] to [[quantum field theory]]:

from \hyperlink{TorreVaradarajan98}{Torre-Varadarajan 98 p.2}:

The idea of evolving a quantum field from any [[Cauchy surface]] to any other seems to have originated in the mid 1940's with the work of Tomonaga 1 and Schwinger 2 on relativisticquantum field theory. Tomonaga and Schwinger wanted an invariant generalization of the Schr\"o{}dinger equation, which describes time evolution of the state of a quantum field relative to a fixed inertial reference frame. By allowing for all possible Cauchy surfaces in the description of dynamical evolution one easily accommodates all possible notions of time for all possible inertial observers. Thus a dynamical formalism incorporating arbitrary Cauchy surfaces does allow for an invariant generalization of the Schr\"o{}dinger equation. Since, the space of Cauchy surfaces is infinite-dimensional, it is impossible to describe time evolution along arbitrary surfaces by using a single time parameter. In essence, one needs a distinct time parameter for every possible foliation of spacet ime. As shown by Tomonaga and Schwinger, if one formulates dynamics in terms of general Cauchy surfaces, the resulting dynamical evolution equation is, formally, a functional differential equation, which is usually called the ``Tomonaga-Schwinger equation''.

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

\begin{itemize}%
\item [[Schrödinger picture]]


\item [[Wheeler superspace]]



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

Named after [[Shin'ichirō Tomonaga]] and [[Julian Schwinger]].

\begin{itemize}%
\item [[Charles Torre]], M. Varadarajan, \emph{Functional Evolution of Free Quantum Fields}, Class.Quant.Grav. 16 (1999) 2651-2668 (\href{https://arxiv.org/abs/hep-th/9811222}{arXiv:hep-th/9811222})

\end{itemize}


\end{document}