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\begin{document}

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\section*{kinematics and dynamics}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{formalization}{Formalization}\dotfill \pageref*{formalization} \linebreak
\noindent\hyperlink{in_fqft}{In FQFT}\dotfill \pageref*{in_fqft} \linebreak
\noindent\hyperlink{in_aqft}{In AQFT}\dotfill \pageref*{in_aqft} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_models}{In $\sigma$-models}\dotfill \pageref*{in_models} \linebreak
\noindent\hyperlink{charged_particle}{Charged particle}\dotfill \pageref*{charged_particle} \linebreak


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

In the context of [[mechanics]] (broadly construed), one distinguishes between \emph{kinematics} and \emph{dynamics}:

\begin{enumerate}%
\item \textbf{Kinematics} concerns (only) the [[physical fields]], [[states]] and [[observables]], including the [[spaces]] and [[algebras]] (such as a [[phase space]] or [[Hilbert space]] [[space of states|of (pure) states]] and an appropriate [[algebra of observables]]) into which they are organised;


\item \textbf{Dynamics} additionally treats the evolution of the system in [[time]] or even [[spacetime]]: as given by a [[Lagrangian]] and [[action functional]] and as given by the [[action]] of [[Hamiltonian]] [[quantum observables]] on physical states.

In the [[Schrödinger picture]], we think of the [[states]] as evolving, while the [[observables]] evolve in the [[Heisenberg picture]]. In the [[interaction picture]] we think of the states as evolving with respect to a given time evolution and the observables to evolve, too, with respect to a [[perturbation theory|perturbation]] of this time evolution.



\end{enumerate}
\hypertarget{formalization}{}\subsection*{{Formalization}}\label{formalization}

The notions of kinematics and dynamics may be formally defined in the two formalizations of [[quantum field theory]]: [[FQFT]] and [[AQFT]].

\hypertarget{in_fqft}{}\subsubsection*{{In FQFT}}\label{in_fqft}

Consider a [[quantum field theory]] as given by a [[strong monoidal functor]]

\begin{displaymath}
Z : Bord_n^S \to \mathcal{C}
\end{displaymath}
from a [[category of cobordisms]] with $S$-[[structure]] (for instance [[conformal structure]] or [[Riemannian structure]]) to some [[symmetric monoidal category]] $\mathcal{C}$.

Then:

\begin{itemize}%
\item the \emph{kinematics} of $Z$ is $Z_0 : Ob(Bord_n^S) \to Ob(\mathcal{C})$, the action of the [[functor]] $Z$ on [[object]]s.


\item with that given, the \emph{dynamics} of $Z$ is $Z_1 : Mor(Bord_n^S) \to Mor(\mathcal{C})$, the action of the functor $Z$ on [[morphism]]s.



\end{itemize}
This means that as we regard an $n$-dimensional [[QFT]] as an [[FQFT|extended QFT]] given by an [[n-functor]]

\begin{displaymath}
Z : Bord_n^S \to \mathcal{C}
\end{displaymath}
to from a [[(infinity,n)-category of cobordisms]] with $S$-structure to some [[symmetric monoidal (infinity,n)-category]] $\mathcal{C}$ the dichotomy between kinematics and dynamics may be regarded as being blurred a bit: we can regard the action $Z_0$ on [[object]]s as the genuine kinematics and the action $Z_n$ on [[n-morphism]]s as the genuine dynamics, and then the actions $Z_{1 \leq k \leq n-1}$ as interpolating between these two notions.

\hypertarget{in_aqft}{}\subsubsection*{{In AQFT}}\label{in_aqft}

(\ldots{})

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

\hypertarget{in_models}{}\subsubsection*{{In $\sigma$-models}}\label{in_models}

We discuss the notions of kinematics and dynamics for [[sigma-model]] [[QFT]]s.

\hypertarget{charged_particle}{}\paragraph*{{Charged particle}}\label{charged_particle}

Given a [[Riemannian manifold|Riemannian]] [[target space]] $(X,g)$ and a [[background gauge field]]s given by a [[circle n-bundle with connection|circle bundle with connection]] $(P \to X, \nabla)$, the corresponding [[sigma-model]] quantum field theory is, as an [[FQFT]], a [[functor]]

\begin{displaymath}
Z : Bord_1^{Riemn} \to Hilb
  \,.
\end{displaymath}
It sends

\begin{itemize}%
\item each [[object]] -- a [[point]] -- to the space $\Gamma(P \times_{U(1)} \mathbb{C})$ of square integrable [[section]]s of the [[line bundle]] [[associated bundle|associated]] with the background gauge field;


\item each [[morphism]] $(\bullet \stackrel{t}{\to} \bullet)$ to the [[operator]] $\exp(i t \nabla^* \nabla)$, where $\nabla^* \nabla$ is the [[Laplace-Beltrami operator]] of the [[covariant derivative]] $\nabla$.



\end{itemize}
So in terms of the background field data we have:

\begin{itemize}%
\item the kinematics is encoded in a smooth [[principal bundle]] -- hence a [[cocycle]] in [[smooth infinity-groupoid -- structures|smooth cohomology]];


\item the dynamics is encoded in a [[connection on a bundle|connection]] on this bundle -- hence a refinement of the above cocycle to [[ordinary differential cohomology]].



\end{itemize}
[[!redirects kinematics and dynamics]] [[!redirects dynamics and kinematics]]

[[!redirects kinematics]] [[!redirects dynamics]]



\end{document}