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\begin{document}

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\section*{action functional}

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

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

[[!include variational calculus - contents]]

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

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{LocalActionFunctional}{Local action functionals (traditional theory)}\dotfill \pageref*{LocalActionFunctional} \linebreak
\noindent\hyperlink{ExtendedLocalInGaugeTheory}{Extended local action functionals in (higher) gauge theory}\dotfill \pageref*{ExtendedLocalInGaugeTheory} \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{idea}{}\subsection*{{Idea}}\label{idea}

In [[physics]] the [[dynamics]] of a [[physical system]] may be encoded by a [[nonlinear functional]] -- called the \textbf{action functional} -- on its [[configuration space]]:

\begin{itemize}%
\item in [[classical mechanics]] and [[classical field theory]] -- by the \textbf{action principle} or \textbf{[[principle of least action]]} -- the extrema of the action functional -- obtained by [[variational calculus]] and given by [[Euler-Lagrange equation]]s -- encode the physically observable configurations ;


\item in [[quantum mechanics]] and [[quantum field theory]] the evolution of the [[quantum state]]s is encoded by the integral -- the [[path integral]] -- of the exponentiated action functional over the space of field configurations.



\end{itemize}
For emphasis the description of dynamics by action functionals is called the \textbf{Lagrangean} approach. Another formulation of dynamics in physics that does not involve an action functional explicitly is [[Hamiltonian mechanics]] on [[phase space]]. At least in certain classes of cases the relation and equivalence of both approaches is understood. Generally the formulation of [[quantum field theory]] in terms of action functionals suffers from a lack of precise understanding of what the [[path integral]] over the action functional really means.

$\backslash$begin\{imagefromfile\} ``file\_name'': ``action\_functional\_on\_a\_napkin.jpg'', ``float'': ``left'', ``margin'': \{ ``top'': 0, ``right'': 10, ``bottom'': 10, ``left'': 0, ``unit'': ``px'' \}, ``alt'': ``Action functional on a napkin'', ``caption'': ``Taken from A Zee, Fearful Symmetry'' $\backslash$end\{imagefromfile\}

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

Let $\mathbf{H}$ be the ambient [[(∞,1)-topos]] with a [[natural numbers object]] and equipped with an additive [[continuum]] [[line object]] $\mathbb{A}^1$ (see there). Let $C \in \mathbf{H}$ be the [[configuration space]] of a physical system. Then an \textbf{action functional} is a morphism

\begin{displaymath}
\exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbb{A}^1 / \mathbb{Z}
  \,
\end{displaymath}
(here $\hbar$ refers to [[Planck's constant]]).

If $\mathbf{H}$ is a [[cohesive (∞,1)-topos]] then there is an  of the action functional to a morphism

\begin{displaymath}
\mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbf{\flat}_{dR}\mathbb{A}^1/\mathbb{Z}
  \,.
\end{displaymath}
The [[equation]]

\begin{displaymath}
\mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) = 0
\end{displaymath}
is the [[Euler-Lagrange equation]] of the system. It characterizes the [[critical locus]] of $S$ is the [[covariant phase space]] inside the configuration space: the space of classically realized trajectories/histories of the system. If $\mathbf{H}$ models [[derived geometry]] then this critical locus is presented by a [[BRST-BV complex]].

\hypertarget{LocalActionFunctional}{}\subsubsection*{{Local action functionals (traditional theory)}}\label{LocalActionFunctional}

An action functional is called \textbf{local} if it arises from integration of a [[Lagrangian]].

In traditional theory this is interpreted as follows: an action functional $S : C \to \mathbb{A}^1$ is called local if

\begin{itemize}%
\item the [[configuration space]] $C$ is the space $C = \Gamma_X(E)$ of [[section]]s of a [[fiber bundle]] $E \to X$ over some parameter space ([[spacetime]] $X$);


\item there is a [[Lagrangian]] density $J_\infty(E) \to \Omega^{\dim X}(X)$ on the [[jet bundle]] of $E$;


\item on a section/field configuration $\phi : X \to E$ the action $S$ takes the value

\begin{displaymath}
S(\phi) = \int_X L(j_\infty(\phi))
  \,,
\end{displaymath}
where $j_\infty(\phi) = (\phi, \partial_i \phi, \cdots)$ is the jet-prolongation of $\phi$ (the collection of all its higher partial derivatives).



\end{itemize}
Consider action functional for on a configuration space of [[smooth function]]s from the line to a [[smooth manifold]] $X$.

We can consider

\begin{enumerate}%
\item $S(q) = \int_a^b L(q,\dot{q}) \,\mathrm{d}t$, where $q$ is a path through configuration space, on the time interval $[a,b]$, with derivative $\dot{q} = \mathrm{d}q/\mathrm{d}t$. When minimising the action, we fix the values of $q(a)$ and $q(b)$.
\item $L(q,\dot{q}) = \int_{S} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$, where now $q$ is a configuration of fields on $S$, which is a region of space. We fix boundary conditions on the boundary of $S$ (typically that $q$ and $\dot{q}$ go to zero if $S$ is all of space).
\item $S(q) = \int_{R} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t$, where now $q$ is a configuration of fields on $R$, which a region of spacetime, with time derivative $\dot{q} = \partial{q}/\partial{t}$. We fix boundary conditions on the boundary of $R$.

\end{enumerate}
The formulation of (3) above is still not manifestly coordinate independent. However, $\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t$ is simply the [[volume form]] on spacetime and $\dot{q}$ is merely one choice of coordinate on [[configuration space|state space]] and could just as easily be replaced by a derivative with respect to any timelike coordinate on spacetime (or drop coordinates altogether).

\hypertarget{ExtendedLocalInGaugeTheory}{}\subsubsection*{{Extended local action functionals in (higher) gauge theory}}\label{ExtendedLocalInGaugeTheory}

For [[gauge theories]] and [[higher gauge theories]] the configuration spaceds of the physical system are in general not plain [[manifolds]] or similar, but are [[orbifolds]] or more generally [[smooth groupoids]], [[smooth ∞-groupoids]]. (An exposition of and introduction to much of the following is at \emph{[[geometry of physics]]}.)

For instance for $G$ a [[Lie group]] and $\mathbf{B}G_{conn}$ the smooth [[moduli stack]] of $G$-[[principal connections]] (see at [[connection on a bundle]]), then the [[smooth groupoid]] of $G$-[[gauge field]] configurations is the [[internal hom]]/[[mapping stack]] $[\Sigma, \mathbf{B}G_{conn}] \in$[[Smooth∞Grpd]] (or some concretification thereof, see at \emph{\href{geometry%20of%20physics#DifferentialModuli}{geometric of physics -- differential moduli}}: this is the [[smooth groupoid]] whose [[objects]] are $G$-[[gauge field]]-configurations on $\Sigma$ ([[connection on a bundle|connections]] on $G$-[[principal bundles]] over $\Sigma$), and whose [[morphisms]] are [[gauge transformations]] between these. The [[infinitesimal space|infinitesimal]] approximation to this [[smooth ∞-groupoid]], its \emph{[[∞-Lie algebroid]]} is the (off-shell) [[BRST complex]] of the theory. The tangent to the $n$-fold [[higher gauge transformations]] becomes the $n$-fold \emph{ghosts} in the BRST complex.

More generall $G$ here can by any [[smooth ∞-group]], such as the [[circle n-group]] $\mathbf{B}^{n-1}U(1)$ or the [[String 2-group]] or the [[Fivebrane 6-group]], and so on, in which case $[\Sigma, \mathbf{B}G_{conn}]$ is the [[smooth ∞-groupoid]] of [[higher gauge theory|higher gauge field]], [[gauge transformations]] between these, [[higher gauge transformations]] between those, and so on.

Notice that this means in particular that in [[higher geometry]] a [[gauge theory]] is a [[sigma-model]] [[quantum field theory]]: one whose [[target space]] is not just a plain [[manifold]] but is a [[moduli stack]] of gauge field configurations.

A [[gauge invariance|gauge invariant]] action functional is then a morphism of [[smooth ∞-groupoids]]

\begin{displaymath}
\exp( i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}] \to U(1)
  \,.
\end{displaymath}
This is of particular interst, again, if it is \emph{local}. In fact, in this context now we can also ask that it is ``extended'' in the sense of [[extended topological quantum field theory]]: that we have an action functional not only in top dimension, being a function, but also in codimension 1, being a [[prequantum bundle]], and in higher codimension, being a [[prequantum n-bundle]].

This is notably the case for all (higher) gauge theoris of [[schreiber:infinity-Chern-Simons theory]] type, such as ordinary [[Chern-Simons theory]] and such as ordinary [[Dijkgraaf-Witten theory]], as well as its higher generalizations. In these cases the action functional $\exp(i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}]$ arises itself from [[transgression]] of an [[extended Lagrangian]] that is defined on the universal [[moduli stack]] of gauge field configurations $\mathbf{B}G_{conn}$ itself, namely from a [[universal characteristic class]] in higher nonabelian [[differential cohomology]] of the form

\begin{displaymath}
\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}
  \,.
\end{displaymath}
Here $\mathbf{B}^n U(1)_{conn}$ is the universal [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack]] for [[circle n-bundles with connection]]. Such a morphism of moduli stacks locally takes a [[connection on a bundle|connection]] [[differential form]] $A$ to a [[Chern-Simons form]] $CS(A)$, but globally it sends the underlying [[principal bundle]] to a [[circle n-group|circle (n-1)-group]] [[principal ∞-bundle]] and accordingly acts globally on the connection. This is hence a fully local [[Lagrangian]]: an \emph{[[extended Lagrangian]]}. Alternatively, one may think of this whole morphism as modulating a [[prequantum circle n-bundle]] on the universal moduli stack $\mathbf{B}G_{conn}$ of [[gauge fields]] itself.

For instance for ordinary [[Chern-Simons theory]] here $n = 3$ $G$ is a [[semisimple Lie group]] and $\mathbf{L}$ is a smooth and differential refinement of the [[first Pontryagin class]]/[[second Chern class]], or of an integral multiple of that (the ``level'' of the theory). In this case $\mathbf{L}$ may also be thought of as modulating the universal [[Chern-Simons circle 3-bundle]]. If instead $G$ is a [[discrete group]] then $\mathbf{L}$ is a [[cocycle]] in the $U(1)$-[[group cohomology]] and this is the [[extended Lagrangian]] of [[Dijkgraaf-Witten theory]].

This [[extended Lagrangian]] becomes an extended action functional after [[transgression]]: the operaton of [[fiber integration in ordinary differential cohomology]] refines to a morphism of moduli stacks of the form

\begin{displaymath}
\exp(2 \pi i \int_{\Sigma_k} (-))
  \colon
  [\Sigma_k, \mathbf{B}^n U(1)_{\mathrm{conn}} ]
  \to 
  \mathbf{B}^{n-k}U(1)_{conn}
 \,,
\end{displaymath}
where $\Sigma$ is an [[orientation|oriented]] [[closed manifold]] of [[dimension]] $k$. This morphism locally simply takes a [[differential n-form]] to its ordinary [[integration of differential forms]] over $\Sigma_k$, but globally it takes the correct [[higher holonomy]] of [[circle n-bundles with connection]].

Combining this with an extended [[schreiber:∞-Chern-Simons theory]] [[Lagrangian]] $\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}$ as above yields for each dimension $k$ a [[prequantum circle n-bundle]] on the space of gauge field configurations over $\Sigma_k$, by forming the [[transgression]] composite

\begin{displaymath}
\exp(i S(-))
  \coloneqq
  \exp(2 \pi i \int_{\Sigma_k} [\Sigma_k, \mathbf{L}])
   \;\; \colon \;\;
  [\Sigma_k, \mathbf{B}G_{conn}]
  \stackrel{[\Sigma_k, \mathbf{L}]}{\to}
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
  \stackrel{\exp(2 \pi i \int_{\Sigma_k}(-))}{\to}
  \mathbf{B}^{n-k}U(1)_{conn}
  \,.
\end{displaymath}
This morphism locally takes the local [[differential form]] incarnation $A$ of a [[connection on an ∞-bundle]] to the exponentiation of the [[integration of differential forms]] $\int_\Sigma CS(A)$ of some higher [[Chern-Simons form]], but globally it computes the correct [[higher holonomy]] of the higher [[circle n-bundle with connection]] over the universal moduli stack of fields, as modulated by the extended [[Lagrangian]] $\mathbf{L}$.

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

For [[spacetime]] [[field theory]]:

\begin{itemize}%
\item [[Einstein-Hilbert action]]

\begin{itemize}%
\item [[Einstein-Maxwell theory]]


\item [[Einstein-Yang-Mills theory]]


\item [[Einstein-Yang-Mills-Dirac theory]]


\item [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]



\end{itemize}


\end{itemize}
For [[branes]]:

\begin{itemize}%
\item [[Nambu-Goto action]]


\item [[Dirac-Born-Infeld action]]


\item [[Perry-Schwarz action]]


\item A large class of examples of action functionals arises in [[schreiber:∞-Chern-Simons theory]]. See there for details.



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

[[!include action (physics) - table]]

\begin{itemize}%
\item [[principle of extremal action]], [[Euler-Lagrange equations]]


\item [[path integral]] [[quantization]]


\item [[effective action functional]], [[background field formalism]]


\item [[parent action functional]]



\end{itemize}
[[!include extended prequantum field theory - table]]

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

Lecture notes with more details are in the section \emph{\href{geometry+of+physics#LagrangiansAndActionFunctionals}{Lagrangians and Action functionals}} of

\begin{itemize}%
\item \emph{[[geometry of physics]]} .

\end{itemize}
Discussion of extended higher local action functional for (higher) gauge theories of generalized [[schreiber:∞-Chern-Simons theory]] type are discussed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], \emph{[[schreiber:Extended higher cup-product Chern-Simons theories]]}

\end{itemize}
The extended local action functionals for ordinary 3d [[Chern-Simons theory]]/[[Dijkgraaf-Witten theory]] and for 7d [[String 2-group]] Chern-Simons theory are constructed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech cocycles for differential characteristic classes]]}

\end{itemize}
and discussed further in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], \emph{[[schreiber:7d Chern-Simons theory and the 5-brane]]}

\end{itemize}
A comprehensive discussion in a general context of higher differential geometry is in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
[[!redirects action functional]] [[!redirects action functionals]] [[!redirects action functinal]]

[[!redirects local action functional]] [[!redirects local action functionals]]



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