\def\it{\item{$\bullet$}}
\def\R{{\bf R}}
\def\ev{{\rm ev}}

Peter Teichner.

Classical field theory.

Goal: Describe a mathematical framework in which all real physical theories can be expressed.

A {\it classical\/} field theory consists of the following three data:
\it space-time~$M$ (a Lorentz manifold of dimension $(d-1)+1$
or a Riemannian manifold of dimension~$d$
or an arbitrary semi-Riemannian manifold);
\it field content $\Phi(M)$ (space of sections of a certain bundle over~$M$);
\it classical action~$S\colon\Phi(M)\to\R$.

Roughly speaking, classically one only sees {\it extrema\/} of~$S$.
In quantum field theory a state is a superposition of all fields
with weight $\exp(iS(\Phi)/\hbar)$.
For statistical field theory the probability density is $\exp(-S(\Phi)/T)$.
So the equations become the same if $T=i\hbar$.

Examples:
\item{(a)} Mechanics: A particle moving in a configuration space~$N$.
Here (1)~$M$ is one-dimensional (just time); (2)~$\Phi(M)=C^\infty(M,N)$;
(3)~$S(\phi)=\int_M L(\phi_r(m))dm$,
where $\phi_r(m)$ is the $r$-jet of $\phi$ at point~$m\in M$
and $L\colon J^r(N)\to\R$ is a Lagrangian,
for example: $r=1$, $J^1=TN$, $L(v)=g(v,v)-f(\pi(v))$.
Note that $J^r(N)$ is finite-dimensional manifold.
\item{(b)} Free boson: (1)~$M$ is any oriented semi-Riemannian manifold;
(2)~$\Phi(M)=C^\infty(M,\R)$ (free scalar $\sigma$-model);
(3)~$S(\phi)=\int_M L(\phi_r(m))\omega$,
where $\omega$ is a volume form,
$L(\phi)=\Delta(\phi)\phi+m^2\phi^2$
and $\Delta=d^*d=-(*d*)d\ge0$.
Hence $L(\phi)\omega=(d\phi)\wedge(*d\phi)+m^2\phi\wedge*\phi$.

Key property: {\it Locality\/} of the classical action~$S$.
We have a map $\ev_r\colon M\times\Phi(M)\to J^r(\Phi)$.
We require that $S=\int_M L$,
where $L$ is a form of degree~$d=\dim M$
on~$M\times\Phi(M)$,
which comes from~$J^r(\Phi)$:
$L$ is the pullback of some form of degree~$d$ on the manifold~$J^r(\Phi)$
along the map~$\ev_r$.

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