\def\R{{\bf R}}
\def\YM{{\rm YM}}
\def\GL{{\rm GL}}
\def\Z{{\bf Z}}
\def\U{{\rm U}}
\def\C{{\bf C}}

Konrad Waldorf.

Classical Gauge Theory and Matter Fields.

References:

G.~Naber: Topology, Geometry, and Gauge Fields.

D.~Bleecker: Gauge Theory and Variational Principles.

Notation: $M$ is a spacetime (oriented Lorentz manifold).

Definition.  A Yang-Mills theory over~$M$ consists of a Lie group~$G$ (gauge group),
Ad-invariant bilinear form~$\kappa\colon g\otimes g\to\R$ on the Lie algebra~$g$ of~$G$,
principal $G$-bundle~$P$ over~$M$,
fields are connections $\omega\in\Omega^1(P,g)$,
action functional is $S_\YM(\omega)=2^{-1}\int_M\|F_\omega\|^2_\kappa$.

Yang-Mills equation: $D^\omega*F_\omega=0$ and $D^\omega F_\omega=0$.

Definition: A gauge transformation is a bundle automorphism $g\colon P\to P$.
Remark: We can identify gauge transformations with $G$-equivariant map $\tilde g\colon P\to G$.
We have $r_{\tilde g}=g$.
Equivariance means $\tilde g(p\gamma)=\gamma^{-1}\tilde g(p)\gamma$.
Remark: If $P$ is a principal $G$-bundle, $\rho\colon G\to\GL(V)$,
then we have a bijection between $\Omega^k_\rho(P,V)$ and $\Omega^k(M,P\times_\rho V)$.
We have $\psi\in\Omega^k_\rho(P,V)$ if $r_\gamma^*\psi=\rho(\gamma^{-1})(\psi)$.

Theorem: $S_\YM$ is gauge-invariant: $S_\YM(\omega)=S_\YM(g^*\omega)$.

Definition: A classical electromagnetic field theory is a Yang-Mills theory with $G=U(1)$.
If we remove a line from 3-dimensional space, then we observe Aharonov-Bohm effect.
If we remove a point, then we observe magnetic monopoles.

Definition: If $G$ is a gauge group, then a matter type for~$G$
is a quadruple $T=(V,h,\rho,f)$ such that $V$ is a finite-dimensional real vector space
(internal state space), $h\colon V\otimes V\to\R$ is a real bilinear form on~$V$,
$\rho\colon G\to\GL(V)$ is an isometric representation of~$G$: $h(\rho(g)v,\rho(g)w)=h(v,w)$
(transformation behavior), $f\colon V\to\R$ is a smooth $\rho$-invariant function:
$f(\rho(g)v)=f(v)$ (self-interaction potential).

Definition: Let $(P,G)$ be a Yang-Mills theory over~$M$, $T$ a matter type for~$G$.
A field for~$P$ of type~$T$ is a section $\phi\colon M\to P\times_\rho V$.
The action functional is $S_T(\omega,\phi)=2^{-1}\int_M\|D^\omega\phi\|^2_h+*(f(\phi))$.

Remark: We have two options: (a)~Keep $\omega$ fixed, consider $S_\omega(\phi)=S_T(\omega,\phi)$.
(b)~Consider $S(\omega,\phi)=S_\YM(\omega)+S_T(\omega,\phi)$.
Notice that $(\omega,\phi)$ extremizes~$S$ iff (1)~$\phi$ extremizes $S_\omega$ (Euler-Lagrange);
(2)~$\delta^\omega F_\omega=J^\omega(\phi)$ (inhomegeneous field equation).
Here $\delta^\omega=*D^\omega*$.

Example~1: $G=\{e\}$, $P=M$.  A scalar field is a field of matter type $(\R,h,{\rm id},f)$
with $f(x)=-m^2x^2$.  $S(\phi)=\int_M\|d\phi\|+m^2\phi^2$.
Euler-Lagrange equation in this case is Klein-Gordon equation
$(\Delta^2+m^2)\phi=0$.

Example~2: $\pi^-$-meson: charge $n\in\Z$, no spin.
Yang-Mills theory $(P,\U(1))$.
Matter type $(\C,h,\rho_n,f)$.
Here $\rho_n$ is the $n$th power represetation of~$\U(1)$ on~$\C$,
$f(z)=-m^2|z|^2$ and $h(z_1,z_2)$ is left as an exercise.
Euler-Lagrange equation is $(\Delta^\omega+m^2)\phi=0$.
Here $\Delta^\omega=\delta^\omega D^\omega$.

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