\def\Cl{\mathop{\rm Cl}}
\def\CL{\mathop{{\bf C\/}l}}
\def\R{{\bf R}}
\def\Spin{{\rm Spin}}
\def\GL{{\rm GL}}

Konrad Waldorf.

Charged spinors.

Definition.  A spinor is a field for a spin structure~$SM$ on~$M$
of matter type~$(V,\rho,h,f)$,
where $(V,\rho)$ is a representation of~$\Cl(p,q)$ on~$\CL(p,q)$,
$h\colon V\times V\to\R$ is a bilinear form,
$f\colon V\to\R$ is $\rho$-invariant.
A field is a section $\psi\colon M\to SM\times_\rho V=\Sigma M$ of the spinor bundle.
We have an action functional $S(\psi)=\int_M D\psi\wedge_h*\psi+*(f\circ\psi)$.
Equation of motion is $D\psi+im\psi=0$ (the Dirac equation).  (Here $f(v)=-mh(v,v)$.)

We want to couple this stuff to electromagnetic field.

Bundle splicing: If $G_k$ is a Lie group
and $P_k$ is a principal $G_k$-bundle over~$\Pi$,
then $P_1\circ P_2=P_1\times_M P_2$ is a principal $G_1\times G_2$-bundle over~$M$.
If we have connections $\omega_k$ on~$P_k$, then
$\omega_1\circ\omega_2=p_1^*\omega_1\oplus p_2^*\omega_2\in\Omega^1(P_1\circ P_2,g_1\oplus g_2)$.
If $\rho_k$ is a representation of $G_k$ such that
$\rho_1(g_1)\circ\rho_2(g_2)=\rho_2(g_2)\circ\rho_1(g_1)$,
then $\rho_1\times\rho_2$ is a representation of~$G_1\times G_2$.

Setup for charged spinors.
We have a spin structure~$SM$ on~$M$ with Levi-Civita connection,
Yang-Mills theory $(G,P)$ and connection~$\omega$ on~$P$.
We have a representation $\rho_{SM}\colon\Spin(p,q)\to\GL(V)$ and $\rho_P\colon G\to\GL(V)$,
a bundle $\Sigma P=(SM\circ P)\times_{\rho_{SM}\times\rho_P}V$
and a Dirac operator on this bundle $D^\omega\colon\Gamma(M,\Sigma P)\to\Gamma(M,\Sigma P)$.

Definition: A charged spinor is a field for $SM\circ P$
of matter type~$(V,\rho_{SM}\times\rho_P,h,f)$.
Action functional is
$S(\psi,\omega)=\int_M D^\omega(\psi)\wedge*\psi+*(f\circ\psi)+S_{\rm YM}(\omega)$.
Dirav equation $D^\omega\psi+im\psi=0$ and $D^\omega*F_\omega=J(\psi)$.

Electrons: $V=\Sigma=\Sigma^+\oplus\Sigma^-$ and $\rho_P(z)(v)=z^n\cdot v$, where $n$ is an integer
number (charge).  Dirac equation is $D^\omega\psi+im\psi=0$.

\bye
