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\begin{document}

A principle bundle, P(M,G) has an associated vector bundle, \( E(M,V_{p}) \) Where
the \( V_{p} \)is a vector space with \( dim(V_{p})=d_{p} \) = dimension of pth representation of
the group.

The transition functions are derivable from the principle bundle connections:
\( g_{i}=t_{ij}g_{j} \). Gives \( v_{i}=\rho (t_{ij})v_{j} \)

Let \( \nabla ^{i}= \)covariant derivative on \( E(M,V_{p}) \) on a local section.

\( (\nabla _{\mu }z)^{i}=\partial _{\mu }z^{i}+A_{u}^{a}\rho (T_{\alpha })^{i}_{j}z^{j} \). This could be written as:\( \nabla _{\mu }=\partial _{\mu }+A_{u}^{a} \) if the \( \rho  \)is understood.

Then \( \nabla =dx^{\mu }\nabla _{u}=d+A \). Also, \( [\nabla _{u},\nabla _{v}]=F_{uv} \)

For quarks, use SU(3) principle bundle. Quarks have a 3-d representation.
Then \( \gamma ^{\mu }\nabla _{u}\psi =0 \) is the dirac equation with \( (\nabla _{u}\psi )^{i}=\partial _{\mu }\psi ^{i}+(A_{\mu })^{i}_{j}\psi ^{j} \). Physicists often insert i's to
make A hermitian and put in scaling factors.

\end{document}