Let's only assume the case of electromagnetism. If one varies the Yang-Mills-Functional one gets the Yang-Mills-equation $*d*F=0$. The whole theory can be geometrically described on principal bundles. On the associated bundle, the Klein-Gordon- or Dirac-equation also has a natural geometrical interpretation.

But how does one deal with sources in the formalism of gauge theory on bundles? What is the geometrical/topological interpretation of the full Maxwell equations $*d*F=j$, $dF=0$, including the source term? Can the source be viewed as e.g. monopoles, giving rise to non trivial bundles, imitating the source term?


1 Answer 1


Because gauge invariance requires covariant "conservation" $$ 0=\nabla_\mu J^\mu\equiv \partial_\mu J^\mu +[A_\mu, J^\mu], $$ the sources $J^\mu$ in non-abelian theory cannot be $c$-numbers. Instead they are Wilson Loops. You can also make them classical but with internal dynamics, as in the Wong equations.

Notice that the "charge" of a source in Yang-Mills theory is the representation of the gauge group in which the Wilson line belongs. For example in the SU(3) colour theory quarks have charge "${\bf 3}$" because they have 3-d representation and gluons have charge "${\bf 8}$" because they are in the 8-dimensional adjoint representation. "Addition" of the charges is the Clebsch-Gordan decomposition of the tensor product of the representations such as $$ {\bf 8}\times {\bf 8}= {\bf 1}+ {\bf 8}+{\bf 8}+ {\bf 10}+\overline {\bf 10}+{\bf 27}. $$ In particular gauge "charge" must come in discrete lumps. You cannot have a smooth charge distribution.

Working out the effect of the sources on the gauge bundle is non trivial. For an abelian theory they just induce a holonomy. For non-abelian theories the result is highly non-trivial. The simplest case that I know of results in the Knizhnik-Zamolodchikov equations of conformal field theory, and its spacetime interpretation as Witten's knot invariants.

  • $\begingroup$ Thanks. Can you maybe provide a reference how dynamics is written with Wilson loops, e.g. in the case of electromagnetism. And how does one write the equations of motions in terms of Wilson loops. The only equation of motion I am aware of is the Maxwell-Dirac-equation system. $\endgroup$
    – NicAG
    Apr 20, 2021 at 14:16
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    $\begingroup$ You can try these notes: eduardo.physics.illinois.edu/phys583/ch18.pdf $\endgroup$
    – mike stone
    Apr 20, 2021 at 14:25
  • $\begingroup$ Is it possible to do the theory also analytically or is it only possible to formulate it as lattice theory $\endgroup$
    – NicAG
    Apr 20, 2021 at 14:47
  • $\begingroup$ There is some analytic work in 2d gauge theories, but I'm only comfortable on a lattice :) $\endgroup$
    – mike stone
    Apr 20, 2021 at 16:07

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