Why is it interesting?

Layman

Student

"One of the most insightful treatment of ghosts in quantum field theory appears in lecture notes for the Basko Polje Summer School (1976) by Benny Lautrup entitled Of Ghoulies and Ghosties." http://scipp.ucsc.edu/~haber/ph218/

Researcher

The group of gauge transformations $G$ means the bundles automorphisms which preserve the Lagrangian. (Source)

The gauge group is simply one fibre of the bundle, i.e. for example, $SU(2)$.

We denote the space of all connections by $A$. Now, to get physically sensible results we must be careful with these different notions:

Integration should therefore be carried out on the quotient space $\mathcal{G}=A/G$. Now $A$ is a linear space but $\mathcal{G}$ is only a manifold and has to be treated with more respect. Thus for integration purposes a Jacobian term arises which, in perturbation theory, gives rise to the well-known Faddeev-Popov "ghost" particles. Nonperturbatively it seems reasonable that global topological features of $\mathcal{G}$ will be relevant.

Geometrical Aspects of Gauge Theories by M. F. Atiyah

Examples

Example1

Example2:

History