Concept of nonintegrable phase factors and global formulation of gauge fields

Abstract

Through an examination of the Bohm-Aharonov experiment an intrinsic and complete description of electromagnetism in a space-time region is formulated in terms of a nonintegrable phase factor. This concept, in its global ramifications, is studied through an examination of Dirac s magnetic monopole field. Generalizations to non-Abelian groups are carried out, and result in identification with the mathematical concept of connections on principal fiber bundles.

Multi-monopoles and magnetic bags

“…This is an instable stationary point of the potential in Figure 1 with energy density ε 0 = λv 4 /8. This vacuum is in an abelian Coulomb phase and the magnetic field is given by the curl of the Wu-Yang vector potential [19]:…”

Hamilton's formalism for systems with constraints

“…Alternatively we can introduce gauge invariant variables, e.g. the Wilson loops [48], (see the lecture of R. Loll) or fix the gauge. To fix the gauge freedom we need 2N gauge fixing conditions on the phase space variables (A, π).…”

Magnetic charge lattices, moduli spaces and fusion rules

“…This consistency condition can be derived from the bundle description [20]. One can work in a gauge where the magnetic field has the form…”

“…More importantly they are singular at a point. As a direct generalization of the Wu-Yang description of U (1) monopoles [20], singular monopoles in Yang-Mills theory with gauge group H correspond to a connection on an H-bundle on a sphere surrounding the singularity. The H-bundle may be topologically non-trivial, but in addition the monopole connection equips the bundle with a holomorphic structure.…”