The Berry Phase and Monopoles in Non-Abelian Gauge Theories

Abstract

We consider the quantum mechanical notion of the geometrical (Berry) phase in SU(2) gauge theory, both in the continuum and on the lattice. It is shown that in the coherent state basis eigenvalues of the Wilson loop operator naturally decompose into the geometrical and dynamical phase factors. Moreover, for each Wilson loop there is a unique choice of U(1) gauge rotations which do not change the value of the Berry phase. Determining this U(1) locally in terms of infinitesimal Wilson loops we define monopole-like defects and study their properties in numerical simulations on the lattice. The construction is gauge dependent, as is common for all known definitions of monopoles. We argue that for physical applications the use of the Lorenz gauge is most appropriate. And, indeed, the constructed monopoles have the correct continuum limit in this gauge. Physical consequences are briefly discussed.

“Branes” in Lattice Su(2) Gluodynamics

“…Both predictions (23) and (24) are in perfect agreement with the data 21,9,10 . Thus, we can say that the simplest vacuum loop corresponding to the monopole field has been directly observed on the lattice.…”

Magnetic monopoles, alive

“…[10]. The basic idea behind this construction is to make monopoles as much geometrical objects as possible.…”

“…The starting point of [10] is somewhat different. Namely, it is the observation that each Wilson loop defines in a natural way its own U(1).…”

“…Moreover, the coherent states can be explicitly constructed in terms of the link matrices, for details and further references see [10]. As a result, for any given lattice fields configuration, one can determine k and detect the Dirac strings in this way.…”

“…In this mini-review we will emphasize some new points related to each of the items (a)-(c) listed above and which are based mostly on the original papers [9,10,11]. The new points, although they refer to various topics, are unified by a common approach.…”