Some comments on canonical gauge theories with boundaries

Tuesday, Sep 24th

Alejandro Corichi, UNAM Morelia
Title: Some comments on cannonical gauge theories with boundaries
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By Jorge Pullin, LSU


When field theories are usually formulated, it is assumed that the portion of space they live in is infinite. There are no boundaries to space, or more precisely, they are placed at infinity. Many physical situations, however, concern situations  with a bounded finite domain. The formulation of field theories requires modifications when boundaries are introduced. The equations of field theories are typically derived through a procedure from a function called the action. Extra terms have to be added to the action if one has a situation with finite boundaries.

Ordinary field theories like electromagnetism have infinitely many degrees of freedom. This means its variables are fields that are functions of space and one can view the value at each point in space as a different degree of freedom. In that sense they are generalizations of usual mechanical systems, which have a finite number of degrees of freedom. Topological field theories are models that, although they have variables that are fields that are functions of space, its equations imply that one only has a finite number of degrees of freedom. This implies many simplifications, and they have proved useful as models in which to test quantization techniques, avoiding the many complexities introduced by having an infinite number of degrees of freedom. Both ordinary and topological field theories are typically described in terms of redundant variables. That means that a several mathematical configurations correspond to a physical configuration. This implies there are symmetries in the theory, this is what is usually called "gauge" symmetries. This talk addressed the issue of topological field theories with boundaries and also ordinary field theories coupled to topological theories. A procedure to treat them was given and shown to work in a pair of examples. Some of the techniques would be of interest to explore fields in the vicinity of black holes, where the horizon (the surface beyond which nothing can return) acts as a natural boundary for fields living in the vicinity of the black hole.