On the Support of Physical Measures in Gauge Theories

TL;DR

This paper proves that in two-dimensional Yang-Mills theory, the physical measure is singular relative to the kinematical measure, challenging standard methods and extending results to more general confining theories.

Contribution

It provides an explicit decomposition of the gauge orbit space and demonstrates the singularity of the physical measure, offering new insights into gauge theory measures.

Findings

Physical measure is purely singular with respect to the kinematical measure.

Standard exponential of minus the action method is not applicable.

Results extend to more general confining theories.

Abstract

It is proven that the physical measure for the two-dimensional Yang-Mills theory is purely singular with respect to the kinematical Ashtekar-Lewandowski measure. For this, an explicit decomposition of the gauge orbit space into supports of these two measures is given. Finally, the results are extended to more general (e.g. confining) theories. Such a singularity implies, in particular, that the standard method of determining the physical measure via ``exponential of minus the action times kinematical measure'' is not applicable.