Hall 123

Gauge theories, spin models and hydrogen atoms

Sreeraj T. P.

S. N. Bose Institute

Why are the effects of strong force safely hidden inside the nucleus of the atom and not apparent in everyday life like electromagnetism or gravity? Showing that the force carrying particle described by a non-abelian gauge theory is massive leading to the short range of strong force is a challenging open problem. The solution of this `mass gap' problem requires a better understanding of the low energy behavior of gauge theories. The conventional formulation of non-abelian gauge theories which describe strong force is in terms of massless gluon fields. These are clearly not the right fields in which to understand its low energy behavior. Most of the difficulties in gauge theories emerge because of the existence of a huge number of redundant gauge degrees of freedom. So, it is desirable to reformulate pure gauge theories in terms of gauge invariant Wilson loops. However, overcompleteness of Wilson loops has been a long standing problem in all loop approaches to lattice gauge theories. Starting with the standard formulation of SU(N) gauge theories on a lattice, we construct a series of canonical transformations to systematically reformulate SU(N) lattice gauge theory in terms of loops without introducing any loop redundancy. All redundant local gauge degrees of freedom are naturally removed leading to a resultant dual SU(N) spin model with global SU(N) invariance and inverse coupling. In 2+1 dimensions, this is an exact non-abelian generalisation of the duality between $Z_2$ Gauge theory and Ising model due to Wegner (1971). In the weak coupling (continuum) limit, this dual SU(N) spin model has only nearest neighbouring interactions. Further, in SU(2) case, our canonical or duality transformation helps us establish an exact isomorphism between the physical (loop) Hilbert space of SU(2) Lattice gauge theory and that of a collection of Hydrogen atoms.