Abstract
We develop the dual description of ($2+1$)-dimensional SU(2) lattice gauge theory as interacting ``Abelian-like'' electric loops by using Schwinger bosons. ``Point splitting'' of the lattice enables us to construct explicit Hilbert space for the gauge invariant theory which in turn makes dynamics more transparent. Using path integral representation in phase space, the interacting closed loop dynamics is analyzed in the weak coupling limit to get the mass gap.
Highlights
There had been many attempts in the past to reformulate gauge theories in terms of gauge invariant Wilson loops [1,2,3,4] both in the continuum as well as on the lattice
A complete gauge invariant Hilbert space could be constructed in the electric space [5] by solving the local Gauss law at each site on the lattice
This leads to complicated dynamics as the matrix element of the Hamiltonian in this basis is given by higher Wigner coefficients [5,6]
Summary
There had been many attempts in the past to reformulate gauge theories in terms of gauge invariant Wilson loops [1,2,3,4] both in the continuum as well as on the lattice. Loop description of gauge theories on a lattice has been shown to be equivalent to a description based on integer local quantum numbers in dual space satisfying triangle inequalities [5] at each site. The local gauge invariant basis created is parametrized in terms of three quantum numbers at each site of the new lattice which are independent and have to satisfy triangle. Matrix elements of the Hamiltonian are simpler to analyze in this basis especially in the weak coupling limit Such a local, complete basis allows us to construct a gauge invariant path integral for SU(2) LGT in the phase space.
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