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From Lattice Gauge Theories to Hydrogen Atoms

We construct canonical transformations to obtain a complete and most economical realization of the physical Hilbert space ${\cal H}^p$ of pure $SU(2)_{2+1}$ lattice gauge theory in terms of Wigner coupled Hilbert spaces of hydrogen atoms. One hydrogen atom is assigned to every plaquette of the lattice. A complete orthonormal description of the Wilson loop basis in ${\cal H}^p$ is obtained by all possible angular momentum Wigner couplings of hydrogen atom energy eigenstates $\vert n~l~m\rangle$ describing electric fluxes on the loops. The SU(2) gauge invariance implies that the total angular momenta of all hydrogen atoms vanish. The canonical transformations also enable us to rewrite the Kogut-Susskind Hamiltonian in terms of fundamental Wilson loop operators and their conjugate electric fields. The resulting loop Hamiltonian has a global SU(2) invariance and a simple weak coupling ($g^2\rightarrow 0$) continuum limit. The canonical transformations leading to the loop Hamiltonian are valid for any SU(N). The ideas and techniques can also be extended to higher dimension.