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Following Wilson, the spatial components of the Vector Potential are replaced with Wilson Line s over the edges, but the time component is associated with the vertices. However, the Temporal Gauge is often employed, setting the Electric Potential to zero. The Eigenvalue s of the Wilson line Operator s U(e) (where e is the ( Oriented ) edge in question) take on values on the Lie Group G. It is assumed that G is Compact or otherwise, we run into many problems. The conjugate operator to U(e) is the Electric Field E(e) whose eigenvalues take on values in the Lie algebra . The Hamiltonian receives contributions coming from the Plaquette s (the magnetic contribution) and contributions coming from the edges (the electric contribution). Hamiltonian lattice gauge theory is exactly dual to a theory of Spin Network s. See that article for more detail. This involves using the Peter-Weyl Theorem . In the spin network basis, the spin network states are Eigenstate s of the operator . REFERENCES
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