Everything about Hamiltonian Lattice Gauge Theory totally explained
In
physics,
Hamiltonian lattice gauge theory is a calculational approach to
gauge theory and a special case of
lattice gauge theory in which the space is discretized but time is not. The
Hamiltonian is then re-expressed as a function of degrees of freedom defined on a d-dimensional lattice.
Following Wilson, the spatial components of the
vector potential are replaced with
Wilson lines 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
eigenvalues of the Wilson line
operators 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
plaquettes (the magnetic contribution) and contributions coming from the edges (the electric contribution).
Hamiltonian lattice gauge theory is exactly dual to a theory of
spin networks. This involves using the
Peter-Weyl theorem. In the spin network basis, the spin network states are
eigenstates of the operator
.
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