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, 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
g
{\displaystyle {\mathfrak {g}}}
. 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
T
r
[
E
(
e
)
2
]
{\displaystyle Tr[E(e)^{2
