View source for Hamiltonian lattice gauge theory

{{short description|Lattice field theory with only spatial discretization}}

{{One source|date=October 2022}}
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 (quantum mechanics)|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 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 (mathematics)|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 group|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 <math>\mathfrak{g}</math>. 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 network]]s. This involves using the [[Peter–Weyl theorem]]. In the spin network basis, the spin network states are [[eigenstate]]s of the operator <math>Tr[E(e)^2]</math>.

==References==
*{{cite journal | last1=Kogut | first1=John |author-link=John Kogut| last2=Susskind | first2=Leonard |author-link2=Leonard Susskind| title=Hamiltonian formulation of Wilson's lattice gauge theories | journal=Physical Review D | publisher=American Physical Society (APS) | volume=11 | issue=2 | date=1975-01-15 | issn=0556-2821 | doi=10.1103/physrevd.11.395 | pages=395–408| bibcode=1975PhRvD..11..395K }}

[[Category:Lattice field theory]]


{{lattice-stub}}