In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Hamiltonian lattice gauge theory is a calculational approach to
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
and a special case of
lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice.
Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum elec ...
in which the space is discretized but time is not. The
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
are replaced with
Wilson line
Thomas Wilson Sons & Co. was a British shipping company, founded in 1840,Greenway (1986), p. 48 It evolved from a joint venture formed by merchants Thomas Wilson, John Beckinton and two unrelated partners named Hudson in 1822.
The company expan ...
s over the edges, but the time component is associated with the vertices. However, the
temporal gauge
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
is often employed, setting the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
to zero. The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the Wilson line
operators U(e) (where e is the (
oriented) edge in question) take on values on the
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
G. It is assumed that G is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, otherwise we run into many problems. The conjugate operator to U(e) is the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
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 network
In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear f ...
s. This involves using the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
. In the spin network basis, the spin network states are
eigenstates of the operator