Editing Lattice gauge theory (section)

==Basics==
In lattice gauge theory, the spacetime is [[Wick rotated]] into [[Euclidean space]] and discretized into a lattice with sites separated by distance <math>a</math> and connected by links. In the most commonly considered cases, such as [[lattice QCD]], [[fermion]] fields are defined at lattice sites (which leads to [[fermion doubling]]), while the [[Gauge boson|gauge fields]] are defined on the links.  That is, an element ''U'' of the [[compact group|compact]] [[Lie group]] ''G'' (not [[Lie algebra|algebra]]) is assigned to each link.  Hence, to simulate QCD with Lie group [[Special unitary group|SU(3)]], a 3×3 [[unitary matrix]] is defined on each link.  The link is assigned an orientation, with the [[inverse element]] corresponding to the same link with the opposite orientation. And each node is given a value in <math>\mathbb{C}^3</math> (a color 3-vector, the space on which the [[fundamental representation]] of SU(3) acts), a [[bispinor]] (Dirac 4-spinor), an ''n<sub>f</sub>'' vector, and a [[Grassmann number|Grassmann variable]].

Thus, the composition of links' SU(3) elements along a path (i.e. the ordered multiplication of their matrices) approximates a [[path-ordered exponential]] (geometric integral), from which [[Wilson loop]] values can be calculated for closed paths.