Editing Lattice gauge theory (section)

==Yang–Mills action==
The [[Yang–Mills theory|Yang–Mills]] action is written on the lattice using [[Wilson loop]]s (named after [[Kenneth G. Wilson]]), so that the limit <math>a \to 0</math> formally reproduces the original continuum action.<ref name="wilson" />  Given a [[faithful representation|faithful]] [[irreducible representation]] ρ of ''G'', the lattice Yang–Mills action, known as the [[Wilson action]], is the sum over all lattice sites of the (real component of the) [[trace (matrix)|trace]] over the ''n'' links ''e''<sub>1</sub>, ..., ''e''<sub>n</sub> in the Wilson loop,

:<math>S=\sum_F -\Re\{\chi^{(\rho)}(U(e_1)\cdots U(e_n))\}.</math>

Here, χ is the [[character (mathematics)|character]].  If ρ is a [[real representation|real]] (or [[pseudoreal representation|pseudoreal]]) representation, taking the real component is redundant, because even if the orientation of a Wilson loop is flipped, its contribution to the action remains unchanged.

There are many possible Wilson actions, depending on which Wilson loops are used in the action. The simplest Wilson action uses only the 1×1 Wilson loop, and differs from the continuum action by "lattice artifacts" proportional to the small lattice spacing <math>a</math>.  By using more complicated Wilson loops to construct "improved actions", lattice artifacts can be reduced to be proportional to <math>a^2</math>, making computations more accurate.