As we mentioned in the previous chapter, the definition of Yang–Mills theories in the continuum in terms of lpops requires a regularization and the resulting eigenvalue equations are, in the non-Abelian case, quite involved. Lattice techniques appear to be a natural way to deal with both these difficulties. First of all since on a lattice there is a minimum length (the lattice spacing), the theory is naturally regularized. An important point is that this is a gauge invariant regularization technique. Secondly, formulating a theory on a lattice reduces an infinite-dimensional problem to a finite-dimensional one. It is set naturally to be analyzed using a computer.
Apart from these technical advantages, the reader may find interest in this chapter from another viewpoint. In terms of lattices one can show explicitly in simple models many of the physical behaviors of Wilson loops that we could only introduce heuristically in previous chapters.
Lattice gauge theories were first explored in 1971 by Wegner . He considered a usual Ising model with up and down spins but with a local symmetry. He associated a spin to each link in the lattice and considered an action that was invariant under a spin-flip of all the spins associated with links emanating from a vertex. He noted that this model could undergo phase transitions, but contrary to what happens with usual Ising models, his model did not magnetize. The absence of the magnetization posed him with the problem of distinguishing the phases of the theory.