Lattice Gauge Theory - Wilson Action

[istaslis]

I have one question, which can be very simple, but i can't answer it.
I have Wilson action for ordinary SU(2) gauge field. In all of the books I had read, the proof, that calculations in lattice theory is true - is equality the Wilson action in continuum limit to the continuum action. We use that exponents in [tex]U_\mu=\exp(iaA_\mu)[/tex] are small (because a is the vanishing lattice spacing) for uniting them into one and so on. And we can derive that [tex] S_W=\sum{1-U_{\mu \nu}} \rightarrow \sum{F_{ \mu \nu} F_{ \mu \nu } } [/tex] . But in real experiments I see that exponents are not small! And I am at a loss how we can use it and why is this procedure true after calculations.
Thank you!

 

[LightHero]

The Wilson action is a lattice approximation of the continuum action. This means that the lattice spacing (a) is chosen to be small enough so that the results of the lattice calculations are close enough to the results of the continuum theory. The exponents in U_mu may not be small, but the difference between the results of the lattice and continuum theories still remains small. This is because the lattice spacing is chosen to be small enough so that the exponentials can be expanded out to higher order terms, which will have a smaller effect on the overall result.