Lattice Approximation

Abstract

The lattice approximation gives a discrete finite dimensional approximation to Euclidean field theories. It is a useful ultraviolet cutoff. In contradistinction to other ultraviolet regularizations in Euclidean field theories it breaks the Euclidean invariance. The virtue of the lattice approximation is that it preserves the Osterwalder-Schrader (OS) positivity. If the Euclidean invariance is restored in the limit when the lattice spacing tends to zero then the main requirements for the continuation to QFT in Minkowski space will be satisfied. We define the lattice approximation for scalar fields in d dimensions. A relation to the continuous spin Ising model of statistical mechanics is explained. We show the OS positivity. We represent the lattice field by the continuum free field so that the continuum limit can be proved in perturbation theory. We develop the heat kernel representation on the lattice in terms of the compound Poisson process obtaining the polymer representation of the Higgs model discussed earlier in the continuum. We discuss the lattice approximation of gauge theories as a way to make the gauge field quantization a mathematically precise theory. We discuss the strong coupling expansion leading to the Higgs mechanism and to the fulfillment of the Wilson criterion for quark confinement.