The P ( d ) , Euclidean quantum field theory as classical statistical mechanics

To date in constructive field theory, two types of ultraviolet cutoffs have been used [32]: 1) A smeared field, i.e., g(x) is replaced by $,(x) = \ h(x y)$(y)dy, where h is some smooth positive approximation to the delta function; 2) Box and sharp momentum cutoff, i.e., g(x) is replaced by the periodic field Q ~ , ~ ( X ) , which is obtained by approximating the Fourier integral defining g(x) by a finite Fourier sum. In this section we wish to introduce a new cutoff method in which Rd (space-imaginary time) is replaced by a lattice and the Laplacian A in the Euclidean propagator is approximated by a finite difference operator. The effect of this approximation is to replace the measure (11.25) by a perturbed Gaussian measure in a finite number of variables, q,, . . . ,q,, each associated with a lattice point: