| Abstract: | We construct an effective lattice action for a continuum theory, by fixing a set of collective coordinates which play the role of lattice variables. As opposed to Symanzik's improvement program our method involves no expansion in powers of the lattice spacing; in other words it simultaneously yields all “irrelevant” operators generated by the renormalization group to a given order in the continuum coupling constant. We are thus able to rigorously establish that the effective lattice action, for both smooth and singular collective coordinates, is local in the sense that long-range couplings decay exponentially over a distance independent of the mass gap of the theory; for asymtotically free theories this is interpreted as an existence proof of Wilson's infrared stable trajectories. Our methods are for convenience described in the context of dimensional φ4, but can be easily extended to any theory with a set of collective coordinates which (i) are renormalizable and (ii) provide an infrared cutoff. Application to the 2-dimensional O(N) σ-model is, in particular, discussed; the technical problems of renormalization posed by gauge invariance are, on the other hand, not dealt with in this paper, although our treatment of singular coordinates is meant as a prelude to them. A by-product of our proofs is the derivation of an interesting factorization property of Zimmermann-subtracted diagrams. |
