Continuum limit of lattice dynamics (Q1077056)

The starting point for quantum theory of interacting fields is usually the ''free'' or ''Fock'' representation of the algebra of physical observables. Next, when the cut-off in nonlinear terms is removed, it appears necessary to pass to another representation of this algebra. Such a procedure is unsatisfactory in some respects. Logically, the choice of a particular ''initial'' representation is unjustified. We cannot be sure that it is ''a good starting point''. Modifications (normal ordering, etc.) needed in originally ill-defined expressions are not physically clear. The change of representation is the main source of divergencies in quantum field theory. Is it possible to construct the quantum dynamics directly for algebras of observables, without any particular representation ? In this paper the author presents a scheme of such construction, based on finite lattice approximation. Transition from one Cauchy hypersurface in space-time to another (the evolution) is defined by passing with a lattice to the continuum limit.