The Ultraviolet Structure of Quantum Field Theories. Part 1: Quantum Mechanics

Abstract: This paper fires the opening salvo in the systematic construction of the lattice-continuum correspondence, a precise dictionary that describes the emergence of continuum quantum theories from finite, nonperturbatively defined models ("lattice theories"). Here the focus will be on quantum field theory in (0+1)D, i.e. quantum mechanics. The main conceptual achievement is an explicit and systematic procedure for reducing a theory with a large but finite Hilbert space to a subtheory in which wavefunctions satisfy prescribed smoothness and compactness constraints. This reduction, here named taming, in effect defines quantum mechanics on a continuum target space. When appropriate lattice theories are tamed, many familiar continuum notions explicitly emerge, e.g. canonical commutation relations, contact terms in correlation functions, continuous spacetime symmetries, and supersymmetry algebras. All of these are thus "put on the lattice" using the present framework. This analysis also leads to further insights into old subjects: for example, it is proven that any supersymmetric lattice theory must have a vanishing Witten index.