On the origin of spacetime topology and some generalizations of quantum field theory

Abstract

The research effort reported in this paper is directed, in a broad sense, towards understanding the small-scale structure of spacetime. The fundamental question that guides our discussion is `what is the physical content of spacetime topology?'. In classical physics, this question has a natural and simple answer: spacetime (as a topological space) is a book-keeping device that we invent to make the description of classical fields (the observables) easier. More precisely, if spacetime, , has sufficiently regular topology, and if sufficiently many fields exist to allow us to observe all continuous functions on X, then this collection of continuous functions uniquely determines both the set of points X and the topology on it. Naturally, however, this answer does not yield any clues into why our spacetime is observed to have a very special (i.e. smooth manifold) topology down to the smallest scales we can probe, or into whether this smooth manifold structure persists indefinitely, at length scales smaller than the smallest observed so far. To explore these queries, we are led to consider the original question in the context of quantum, rather than classical field theory. After all, in the real world physical fields (the observables) are not classical (continuous functions) but quantum operators, and the fundamental observable is not the collection of all continuous functions but the local algebra of quantum field operators. Presently the only examples of local quantum field algebras that we know how to construct rigorously (apart from some two-dimensional models) are the operator algebras of free (linear) quantum fields propagating on a smooth, globally hyperbolic spacetime. Since this class of examples is too small, we find it necessary to generalize the algebraic notion of `quantum field' in such a way that it becomes possible to talk about quantum field theory on an arbitrary (not necessarily smooth) topological space on which no notion of spacetime metric exists a priori. (One interesting offshoot of this generalization is an algebraic framework for linear quantum field theory on non-globally-hyperbolic spacetimes (e.g. spacetimes with naked singularities or closed time-like curves), which is the subject of a separate paper that appears elsewhere in this journal.) In pursuing the original problem further, we develop a still wider generalization of quantum field theory; this ultimate generalization dispenses with the fixed background topological space altogether and proposes that the fundamental spacetime observable is a lattice (or more specifically a `frame', in the sense of set theory) of closed subalgebras of an abstract algebra. Our discussion concludes with the definition and some elementary properties of these `quantum lattices' and `quantum frames'.