Discrete differential calculus: Graphs, topologies, and gauge theory

Abstract

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a 'reduction' of the 'universal differential algebra' and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a 'Hasse diagram' determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two-point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an 'internal' discrete space (à la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a 'symmetric lattice' is studied which (in a certain continuum limit) turns out to be related to a 'noncommutative differential calculus' on manifolds.

Non-commutative geometry of finite groups

“…If the number N of elements is prime, then the only irreducible group structure is Z Z N , the additive abelian group of integers modulo N. Differential calculi on discrete groups have been studied in [8,9]. More generally, differential calculus on discrete sets has been developed in [4,5].…”

“…A commutative algebra of particular interest in this context is the algebra of functions on a finite (or discrete) set. A differential calculus on a finite set provides the latter with a structure which may be viewed as a discrete counterpart to that of a (continuous) differentiable manifold [4,5]. It has been shown in [4] that (first order) differential calculi on discrete sets are in correspondence with (di)graphs with at most two (antiparallel) arrows between any two vertices.…”

“…A differential calculus on a finite set provides the latter with a structure which may be viewed as a discrete counterpart to that of a (continuous) differentiable manifold [4,5]. It has been shown in [4] that (first order) differential calculi on discrete sets are in correspondence with (di)graphs with at most two (antiparallel) arrows between any two vertices. This relation with graphs and networks suggests applications of the formalism to dynamics on networks [5] and the universal dynamics considered in [6], for example.…”

A No-Go Theorem for Compatibility Between Involutions of First Order Differentials on a Lattice and Continuum Limit

“…In [6], dx µ and dx * µ are algebraically independent generators of first order differential forms; this so-called nearest symmetric reduction has also been mentioned as an example in [7]. Due to the above-proved no-go theorem, coordinate functions have to be suppose to be algebraic independent to their involutive images, if antihomomorphic rule and continuum limit of involution are regarded as being more natural and more necessary, which in physics implies that a connection 1-form, thus a gauge field, has two components along one direction!…”

Distances on a lattice from non-commutative geometry

“…In the case of an hypercubic lattice, we find that points belonging to a straight line parallel to a coordinate axis ar not 'on a line', in the sense that distances between sites two apart is not twice the distance between prime neighbours, and in general the distances are not additive, although they become so in the limit of large separations. † For a review of approaches to lattice physics based on noncommutative geometry see [6] and references therein.…”