Aristophanes Dimakis

Noncommutative differential calculus and lattice gauge theory

The authors study consistent deformations of the classical differential calculus on algebras of functions (and, more generally, commutative algebras) such that differentials and functions satisfy nontrivial commutation relations. For a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice. These differential calculi generate a lattice on a space continuum. The whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus. In this framework one just has to express the Lagrangian for the continuum theory in terms of differential forms. This expression then also makes sense for the deformed differential calculus. There is a natural integral associated with the latter. Integration of the Lagrangian over a space continuum then produces the correct lattice action for a large class of theories. This is explicitly shown for the scalar field action and the action for SU(m) gauge theory.

Discrete differential calculus: Graphs, topologies, and gauge theory

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 (a la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, a ‘‘symmetric lattice’’ is also studied which (in a certain continuum limit) turns out to be related to a ‘‘noncommutative differential calculus’’ on manifolds.

Differential calculi and linear connections

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one‐to‐one correspondence, between the module structure of the 1‐forms and the metric torsion‐free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1‐forms.

Discrete Riemannian geometry

Within a framework of noncommutative geometry, we develop an analog of (pseudo-) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs (the vertices of the latter are given by the elements of the finite set). Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (nonlocal) tensor product over the algebra of functions, as considered previously by several authors. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on...

Quantum mechanics on a lattice and q-deformations

Abstract Based on previous work on a relation between noncommutative differential calculus and lattice structures it is shown that q-calculus arises as an exponentiated form of lattice calculus. In particular, quantum mechanics on a lattice can also be described as “q-deformed quantum mechanics”.

Bi-differential calculi and integrable models

The existence of an infinite set of conserved currents in completely integrable classical models, including chiral and Toda models as well as the KP and self-dual Yang-Mills equations, is traced back to a simple construction of an infinite chain of closed (respectively, covariantly constant) 1-forms in a (gauged) bi-differential calculus. The latter consists of a differential algebra on which two differential maps act. In a gauged bi-differential calculus these maps are extended to flat covariant derivatives.

From continuum to lattice theory via deformation of the differential calculus

Abstract It is shown that a lattice (gauge) theory can be obtained from a continuum theory via deformation of the standard differential calculus in such a way that functions and differentials no longer commute. Any lagrangian for a continuum theory which can be expressed in terms of differential forms is also defined for the deformed differential calculus. Using an integral which is naturally associated with the deformed differential calculus, one obtains an action for a lattice theory.

Moyal Deformation, Seiberg–Witten Maps, and Integrable Models

A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.

Discrete differential manifolds and dynamics on networks

A discrete differential manifold is a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time. Several examples are presented and a notion of differentiability of maps between discrete differential manifolds is introduced. Particular attention is given to differentiable curves in such spaces. Every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity.

The Korteweg–de-Vries equation on a noncommutative space-time

Abstract We construct a deformation quantized version (ncKdV) of the KdV equation which possesses an infinite set of conserved densities. Solutions of the ncKdV equation are obtained from solutions of the KdV equation via a kind of Seiberg–Witten map. The ncKdV equation is related to a modified ncKdV equation by a noncommutative Miura transformation.