Symmetries of Discrete Systems

In this series of lectures presented at the CIMPA Winter School on Discrete Integrable Systems in Pondicherry, India, in February, 2003 we give a review of the application of Lie point symmetries, and their generalizations to the study of difference equations. The overall theme of these lectures could be called "continuous symmetries of discrete equations".Comment: 58 pages, 5 figures, Lectures presented at the Winter School on Discrete Integrable Systems in Pondicherry, India, February 200

Orthogonal nets and Clifford algebras

A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is discussed in a conformal setting. This way, the notions of ``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal systems'' are obtained --- the latter as a generalization of discrete orthogonal systems in Euclidean space. The relation of a Cauchy problem for discrete orthogonal nets and a permutability theorem for the Ribaucour transformation of smooth orthogonal systems is discussed.Comment: Plain TeX, 16 pages, 4 picture

Discrete-time multi-scale systems

We introduce multi-scale filtering by the way of certain double convolution systems. We prove stability theorems for these systems and make connections with function theory in the poly-disc. Finally, we compare the framework developed here with the white noise space framework, within which a similar class of double convolution systems has been defined earlier

Canonical quantization of constrained theories on discrete space-time lattices

We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.Comment: 18 pages, RevTex, one figur