We formulate generalized uncertainty relations in a crystal-like universe—a “world crystal”—
whose lattice spacing is of the order of Planck length. In the particular case when energies …

This review is based on two lectures given at the 2000 TMR school in Torino (TMR school
on contemporary String Theory and Brane Physics, 26 January-2 February 2000, Torino …

We present a lattice formulation of non-commutative Yang-Mills theory in arbitrary even
dimensionality. The UV/IR mixing characteristic of noncommutative field theories is …

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 …

We consider the U (1) gauge theory on a four-dimensional torus, where the instanton
number is restricted to an integral multiple of p. This theory possesses the nontrivial higher …

We study metric properties stemming from the Connes spectral distance on three types of
non-compact non-commutative spaces which have received attention recently from various …

We develop differential calculus and gauge theory on a finite set G. An elegant formulation is
obtained when G is supplied with a group structure and in particular for a cyclic group …

Following the general principles of noncommutative geometry, it is possible to define a
metric on the space of pure states of the noncommutative algebra generated by the …

A finite set can be supplied with a group structure which can then be used to select (classes
of) differential calculi on it via the notions of left-, right-and bicovariance. A corresponding …

We construct a non-trivial principal bundle on T 4 from the compact U (1) lattice gauge field
by generalizing Lüscher's constriction so that the cocycle condition contains elements (the't …