The first part of this paper is a review of the author’s work with S. Bahcall which gave an elementary derivation of the Chern Simons description of the Quantum Hall effect for filling fraction 1/n. The notation has been modernized to conform with standard gauge theory conventions. In the second part arguments are given to support the claim that abelian non– commutative Chern Simons theory at l...

We propose a general scheme for the “logic” of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non Commutative Geometry. It involves Baire*-algebras, the non-commutative version of measurable functions, arising as envelope of the C*-algebras identifying the topology of the (non-commutative) phase space. We outline some conseq...

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of...

We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to ...

The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative ge...

Hopf-Galois extensions are known to be the right generalizations of both Galois field extensions and principal G-bundles in the framework of non-commutative associative algebras. An abundant literature has been devoted to them by Hopf algebra specialists (see [Mg], [Sn1], [Sn2] and references therein). Recently there has been a surge of interest in Hopf-Galois extensions among mathematicians an...

In this paper we show the deep connection between the WignerMoyal approach and the Bohm approach to quantum mechanics. We point out that the key equations used in the Bohm approach were already contained in Moyal’s classic 1949 paper. Furthermore we argue that these two approaches can be seen as different but related aspects of standard quantum formalism when the algebraic approach, rather than...

We re-examine various issues surrounding the definition of twisted quantum field theories on flat noncommutative spaces. We propose an interpretation based on nonlocal commutative field redefinitions which clarifies previously observed properties such as the formal equivalence of Green’s functions in the noncommutative and commutative theories, causality, and the absence of UV/IR mixing. We use...