The critical properties of the real φ scalar field theory are studied numerically on the fuzzy sphere. The fuzzy sphere is a finite matrix (non–commutative) approximation of the algebra of functions on the usual two dimensional sphere. It is also one of the simplest examples of a non–commutative space to study field theory on. Aside from the usual disordered and uniform phases present in the co...

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...

We say that a C∗-algebra X has the approximate n-th root property (n ≥ 2) if for every a ∈ X with ‖a‖ ≤ 1 and every ε > 0 there exits b ∈ X such that ‖b‖ ≤ 1 and ‖a − bn‖ < ε. Some properties of commutative and non-commutative C∗-algebras having the approximate nth root property are investigated. In particular, it is shown that there exists a non-commutative (resp., commutative) separable unita...

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 ...

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, is called inner or outer if multiplication by this defines an isometry dense range, respectively, as linear operator on Hardy Space, H2, functions with square-summable Taylor series. This factorization can be further refined; θ decomposes uniquely p...

We discuss non-commutative field theories in coordinate space. To do so we introduce pseudo-localized operators that represent interesting position dependent (gauge invariant) observables. The formalism may be applied to arbitrary field theories, with or without supersymmetry. The formalism has a number of intuitive advantages. First it makes clear the appearance of new degrees of freedom in th...

We consider the Standard Model on a non-commutative space and expand the action in the non-commutativity parameter θ . No new particles are introduced, the structure group is SU(3)×SU(2)×U(1). We derive the leading order action. At zeroth order the action coincides with the ordinary Standard Model. At leading order in θ we find new vertices which are absent in the Standard Model on commutative ...

We compute at the one-loop order the β-functions for a renormalisable noncommutative analog of the Gross Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The β-function for the non-commutative counterpart of the Thirring model is found ...

We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adèle class space of the field of rational numbers by the maximal compact subgroup of the idèle class group, which we had previously shown to yield the correct counting function to obtain the co...