Internal Space for the Noncommutative Geometry Standard Model and Strings

In this paper I discuss connections between the noncommutative geometry approach to the standard model on one side, and the internal space coming from strings on the other. The standard model in noncommutative geometry is described via the spectral action. I argue that an internal noncommutative manifold compactified at the renormalization scale, could give rise to the almost commutative geometry required by the spectral action. I then speculate how this could arise from the noncommutative geometry given by the vertex operators of a string theory.Comment: 1+22 pages. More typos and misprints correcte

The Kirillov picture for the Wigner particle

We discuss the Kirillov method for massless Wigner particles, usually (mis)named "continuous spin" or "infinite spin" particles. These appear in Wigner's classification of the unitary representations of the Poincar\'e group, labelled by elements of the enveloping algebra of the Poincar\'e Lie algebra. Now, the coadjoint orbit procedure introduced by Kirillov is a prelude to quantization. Here we exhibit for those particles the classical Casimir functions on phase space, in parallel to quantum representation theory. A good set of position coordinates are identified on the coadjoint orbits of the Wigner particles; the stabilizer subgroups and the symplectic structures of these orbits are also described.Comment: 19 pages; v2: updated to coincide with published versio

From Peierls brackets to a generalized Moyal bracket for type-I gauge theories

In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang--Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to i hbar times the Peierls bracket to lowest order in hbar.Comment: 14 pages, Late

Noncommutative differential calculus for Moyal subalgebras

We build a differential calculus for subalgebras of the Moyal algebra on R^4 starting from a redundant differential calculus on the Moyal algebra, which is suitable for reduction. In some cases we find a frame of 1-forms which allows to realize the complex of forms as a tensor product of the noncommutative subalgebras with the external algebra Lambda^*.Comment: 13 pages, no figures. One reference added, minor correction

Non-commutative geometry and the standard model vacuum

The space of Dirac operators for the Connes-Chamseddine spectral action for the standard model of particle physics coupled to gravity is studied. The model is extended by including right-handed neutrino states, and the S0-reality axiom is not assumed. The possibility of allowing more general fluctuations than the inner fluctuations of the vacuum is proposed. The maximal case of all possible fluctuations is studied by considering the equations of motion for the vacuum. Whilst there are interesting non-trivial vacua with Majorana-like mass terms for the leptons, the conclusion is that the equations are too restrictive to allow solutions with the standard model mass matrix.Comment: 21 pages. v2: some comments improve

Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing

We show that the Ultraviolet/Infrared mixing of noncommutative field theories with the Gronewold-Moyal product, whereby some (but not all) ultraviolet divergences become infrared, is a generic feature of translationally invariant associative products. We find, with an explicit calculation that the phase appearing in the nonplanar diagrams is the one given by the commutator of the coordinates, the semiclassical Poisson structure of the non commutative spacetime. We do this with an explicit calculation for represented generic products.Comment: 24 pages, 1 figur

A coordinated control strategy for insulin and glucagon delivery in type 1 diabetes

Type 1 diabetes is an autoimmune condition characterised by a pancreatic insulin secretion deficit, resulting in high blood glucose concentrations, which can lead to micro- and macrovascular complications. Type 1 diabetes also leads to impaired glucagon production by the pancreatic α-cells, which acts as a counter-regulatory hormone to insulin. A closed-loop system for automatic insulin and glucagon delivery, also referred to as an artificial pancreas, has the potential to reduce the self-management burden of type 1 diabetes and reduce the risk of hypo- and hyperglycemia. To date, bihormonal closed-loop systems for glucagon and insulin delivery have been based on two independent controllers. However, in physiology, the secretion of insulin and glucagon in the body is closely interconnected by paracrine and endocrine associations. In this work, we present a novel biologically-inspired glucose control strategy that accounts for such coordination. An in silico study using an FDA-accepted type 1 simulator was performed to evaluate the proposed coordinated control strategy compared to its non-coordinated counterpart, as well as an insulin-only version of the controller. The proposed coordinated strategy achieves a reduction of hyperglycemia without increasing hypoglycemia, when compared to its non-coordinated counterpart

Star Product Geometries

We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. This way symmetry principles can be implemented. We review two main examples [15]-[18]: a) general covariance in noncommutative spacetime. This leads to a noncommutative gravity theory. b) Symplectomorphims of the algebra of observables associated to a noncommutative configuration space. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e., we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutators. New results concerning noncommutative gravity include the Cartan structural equations for the torsion and curvature tensors, and the associated Bianchi identities. Concerning scalar field theories the deformed algebra of classical and quantum observables has been understood in terms of a twist within the algebra.Comment: 27 pages. Based on the talk presented at the conference "Geometry and Operators Theory," Ancona (Italy), September 200

Noncommutative Gauge Field Theories: A No-Go Theorem

Studying the general structure of the noncommutative (NC) local groups, we prove a no-go theorem for NC gauge theories. According to this theorem, the closure condition of the gauge algebra implies that: 1) the local NC $u(n)$ {\it algebra} only admits the irreducible n by n matrix-representation. Hence the gauge fields are in n by n matrix form, while the matter fields {\it can only be} in fundamental, adjoint or singlet states; 2) for any gauge group consisting of several simple-group factors, the matter fields can transform nontrivially under {\it at most two} NC group factors. In other words, the matter fields cannot carry more than two NC gauge group charges. This no-go theorem imposes strong restrictions on the NC version of the Standard Model and in resolving the standing problem of charge quantization in noncommutative QED.Comment: latex, 4 page

Reduction Procedures in Classical and Quantum Mechanics

We present, in a pedagogical style, many instances of reduction procedures appearing in a variety of physical situations, both classical and quantum. We concentrate on the essential aspects of any reduction procedure, both in the algebraic and geometrical setting, elucidating the analogies and the differences between the classical and the quantum situations.Comment: AMS-LaTeX, 35 pages. Expanded version of the Invited review talk delivered by G. Marmo at XXIst International Workshop On Differential Geometric Methods In Theoretical Mechanics, Madrid (Spain), 2006. To appear in Int. J. Geom. Methods in Modern Physic