While there has been growing interest for noncommutative spaces in recent times, most examples have been based on the simplest noncommutative algebra: [x i , x j ] = iθ ij . Here we present new classes of (non-formal) deformed products associated to linear Lie algebras of the kind [x i , x j ] = ic k ij x k . For all possible threedimensional cases, we define a new star product and discuss its properties. To complete the analysis of these novel noncommutative spaces, we introduce noncompact spectral triples, and the concept of star triple, a specialization of the spectral triple to deformations of the algebra of functions on a noncompact manifold. We examine the generalization to the noncompact case of Connes' conditions for noncommutative spin geometries, and, in the framework of the new star products, we exhibit some candidates for a Dirac operator. On the technical level, properties of the Moyal multiplier algebra M (R 2n θ ) are elucidated.

Abstract:We investigate the possible effects on the evolution of perturbations in the inflationary epoch due to short distance physics. We introduce a suitable non local action for the inflaton field, suggested by Noncommutative Geometry, and obtained by adopting a generalized star product on a Friedmann-Robertson-Walker background. In particular, we study how the presence of a length scale where spacetime becomes noncommutative affects the gaussianity and isotropy properties of fluctuations, and the corresponding effects on the Cosmic Microwave Background spectrum.

It shown that the chiral model with SU(3) flavor symmetry predicts a dibaryon state of low mass M (M^2.2 GeV). It is electrically neutral and is an SU(3) singlet with J p =0 + .It corresponds to a six-quark state found in the MIT bag model by Jaffe. It is also shown that there is no stable particlelike state of baryon number 2 which is based on Skyrme's spherically symmetric Ansatz for the chiral field.PACS numbers: ll.30. Rd, 12.35.Ht, 14.20.Pt It is believed that the low-energy properties of QCD are effectively reproduced by the chiral model. The order parameter in this model when we consider only the light quarks is a field U where U(x) is a 3x3 SU(3) matrix. Skyrme pointed out many years ago 1 that this model admits solitons characterized by an integer-valued topological number and proposed to interpret the states with the unit value of this number as the nucleon and its excitations. He also suggested that the topological number t is the baryon number B of the nucleon. This conjecture was confirmed in all essential respects by Balachandran, Nair, Rajeev, and Stern 2 ' 3 who showed that b = const x t, where the constant is completely determined by the detailed assumptions in the treatment of the fermions in the model. Further studies of the chiral model 4,5 which include in particular the topological effects of the Wess-Zumino term also suggest that the \tI = 1 states are indeed fermions. There is thus good support to Skyrme's conjecture that the 1*1 = 1 solitons are baryons and t is related to the baryon number. The conservative assumption at this point would be to identify these states with the baryon octet and assume that t is exactly equal to B. Following Skyrme 1 and Witten, 4 we shall adopt this interpreta-tion for the purposes of this paper 6 ; our conclusions can, however, be readily modified if, as has been suggested, 2 the topological excitations represent a novel family of states.The stable static solutions with |2?| = 1 in the Skyrme model are described by a "spherically symmetric' ' configuration. In this context, spherical symmetry is understood in a generalized sense and depends on the choice of an SU(2) subgroup of the flavor SU(3). There are, however, spherically symmetric configurations which involve instead the SO (3) subgroup of real orthogonal matrices of SU(3) 7 and the major results of this note pertain to these configurations. The associated topological excitations are characterized by \B| = 0,2,4, ... . We show in this note that the lightest dibaryon states in this sequence with B = ± 2 have a mass of the order of 2.2 GeV. They are also expected to be SU(3) singlets with J p =0 + . In this note, we shall also briefly study the \B\ = 2 states based on the SU(2) subgroup and show that the corresponding static configurations are not stable even classically. Therefore we do not expect a dibaryon resonance identifiable with such a configuration. 8 We shall first briefly review the relevant aspects of Skyrme's model for three flavors. It is based on the Lagrangian density J?= -jflTrid^d^U...

We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from ⋆-Poisson brackets to ⋆-commutators. In particular commutation relations among creation and annihilation operators are deduced.