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Register a new userNoncommutative Geometry and Particle Physics
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Fedele Lizzi
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We review the noncommutative approach to the standard model. We start with the introduction if the mathematical concepts necessary for the definition of noncommutative spaces, and manifold in particular. This defines the framework of spectral geometry. This is applied to the standard model of particle interaction, discussing the fermionic and bosonic spectral action. The issues relating to the calculation of the mass of the Higgs are discussed, as well as the role of neutrinos and Wick rotations. Finally, we present the possibility of solving the problem of the Higgs mass by considering a pregeometric grand symmetry.
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The section condition of Double Field Theory has been argued to mean that doubled coordinates are gauged: a gauge orbit represents a single physical point. In this note, we consider a doubled and at the same time gauged particle action, and show that its BRST formulation including Faddeev--Popov ghosts matches with the graded Poisson geometry that has been recently used to describe the symmetries of Double Field Theory. Besides, by requiring target spacetime diffeomorphisms at the quantum level, we derive quantum corrections to the classical action involving dilaton, which might be comparable with the Fradkin--Tseytlin term on string worldsheet.
We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative ma
nifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum isometry group with the universal object in a bigger category, namely the category of `quantum families of smooth isometries, defined along the line of Woronowicz and Soltan. We also construct a spectral triple on the Hilbert space of forms on a noncommutative manifold which is equivariant with respect to a natural unitary representation of the quantum isometry group. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in cite{hajac} as the universal quantum group of holomorphic isometries of the noncommutative torus.
We study some consequences of noncommutativity to homogeneous cosmologies by introducing a deformation of the commutation relation between the minisuperspace variables. The investigation is carried out for the Kantowski-Sachs model by means of a comparative study of the universe evolution in four different scenarios: the classical commutative, classical noncommutative, quantum commutative, and quantum noncommutative. The comparison is rendered transparent by the use of the Bohmian formalism of quantum trajectories. As a result of our analysis, we found that noncommutativity can modify significantly the universe evolution, but cannot alter its singular behavior in the classical context. Quantum effects, on the other hand, can originate non-singular periodic universes in both commutative and noncommutative cases. The quantum noncommutative model is shown to present interesting properties, as the capability to give rise to non-trivial dynamics in situations where its commutative counterpart is necessarily static.
We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.
We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(mathcal{A},H,D)$ and self-adjoint $V in mathcal{A}$, there holds [lim_{hdownarrow 0} h^pmathrm{Tr}(chi_{(-infty,0)}(h^2D^2+V)) = int V_-^{frac{p}{2}}|ds|^p.] where $int$ is Connes noncommutative integral.
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