On a Classification of Irreducible Almost Commutative Geometries

We classify all irreducible, almost commutative geometries whose spectral action is dynamically non-degenerate. Heavy use is made of Krajewski's diagrammatic language. The motivation for our definition of dynamical non-degeneracy stems from particle physics where the fermion masses are non-degenerate.Comment: 47 pages, 9 figure

Spectral action for torsion with and without boundaries

We derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the (skew-symmetric) torsion and the chiral bag conditions on the boundary. The spectral action splits into bulk and boundary parts. In the bulk, we clarify certain issues of the previous calculations, show that many terms in fact cancel out, and demonstrate that this cancellation is a result of the chiral symmetry of spectral action. On the boundary, we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter $\theta$ of the boundary conditions, and show that $\theta=0$ is a critical point of the action in any dimension and at all orders of the expansion.Comment: 16 pages, references adde

Carnot-Caratheodory metric and gauge fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Caratheodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes's distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator ? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Caratheodory distance dh defined by A. In this paper we precise this link, showing that the equality of d and dh strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).Comment: published version with additional figures to make the proof more readable. Typos corrected in this ultimate versio

Chiral Asymmetry and the Spectral Action

We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter

Euclidean Supergravity in Terms of Dirac Eigenvalues

It has been recently shown that the eigenvalues of the Dirac operator can be considered as dynamical variables of Euclidean gravity. The purpose of this paper is to explore the possiblity that the eigenvalues of the Dirac operator might play the same role in the case of supergravity. It is shown that for this purpose some primary constraints on covariant phase space as well as secondary constraints on the eigenspinors must be imposed. The validity of primary constraints under covariant transport is further analyzed. It is show that in the this case restrictions on the tanget bundle and on the spinor bundle of spacetime arise. The form of these restrictions is determined under some simplifying assumptions. It is also shown that manifolds with flat curvature of tangent bundle and spinor bundle and spinor bundle satisfy these restrictons and thus they support the Dirac eigenvalues as global observables.Comment: Misprints and formulae corrected; to appear in Phys. Rev.

Spectral noncommutative geometry and quantization: a simple example

We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over this phase space and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two ``spacetime points''), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operator *D on the direct product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization of the family of the triples (A, H, D).Comment: 7 pages, no figure

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log \lambda$ converges to an explicit constant ($\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\"odinger dynamics generated by the Fibonacci Hamiltonian.Comment: 23 page

The Spectral Action for Dirac Operators with skew-symmetric Torsion

We derive a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion. We find that the torsion becomes dynamical and couples to the traceless part of the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard Model of particle physics in presence of torsion from the Chamseddine-Connes Dirac operator.Comment: Longer introduction and conclusion adde

On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

Local covariant quantum field theory over spectral geometries

A framework which combines ideas from Connes' noncommutative geometry, or spectral geometry, with recent ideas on generally covariant quantum field theory, is proposed in the present work. A certain type of spectral geometries modelling (possibly noncommutative) globally hyperbolic spacetimes is introduced in terms of so-called globally hyperbolic spectral triples. The concept is further generalized to a category of globally hyperbolic spectral geometries whose morphisms describe the generalization of isometric embeddings. Then a local generally covariant quantum field theory is introduced as a covariant functor between such a category of globally hyperbolic spectral geometries and the category of involutive algebras (or *-algebras). Thus, a local covariant quantum field theory over spectral geometries assigns quantum fields not just to a single noncommutative geometry (or noncommutative spacetime), but simultaneously to ``all'' spectral geometries, while respecting the covariance principle demanding that quantum field theories over isomorphic spectral geometries should also be isomorphic. It is suggested that in a quantum theory of gravity a particular class of globally hyperbolic spectral geometries is selected through a dynamical coupling of geometry and matter compatible with the covariance principle.Comment: 21 pages, 2 figure