Understanding truncated non-commutative geometries through computer simulations
Lisa Glaser, Abel Stern
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This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Journal of Mathematical Physics 61, 033507 (2020) and may be found at https://doi.org/10.1063/1.5131864
When aiming to apply mathematical results of non-commutative geometry to physical problems the question arises how they translate to a context in which only a part of the spectrum is known. In this article we aim to detect when a finite-dimensional triple is the truncation of the Dirac spectral triple of a spin manifold. To that end, we numerically investigate the restriction that the higher Heisenberg equation [A. H. Chamseddine, A. Connes, and V. Mukhanov, Journal of High Energy Physics, 98 (2014)] places on a truncated Dirac operator. We find a bounded perturbation of the Dirac operator on the Riemann sphere that induces the same Chern class.
Cette these etudie l'aspect metrique de la geometrie non commutative a travers la formulation de Connes de la distance entre etats d'une algebre. Sont etudies des exemples d'espaces finis et le modele standard ou le champs de Higgs s'interprete comme coefficient d'une metrique dans une dimension interne discrete.
We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra $\mathcal{S}(\mathbb{R}^{1,1}) \otimes M_2(\mathbb{C})$, which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space.
We classify the twists of almost commutative spectral triples that keep the Hilbert space and the Dirac operator untouched. The involved twisting operator is shown to be the product of the grading of a manifold by a finite dimensional operator, which is not necessarily a grading of the internal space. Necessary and sufficient constraints on this operator are listed.
We analyze the leading terms of the spectral action for a model of noncommutative geometry, which is a product of $4$-dimensional Riemannian manifold with a two-point space exploring the previously neglected case when the metrics over each sheet are different. Assuming the Friedmann-Lemaître-Robertson-Walker type of the metric for both sheets we obtain the action, which in addition to the the usual cosmological constant terms and the Einstein-Hilbert term involves a nonlinear interaction term. We study qualitative picture of potential consequences of such term in the basic cosmological models.
In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We attempt to flesh out the details of Connes' ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemannian spin manifolds. Throughout we are as explicit and elementary as possible.
A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the Dirac operator. Examples include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are investigated in detail, and it is shown that the fuzzy analogues correspond to two spinor fields on the commutative sphere. In some cases it is necessary to add a mass mixing matrix to the commutative Dirac operator to get a precise agreement for the eigenvalues.
We explore the geometric implications of introducing a spectral cut-off on Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are \emph{truncated} by spectral projections of Dirac-type operators. We prove that the underlying Riemannian manifold is the Gromov-Hausdorff limit of the metric spaces we associate to its truncations. This leads us to propose a computational algorithm that allows us to recover these metric spaces from the finite-dimensional truncated spectral data. We subsequently develop a technique f…
We present a short introductory overview of the non-commutative extensions of several classical physical theories. After a general discussion of the reasons that suggest that the non-commutativity is a major issue that will eventually lead to the unification of gravity with other fundamental interactions, we display examples of non-commutative generalizations of known geometries. Finally we discuss the general properties of the algebras that could become generalizations of algebras of smooth functions on Minkowskian (Riemannian) manifolds, needed for the description of Quantum Gravity.
This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
In the context of non-commutative geometries, we develop a group Fourier transform for the Lie group SU(2). Our method is based on the Schwinger representation of the Lie algebra su(2) in terms of spinors. It allows us to prove that the non-commutative R^3 space dual to the SU(2) group is in fact of the Moyal-type and endowed with the Voros star-product when expressed in the spinor variables. Finally, from the perspective of quantum gravity, we discuss the application of these new tools to group field theories for spinfoam models and their interpretation as non-commutative field theories with …
REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a short discussion on the uni-modulare case, and the computation for $H= U_q(sl_2)$. THE OLD ABSTRACT: We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a twisted antipode. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the …
Two examples of spectral triples with non-integer dimension spectrum are considered. These triples involve commutative C*-algebras. The first example has complex dimension spectrum and trivial differential algebra. The other is a parameter dependent deformation of the canonical spectral triple over S1. Its dimension spectrum includes real non-integer values. It has a non-trivial differential algebra and in contrast with the one dimensional case there are no junk forms for a non-vanishing deformation parameter. The distance on this space depends non-trivially on this parameter.
We recall a construction of non-commutative algebras related to a one-parameter family of (deformed) spheres and tori, and show that in the case of tori, the *-algebras can be completed into C*-algebras isomorphic to the standard non-commutative torus. As the former was constructed in the context of matrix (or fuzzy) geometries, it provides an important link to the framework of non-commutative geometry, and opens up for a concrete way to study deformations of non-commutative tori. Furthermore, we show how the well-known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of f…
Region-of-Interest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only x-ray projections intersecting $C$ with the goal to reduce overall radiation exposure when only a small specific region of the body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve $Γ$ in $\mathbb{R}^3$ verifying classical Tuy's condition. In this situation, the {\it non-trucated} cone-beam transform $D f$ of smooth densities $f$ admits an explicit inverse $Z$; however $Z$ cannot directly reconstruct $f$ from ROI-truncated projections. To deal w…
The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzi…
We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)--commutative geometries. By selecting gauging data these geometries are realized by matrices through an explicit construction or a Kan-extension. We describe the changes in gauge via the action of a regauging groupoid. It acts via matrices that give rise to a noncommutative 2--cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of regaugings. In the commutative case, …
The behaviour of solutions to the partial differential equation $(D + λW)f_λ= 0$ is discussed, where $D$ is a normal hyperbolic partial differential operator, or pre-normal hyperbolic operator, on $n$-dimensional Minkowski spacetime. The potential term $W$ is a $C_0^\infty$ kernel operator which, in general, will be non-local in time, and $λ$ is a complex parameter. A result is presented which states that there are unique advanced and retarded Green's operators for this partial differential equation if $|λ|$ is small enough (and also for a larger set of $λ$ values). Moreover, a scattering oper…
The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an L$_\infty^{full}$ algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory. We derive the minimal set of non-vanishing $\ell$-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L$_\infty$ algebras. On the other hand, it can be used as a starting point for bootstrapping the full non-commutative gauge theory. The first few brackets of such a theo…
This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September 1-10, 1997. In the published version, an epilogue of recent developments and many new references from 1998-2006 have been added. 1. Commutative geometry from the noncommutative point of view. 2. Spectral triples on the Riemann sphere. 3. Real spectral triples, the axiomatic foundation. 4. Geometries on the noncommutative torus. 5. The noncommutative integral. 6.…
