Noncommutative Spectral Geometry Research Articles

Quantum Gravity and Cosmology: an intimate interplay

I will briefly discuss three cosmological models built upon three distinct quantum gravity proposals. I will first highlight the cosmological rôle of a vector field in the framework of a string/brane cosmological model. I will then present the resolution of the big bang singularity and the occurrence of an early era of accelerated expansion of a geometric origin, in the framework of group field theory condensate cosmology. I will then summarise results from an extended gravitational model based on non-commutative spectral geometry, a model that offers a purely geometric explanation for the standard model of particle physics.

Solar System and stellar tests of noncommutative spectral geometry

By using purely geometric forces on a noncommutative spacetime, noncommutative spectral geometry (NCSG) was proposed as a possible way to unify gravitation with the other known fundamental forces. The correction of the NCSG solution to Einstein's general relativity (GR) in the four-dimensional spacetime can be characterized by a parameter \(\beta\sim 1/\sqrt{f_{0}}\), where \( f_{0}\) denotes the coupling constants at the unification. The parameter \( \beta\) contributes a Yukawa-type correction \(\mathrm{exp}(-\beta r)/r\) to the Newtonian gravitational potential at the leading order, which can be interpreted as either the massive component of the gravitational field or the typical range of interactions carried by that component of the field. As an extension of previous works, we mainly focus on the Solar System and stellar tests of the theory, and the constraints on \(\beta\) obtained by the present work is independent of the previous ones. In the Solar System, we investigate the effects of the NCSG on the perihelion shift of a planet, deflection of light, time delay at superior conjunction (SC) and inferior conjunction (IC), and the Cassini experiment by modeling new observational results and adopting new datasets. In the binary pulsars system, based on the observational data sets of four systems of binary pulsars, PSR B1913+16, PSR B1534+12, PSR J0737-3039, and PSR B2127+11C, the secular periastron precessions are used to constrain this theory. These effects in the scale of the Solar System and binary pulsars were not considered in previous works. We find that the lower bounds given by these experiments are \(\beta \simeq 10^{-9} \sim 10^{-10}\) m-1, considerably smaller than those obtained in laboratory experiments. This confirms that experiments and observations at smaller scales are more favorable for testing the NCSG theory.

Linear stability of noncommutative spectral geometry

We consider the spectral action within the context of a 4-dimensional manifold with torsion and show that, in the vacuum case, the equations of motion reduce to Einstein's equations, securing the linear stability of the theory. To subsequently investigate the nonvacuum case, we consider the spectral action of an almost commutative torsion geometry and show that the Hamiltonian is bounded from below, a result which guarantees the linear stability of the theory.

Localization of quantum topology in the presence of matter and gauge fields

In this paper a toy model of quantum topology is reviewed to study effects of matter and gauge fields on the topology fluctuations. In the model a collection of N one-dimensional manifolds is considered where a set of boundary conditions on states of Hilbert space specifies a set of all topologies perceived by quantum particle and probability of having a specific topology is determined by a partition function over all the topologies in the context of noncommutative spectral geometry. In general the topologies will be fuzzy with the exception of a particular case which is localized by imposing a specific boundary condition. Here fermions and bosons are added to the model. It is shown that in the presence of matter, the fuzziness of topology will be dependent on N, however for large N the dependence is removed similar to the case without matter. Also turning on a particular background gauge field can overcome the fuzziness of topology to reach a localized topology with classical interpretation. It can be seen that for large N more opportunities can be provided for choosing the background gauge field to localize the fuzzy topology.

Aspects of the Bosonic Spectral Action

A brief description of the elements of noncommutative spectral geometry as an approach to unification is presented. The physical implications of the doubling of the algebra are discussed. Some high energy phenomenological as well as various cosmological consequences are presented. A constraint in one of the three free parameters, namely the one related to the coupling constants at unification, is obtained, and the possible role of scalar fields is highlighted. A novel spectral action approach based upon zeta function regularisation, in order to address some of the issues of the traditional bosonic spectral action based on a cutoff function and a cutoff scale, is discussed.

Constraining models of extended gravity using Gravity Probe B and LARES experiments

We consider models of extended gravity and in particular, generic models containing scalar-tensor and higher-order curvature terms, as well as a model derived from noncommutative spectral geometry. Studying, in the weak-field approximation (the Newtonian and post-Newtonian limit of the theory), the geodesic and Lense-Thirring processions, we impose constraints on the free parameters of such models by using the recent experimental results of the Gravity Probe B (GPB) and Laser Relativity Satellite (LARES) satellites. The imposed constraint by GPB and LARES is independent of the torsion-balance experiment, though it is much weaker.

Highlights of non-commutative spectral geometry

A summary of non-commutative spectral geometry as an approach to unification has been presented. The role of the doubling of the algebra, the seeds of quantization and some cosmological implications have been briefly discussed.

Doubling of the algebra and neutrino mixing within noncommutative spectral geometry

We study the physical implications of the doubling of the algebra, an essential element in the construction of the noncommutative spectral geometry model, proposed by Connes and his collaborators as offering a geometric explanation for the standard model of the strong and electroweak interactions. Linking the algebra doubling to the deformed Hopf algebra, we build Bogoliubov transformations and show the emergence of neutrino mixing.

Constraints on noncommutative spectral action from Gravity Probe B and torsion balance experiments

Noncommutative spectral geometry offers a purely geometric explanation for the standard model of strong and electroweak interactions, including a geometric explanation for the origin of the Higgs field. Within this framework, the gravitational, the electroweak and the strong forces are all described as purely gravitational forces on a unified noncommutative space-time. In this study, we infer a constraint on one of the three free parameters of the model, namely the one characterising the coupling constants at unification, by linearising the field equations in the limit of weak gravitational fields generated by a rotating gravitational source, and by making use of recent experimental data. In particular, using data obtained by Gravity Probe B, we set a lower bound on the Weyl term appearing in the noncommutative spectral action, namely β≳10−6m−1. This constraint becomes stronger once we use results from torsion balance experiments, leading to β≳104m−1. The latter is much stronger than any constraint imposed so far to curvature squared terms.

Noncommutative spectral geometry and the deformed Hopf algebra structure of quantum field theory

We report the results obtained in the study of Alain Connes noncommutative spectral geometry construction focusing on its essential ingredient of the algebra doubling. We show that such a two-sheeted structure is related with the gauge structure of the theory, its dissipative character and carries in itself the seeds of quantization. From the algebraic point of view, the algebra doubling process has the same structure of the deformed Hops algebra structure which characterizes quantum field theory.

Noncommutative spectral geometry: a short review

We review the noncommutative spectral geometry, a gravitational model that combines noncommutative geometry with the spectral action principle, in an attempt to unify General Relativity and the Standard Model of electroweak and strong interactions. Despite the phenomenological successes of the model, the discrepancy between the predicted Higgs mass and the current experimental data indicate that one may have to go beyond the simple model considered at first. We review the current status of the phenomenological consequences and their implications. Since this model lives by construction at high energy scales, namely at the Grand Unified Theories scale, it provides a natural framework to investigate early universe cosmology. We briefly review some of its cosmological consequences.

Noncommutative spectral geometry, dissipation and the origin of quantization

We present a physical interpretation of the doubling of the algebra, which is the basic ingredient of the noncommutative spectral geometry, developed by Connes and collaborators as an approach to unification. We discuss its connection to dissipation and to the gauge structure of the theory. We then argue, following 't Hooft's conjecture, that noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.

Noncommutative spectral geometry, algebra doubling, and the seeds of quantization

A physical interpretation of the two-sheeted space, the most fundamental ingredient of noncommutative spectral geometry proposed by Connes as an approach to unification, is presented. It is shown that the doubling of the algebra is related to dissipation and to the gauge structure of the theory, the gauge field acting as a reservoir for the matter field. In a regime of completely deterministic dynamics, dissipation appears to play a key role in the quantization of the theory, according to 't Hooft's conjecture. It is thus argued that the noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.

Cosmological consequences of the noncommutative spectral geometry as an approach to unification

Noncommutative spectral geometry succeeds in explaining the physics of the Standard Model of electroweak and strong interactions in all its details as determined by experimental data. Moreover, by construction the theory lives at very high energy scales, offering a natural framework to address early universe cosmological issues. After introducing the main elements of noncommutative spectral geometry, I will summarise some of its cosmological consequences and discuss constraints on the gravitational sector of the theory.

Non-commutative Weitzenböck geometry, gerbe modules and WZW branes

We study the non-commutative matrix model which arises as the low-energy effective action of open strings in WZW models. We re-derive this fuzzy effective gauge dynamics in two different ways, without recourse to conformal field theory: The first method starts from a linearised version of the WZW σ-model, which is classically equivalent to an action of the Schild type, which in turn can be quantised in a natural way to yield the matrix model. The second method relies on purely geometric symmetry principles—albeit within the non-commutative spectral geometry that is provided by the boundary CFT data: we show that imposing invariance under extended gauge transformations singles out the string-theoretic action up to the relevant order in the gauge field. The extension of ordinary gauge transformations by tangential shifts is motivated by the gerbe structure underlying the classical WZW model and standard within Weitzenbock geometry—which is a natural reformulation of geometry to use when describing strings in targets with torsion.

Dirac Operator on the Podleś Sphere

The spectral problem for the Dirac operator on the Podleś sphere is discussed and solved in one of its possible formulations. The standard constructions for the Dirac operator on classical symmetric spaces and the spectrum of the Dirac operator on the classical two-dimensional sphere are recalled. The problem of defining a spinor structure on a quantum space is discussed and the definitions of a classical spinor structure and Dirac operator according to Ðurdevich are sketched. The Dirac operator for the Podles´ quantum sphere treated as a quotient space of SμU(2) is constructed using the Woronowicz left-covariant calculus over this quantum group. The spectrum of the operator is obtained. Disagreement of its asymptotic behavior with Connes' axiom of noncommutative spectral geometry is stressed.

Adiabatic limits and spectral geometry of foliations

We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundle-like metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution function is obtained. The relationships with the spectral theory of leafwise Laplacian and with the noncommutative spectral geometry of foliations are discussed.

Noncommutative spectral geometry of riemannian foliations

We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.