The Spectral Action Principle

We propose a new action principle to be associated with a noncommutative space . The universal formula for the spectral action is where is a spinor on the Hilbert space, is a scale and a positive function. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling constants identical to those of SU(5) as well as the Higgs self-coupling, to be taken at a fixed high energy scale. Received: 1 October 1996 / Accepted: 15 November 1996     

The Holst Action by the Spectral Action Principle

We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields.     

The Uncanny Precision of the Spectral Action

Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as an action functional for the theory. To demonstrate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic expansion in the case of the round sphere S ³. We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of S ³ × S ¹, where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms a _2n vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a ₄ and a ₆ both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function.     

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.     

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 the presence of torsion from the Chamseddine-Connes Dirac operator.     

Spectral Action Beyond the Weak-Field Approximation

The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality. In the ultraviolet regime, p????, the action decays as 1/p ⁴ in any even dimension.     

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 θ of the boundary conditions, and show that θ = 0 is a critical point of the action in any dimension and at all orders of the expansion.     

Spectral Action for Scalar Perturbations of Dirac Operators

We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey–de Witt coefficients and make explicit calculations for the case of n-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes (J Geom Phys 57:121, 2006) and study the case of SU _q (2).     

Cosmic radiation: The primary particles

Observations of primary cosmic-ray particles entering the upper atmosphere show that the solar system is situated in an isotropic flow of protons and of heavier nuclei, and that these particles carry energy which can vary from small values to values so great that we cannot at present define the upper limit of energy carried by a single particle.

These primary particles are constrained to diffuse through the galaxy as if in a reservoir and must initially have been injected into the reservoir from specific sources. In many ways the turbulent clouds which remain at the site of eruption of supernovae meet the requirements of such sources, and this property can be related to the identification of these objects as sources of strong radio-emission or radio-stars.     

The Seiberg-Witten Map and Topology

The mapping of topologically nontrivial gauge transformations in noncommutative gauge theory to corresponding commutative ones is investigated via the operator form of the Seiberg-Witten map. The role of the gauge transformation part of the map is analyzed. Chern-Simons actions are examined and the correspondence to their commutative counterparts is clarified.     

The Spectral Shift Function and Spectral Flow

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D +  V with D having compact resolvent belonging to a general semifinite von Neumann algebra and the perturbation . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to and V is any bounded self-adjoint operator in . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].     

The Idea of a Cosmic Time

The paper shows the standard definition of time at a distance to be beset with ambiguities that may be solved by making a fresh start taking its point of departure in the idea of a cosmic time as proposed by the British tradition of relativistic cosmology.This paper is written in honour of Franco Selleri, faithful defender of reason in physics, who committed his efforts to the liberation of time from the enslavement to space.     

Cosmic Ether

A prerelativistic approach to particle dynamicsis explored in an expanding Robertson–Walkercosmology. The receding galactic background provides adistinguished frame of reference and a unique cosmictime. In this context the relativistic, purelygeometric space-time concept is criticized. Physicalspace is regarded as a permeable medium, the cosmicether, which effects the world-lines of particles andrays. We study in detail a Robertson–Walkeruniverse with linear expansion factor and negativelycurved, open three-space; we choose the permeabilitytensor of the ether in such a way that the semiclassicalapproximation is exact. Galactic red-shifts depend on therefractive index of the ether. In the local Minkowskianlimit the ether causes a time variation of mass, whichscales inversely proportional to cosmic time. In the globally geodesic rest frames of galacticobservers the ether manifests itself in an unboundedspeed of signal transfer, in bifurcations ofworld-lines, and in time inversion effects.     

Cosmic deuterium

The knowledge of the primordial deuterium to hydrogen ratio provides one of the most reliable tests of the early Universe nucleosynthesis models and a direct estimate of the cosmic baryon density. Evaluations have been traditionally made using D/H estimations in the interstellar medium, extrapolated backwards in time with the use of galactic evolution models. Direct primordial D/H measurements have been carried out only recently in the direction of quasars. These measurements of deuterium abundances along with observations made in the solar system and in the interstellar medium are presented.New results that indicate spatial variations of the deuterium abundance in the interstellar medium at the level of 50% over scales possibly as small as 10 pc, may question our global vision of deuterium evolution until the causes for the origin of these variations are understood. With a conservative point of view, observations thus suggest that the primordial D/H value should be within the range 1. × 10⁻⁵−3. × 10⁻⁴, leading to a relatively low baryon content Universe.Since the actual evolution of deuterium from primordial nucleosynthesis to now is not known in details, more observations, hopefully to be made with the Hubble Space Telescope, FUSE the Far Ultraviolet Spectroscopic Explorer (launched in 1999). or from the ground with the largest telescopes (Keck, VLT,), should reveal the evolution of that key element, and better constrain its primordial abundance.     

Cosmic jets

We discuss time-dependent gravitational fields that “accelerate” free test particles to the speed of light resulting in cosmic double-jet configurations. It turns out that complete gravitational collapse along a spatial axis together with corresponding expansion along the other two axes leads to the accelerated motion of free test particles up and down parallel to the collapse axis such that a double-jet pattern is asymptotically formed with respect to the collapsed configuration.     

Einstein and Cosmic Physics

The article gives an overview on Albert Einstein's activity in relation to geophysics. Various aspects of his fundamental investigations and their significance for geophysical research are discussed.