25 research outputs found
Noncommutative Geometry and the standard model with neutrino mixing
We show that allowing the metric dimension of a space to be independent of
its KO-dimension and turning the finite noncommutative geometry F-- whose
product with classical 4-dimensional space-time gives the standard model
coupled with gravity--into a space of KO-dimension 6 by changing the grading on
the antiparticle sector into its opposite, allows to solve three problems of
the previous noncommutative geometry interpretation of the standard model of
particle physics:
The finite geometry F is no longer put in "by hand" but a conceptual
understanding of its structure and a classification of its metrics is given.
The fermion doubling problem in the fermionic part of the action is resolved.
The spectral action of our joint work with Chamseddine now automatically
generates the full standard model coupled with gravity with neutrino mixing and
see-saw mechanism for neutrino masses. The predictions of the Weinberg angle
and the Higgs scattering parameter at unification scale are the same as in our
joint work but we also find a mass relation (to be imposed at unification
scale).Comment: Typos removed, to appear in JHE
Gravity coupled with matter and foundation of non-commutative geometry
We first exhibit in the commutative case the simple algebraic relations
between the algebra of functions on a manifold and its infinitesimal length
element . Its unitary representations correspond to Riemannian metrics and
Spin structure while is the Dirac propagator ds = \ts \!\!---\!\! \ts =
D^{-1} where is the Dirac operator. We extend these simple relations to
the non commutative case using Tomita's involution . We then write a
spectral action, the trace of a function of the length element in Planck units,
which when applied to the non commutative geometry of the Standard Model will
be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian
coupled to gravity. The internal fluctuations of the non commutative geometry
are trivial in the commutative case but yield the full bosonic sector of SM
with all correct quantum numbers in the slightly non commutative case. The
group of local gauge transformations appears spontaneously as a normal subgroup
of the diffeomorphism group.Comment: 30 pages, Plain Te
Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model
The purpose of this letter is to remove the arbitrariness of the ad hoc
choice of the algebra and its representation in the noncommutative approach to
the Standard Model, which was begging for a conceptual explanation. We assume
as before that space-time is the product of a four-dimensional manifold by a
finite noncommmutative space F. The spectral action is the pure gravitational
action for the product space. To remove the above arbitrariness, we classify
the irreducibe geometries F consistent with imposing reality and chiral
conditions on spinors, to avoid the fermion doubling problem, which amounts to
have total dimension 10 (in the K-theoretic sense). It gives, almost uniquely,
the Standard Model with all its details, predicting the number of fermions per
generation to be 16, their representations and the Higgs breaking mechanism,
with very little input. The geometrical model is valid at the unification
scale, and has relations connecting the gauge couplings to each other and to
the Higgs coupling. This gives a prediction of the Higgs mass of around 170 GeV
and a mass relation connecting the sum of the square of the masses of the
fermions to the W mass square, which enables us to predict the top quark mass
compatible with the measured experimental value. We thus manage to have the
advantages of both SO(10) and Kaluza-Klein unification, without paying the
price of plethora of Higgs fields or the infinite tower of states.Comment: Title change only. The title "A Dress for SM the Beggar" was changed
by the Editor of Physical Review Letter
A Short Survey of Noncommutative Geometry
We give a survey of selected topics in noncommutative geometry, with some
emphasis on those directly related to physics, including our recent work with
Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at
length two issues. The first is the relevance of the paradigm of geometric
space, based on spectral considerations, which is central in the theory. As a
simple illustration of the spectral formulation of geometry in the ordinary
commutative case, we give a polynomial equation for geometries on the four
dimensional sphere with fixed volume. The equation involves an idempotent e,
playing the role of the instanton, and the Dirac operator D. It expresses the
gamma five matrix as the pairing between the operator theoretic chern
characters of e and D. It is of degree five in the idempotent and four in the
Dirac operator which only appears through its commutant with the idempotent. It
determines both the sphere and all its metrics with fixed volume form.
We also show using the noncommutative analogue of the Polyakov action, how to
obtain the noncommutative metric (in spectral form) on the noncommutative tori
from the formal naive metric. We conclude on some questions related to string
theory.Comment: Invited lecture for JMP 2000, 45
A Universal Action Formula
A universal formula for an action associated with a noncommutative geometry,
defined by a spectal triple (\Ac ,\Hc ,D), is proposed. It is based on the
spectrum of the Dirac operator and is a geometric invariant. The new symmetry
principle is the automorphism of the algebra \Ac which combines both
diffeomorphisms and internal symmetries. Applying this to the geometry defined
by the spectrum of the standard model gives an action that unifies gravity with
the standard model at a very high energy scale.Comment: This is a short non technical letter based on the longer version,
hep-th/9606001. Tex file, 10 page
Noncommutative geometry and motives: the thermodynamics of endomotives
We combine aspects of the theory of motives in algebraic geometry with
noncommutative geometry and the classification of factors to obtain a
cohomological interpretation of the spectral realization of zeros of
-functions. The analogue in characteristic zero of the action of the
Frobenius on l-adic cohomology is the action of the scaling group on the cyclic
homology of the cokernel (in a suitable category of motives) of a restriction
map of noncommutative spaces. The latter is obtained through the thermodynamics
of the quantum statistical system associated to an endomotive (a noncommutative
generalization of Artin motives). Semigroups of endomorphisms of algebraic
varieties give rise canonically to such endomotives, with an action of the
absolute Galois group. The semigroup of endomorphisms of the multiplicative
group yields the Bost-Connes system, from which one obtains, through the above
procedure, the desired cohomological interpretation of the zeros of the Riemann
zeta function. In the last section we also give a Lefschetz formula for the
archimedean local L-factors of arithmetic varieties.Comment: 52 pages, amslatex, 1 eps figure, v2: final version to appea
The term a_4 in the heat kernel expansion of noncommutative tori
We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of the modular automorphism of the state that encodes the conformal perturbation of the flat metric. We confirm the validity of the calculated expressions by showing that they satisfy a family of conceptually predicted functional relations. By studying these functional relations abstractly, we derive a partial differential system which involves a natural action of cyclic groups of order 2, 3 and 4 and a flow in parameter space. We discover symmetries of the calculated expressions with respect to the action of the cyclic groups. In passing, we show that the main ingredients of our calculations, which come from a rearrangement lemma and relations between the derivatives up to order 4 of the conformal factor and those of its logarithm, can be derived by finite differences from the generating function of the Bernoulli numbers and its multiplicative inverse. We then shed light on the significance of exponential polynomials and their smooth fractions in understanding the general structure of the noncommutative geometric invariants appearing in the heat kernel expansion. As an application of our results we obtain the a4 term for noncommutative four tori which split as products of two tori. These four tori are not conformally flat and the a4 term gives a first hint of the Riemann curvature and the higher-dimensional modular structure
Connes distance and optimal transport
We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states
Renormalization: a quasi-shuffle approach
In recent years, the usual BPHZ algorithm for renormalization in perturbative
quantum field theory has been interpreted, after dimensional regularization, as
a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs,
with values in a Rota-Baxter algebra of amplitudes. We associate in this paper
to any such algebra a universal semi-group (different in nature from the
Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes
associated to Feynman graphs produces the expected operations: Bogoliubov's
preparation map, extraction of divergences, renormalization. In this process a
key role is played by commutative and noncommutative quasi-shuffle bialgebras
whose universal properties are instrumental in encoding the renormalization
process
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 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 three
sphere. 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 the product
of the three sphere by a circle 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
vanish. This result is confirmed by evaluating the spectral action using the
heat kernel expansion where we check that the higher order terms a4 and a6 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.Comment: 33 pages, 1 figur