arXiv subjects

Fabien Besnard

Publications and source records attributed to Fabien Besnard.

At least 19 recordsLinked to original sources

arXiv2025

Angular dependence and powder average of resonant inelastic X-ray scattering

Resonant Inelastic X-ray scattering (RIXS) is a synchrotron-based spectroscopy that has seen growing interest across a range of scientific disciplines beyond fundamental physics. The interpretation of experimental RIXS data requires theoretical calculations based on the Kramers-Heisenberg formula. However, due to the dependence of RIXS on both the incident and scattered photon properties, a tractable treatment of the angular dependence in this formula has been lacking. In this work, within the electric dipole approximation, we determine the number of fundamental spectra contributing to the RIXS cross-section for all crystallographic point groups. We then derive a general expression for the RIXS cross-section of isotropic samples such as un-textured powders, homogeneous glasses or liquids, explicitly accounting for the polarization and propagation directions of both the incident and scattered photons. Simplified forms of the RIXS expressions are subsequently obtained for most common point groups. Finally, we demonstrate the applicability of our formalism through a case study of uranium 3d4f RIXS.

cond-mat.mtrl-sci↗

arXiv2022

Particle models from special Jordan backgrounds and spectral triples

We put forward a definition for spectral triples and algebraic backgrounds based on Jordan coordinate algebras. We also propose natural and gauge-invariant bosonic configuration spaces of fluctuated Dirac operators and compute them for general, almost-associative, Jordan, coordinate algebras. We emphasize that the theory so obtained is not equivalent with usual associative noncommutative geometry, even when the coordinate algebra is the self-adjoint part of a $C^*$-algebra. In particular, in the Jordan case, the gauge fields are always unimodular, thus curing a long-standing problem in noncommutative geometry.

math-ph↗

arXiv2021

Estimating noncommutative distances on graphs

We report on some findings concerning Connes' noncommutative distance $d$ on a weighted undirected graph $G$. Our main result is the lower bound $\ell/\Delta(G)\le d$ where $\ell$ is the geodesic distance and $\Delta(G)$ the degree of $G$. It is obtained thanks to an auxiliary spectral triple on the collection of the edges of $G$.

math.OA↗

arXiv2020

On symmetry breaking in the B-L extended spectral Standard Model

We apply Connes-Chamseddine spectral action to the $U(1)_{\rm B-L}$- extension of the Standard Model. We show that in order for the scalar potential to reach its minimum for a non-zero value of the new complex scalar field, thus triggering the breaking of B-L symmetry, a constraint on the quartic coupling constants must be satisfied at unification scale. We then explore the renormalization flow of this model in two opposite scenarios for the neutrino sector, and show that this constraint is not compatible with the pole masses of the top quark and SM Higgs boson. We also show that the model suffers from a mass-splitting problem similar to the doublet-triplet splitting problem of Grand Unified Theories. We discuss potential implications for the Noncommutative Geometry program.

hep-ph↗

arXiv2020

Extensions of the noncommutative Standard Model and the weak order one condition

In the derivation of the Standard Model from the axioms of Noncommutative Geometry, the scalar sector is given by a finite Dirac operator which has to satisfy the so-called \emph{first-order condition}. However, the general solution to this constraint still has unphysical terms which must be fine-tuned to zero. Moreover, the first-order condition generally does not survive in extensions to models with gauge groups larger that $U(1)\times SU(2)\times SU(3)$. In this paper we show that in the $U(1)_{\rm B-L}$-extension one can implement a weaker form of the first-order condition which we argue is necessary in order for Noncommutative Gauge Theory to make sense at all, and that this condition reduce the amount of fine-tuning to the off-diagonal terms in the Yukawa mass matrices for the leptons and quarks. We also show that this condition eliminates the Majorana mass terms for right-handed neutrinos when it is applied to the Pati-Salam model.

hep-th↗

arXiv2020

Noncommutative geometry, the Lorentzian Standard Model and its B-L extension

We explore the 1-loop renormalization group flow of two models coming from a generalization of the Connes-Lott version of Noncommutative Geometry in Lorentzian signature: the Noncommutative Standard Model and its B-L extension. Both make predictions on coupling constants at high energy, but only the latter is found to be compatible with the top quark and Higgs boson masses at the electroweak scale. We took into account corrections introduced by threshold effects and the relative positions of the Dirac and Majorana neutrino mass matrices and found them to be important. Some effects of 2-loop corrections are briefly discussed. The model is consistent with experiments only for a very small part of its parameter space and is thus predictive. The masses of the $Z'$ and B-L breaking scalar are found to be of the order $10^{14}$ GeV.

hep-ph↗

arXiv2019

A $U(1)_{B-L}$-extension of the Standard Model from Noncommutative Geometry

We derive a $U(1)_{B-L}$-extension of the Standard Model from a generalized Connes-Lott model with algebra ${\mathbb C}\oplus{\mathbb C}\oplus {\mathbb H}\oplus M_3({\mathbb C})$. This generalization includes the Lorentzian signature, the presence of a real structure, and a weakening of the order $1$ condition. In addition to the SM fields, the model contains a $Z_{B-L}'$ boson and a complex scalar field $\sigma$ which spontaneously breaks the new symmetry. This model is the smallest one which contains the SM fields and is compatible with both the Connes-Lott theory and the algebraic background framework.

hep-th↗

arXiv2019

On the uniqueness of Barrett's solution to the fermion doubling problem in Noncommutative Geometry

A solution of the so-called fermion doubling problem in Connes' Noncommutative Standard Model has been given by Barrett in 2006 in the form of Majorana-Weyl conditions on the fermionic field. These conditions define a ${\cal U}_{J,\chi}$-invariant subspace of the correct physical dimension, where ${\cal U}_{J,\chi}$ is the group of Krein unitaries commuting with the chirality and real structure. They require the KO-dimension of the total triple to be $0$. In this paper we show that this solution is, up to some trivial modifications, and under some mild assumptions on the finite triple, the only one with this invariance property. We also observe that a simple modification of the fermionic action can act as a substitute for the explicit projection on the physical subspace.

hep-th↗

arXiv2019

Algebraic backgrounds: a framework for noncommutative Kaluza-Klein theory

We investigate the representation of diffeomorphisms in Connes' Spectral Triples formalism. By encoding the metric and spin structure in a moving frame, it is shown on the paradigmatic example of spin semi-Riemannian manifolds that the bimodule of noncommutative 1-forms $\Omega^1$ is an invariant structure in addition to the chirality, real structure and Krein product. Adding $\Omega^1$ and removing the Dirac operator from an indefinite Spectral Triple we obtain a structure which we call an \emph{algebraic background}. All the Dirac operators compatible with this structure then form the configuration space of a noncommutative Kaluza-Klein theory. In the case of the Standard Model, this configuration space is stricty larger than the one obtained from the fluctuations of the metric, and contains in addition to the usual gauge fields the $Z_{B-L}'$-boson, a complex scalar field $\sigma$, which is known to be required in order to obtain the correct Higgs mass in the Spectral Standard Model, and flavour changing fields. The latter are invariant under automorphisms and can be removed without breaking the symmetries. It is remarkable that, starting from the conventional Standard Model algebra ${\mathbb C}\oplus {\mathbb H}\oplus M_3({\mathbb C})$, the "accidental" $B-L$ symmetry is necessarily gauged in this framework.

math-ph↗

arXiv2018

Doppler shift in semi-Riemannian signature and the non-uniqueness of the Krein space of spinors

We give examples illustrating the fact that the different space/time splittings of the tangent bundle of a semi-Riemannian spin manifold give rise to non-equivalent norms on the space of compactly supported sections of the spinor bundle, and as a result, to different completions. We give a necessary and sufficient condition for two space/time splittings to define equivalent norms in terms of a generalized Doppler shift between maximal negative definite subspaces. We explore some consequences for the Noncommutative Geometry program.

math.DG↗

arXiv2016

A remark on the mathematics of the seesaw mechanism

To demonstrate that matrices of seesaw type lead to a hieararchy in the neutrino masses, i.e. that there is a large gap in the singular spectrum of these matrices, one generally uses an approximate block-diagonalization procedure. In this note we show that no approximation is required to prove this gap property if the Courant-Fisher-Weyl theorem is used instead. This simple observation might not be original, however it does not seem to show up in the literature. We also sketch the proof of additional inequalities for the singular values of matrices of seesaw type.

hep-ph↗

arXiv2016

On the definition of spacetimes in Noncommutative Geometry, Part I

In this two-part paper we propose an extension of Connes' notion of even spectral triple to the Lorentzian setting. This extension, which we call a spectral spacetime, is discussed in part II where several natural examples are given which are not covered by the previous approaches to the problem. Part I only deals with the commutative and continuous case of a manifold. It contains all the necessary material for the generalization to come in part II, namely the characterization of the signature of the metric in terms of a time-orientation 1-form and a natural Krein product on spinor fields. It turns out that all the data available in Noncommutative Geometry (the algebra of functions, the Krein space of spinor fields, the representation of the algebra on it, the Dirac operator, charge conjugation and chirality), but nothing more, play a role in this characterization. Thus, only space and time oriented spin manifolds of even dimension are considered for a noncommutative generalization in this approach. We observe that these are precisely the kind of manifolds on which the modern theories of spacetime and matter are defined.

math.OA↗

arXiv2016

On the definition of spacetimes in Noncommutative Geometry, part II

In this second part of the paper, we define spectral spacetimes, a noncommutative generalization of Lorentzian orientable spacetimes of even dimension with a spin structure. There are two main differences with spectral triples: the existence of time-orientation 1-forms and the non-existence of a distinguished C*-structure on the algebra. If a "reconstructibility condition" is met, different, yet isomorphic, C*-structures exist. We define a natural notion of stable causality for spectral spacetimes. We give an example of a commutative spectral spacetime which is a Wick rotated version of the spectral triple that must be used in order to recover the usual notion of distance on a finite graph through Connes' distance formula. We show that this spectral spacetime is stably causal iff the time-orientation form induces no cycle. We also provide a noncommutative example, the split Dirac structure, which we study in some details. This structure is defined thanks to a discrete spinor bundle on a finite graph, a discrete connection, and operators at the vertices playing the role of gamma matrices. We give necessary and sufficient conditions for the split Dirac structure to be a spectral spacetime. These includes the natural analogues of the properties defining a spin connection in the continuous case. However the split Dirac structure is not always reconstructible: we prove that this is the case exactly when there exists a parallel timelike vector field on the graph. Nonreconstructible split Dirac structures furnish examples of spectral spacetimes which are not Wick rotations of usual spectral triples, and are thus genuinely Lorentzian. Moreover we show that the Dirac operator of the split Dirac structure is related to a Lorentzian version of a discretization of the Dirac operator introduced earlier by Marcolli and van Suijlekom.

math.OA↗

arXiv2016

Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and quantum electrodynamics

An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and an anti-unitary operator with specific commutation relations. It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product are the sums of the space and time dimensions of its factors. This could provide an interpretation of the presence of such algebras in PT-symmetric Hamiltonians or the description of topological matter. This construction is used to build an indefinite (i.e. pseudo-Riemannian) version of the spectral triples of noncommutative geometry, defined over Krein spaces instead of Hilbert spaces. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.

hep-th↗

arXiv2015

Two roads to noncommutative causality

We review the physical motivations and the mathematical results obtained so far in the isocone-based approach to noncommutative causality. We also give a briefer account of the alternative framework of Franco and Eckstein which is based on Lorentzian spectral triples. We compare the two theories on the simple example of the product geometry of the Minkowski plane by the finite noncommutative space with algebra $M_2({\mathbb C})$.

math.OA↗

arXiv2015

The Standard Model as an extension of the noncommutative algebra of forms

The Standard Model of particle physics can be deduced from a small number of axioms within Connes' noncommutative geometry (NCG). Boyle and Farnsworth [New J. Phys. 16 (2014) 123027] proposed to interpret Connes' approach as an algebra extension in the sense of Eilenberg. By doing so, they could deduce three axioms of the NCG Standard Model (i.e. order zero, order one and massless photon) from the single requirement that the extended algebra be associative. However, their approach was only applied to the finite algebra and fails the full model. By taking into account the differential graded structure of the algebra of noncommutative differential forms, we obtain a formulation where the same three axioms are deduced from the associativity of the extended differential graded algebra, but which is now also compatible with the full Standard Model. Finally, we present a Lorentzian version of the noncommutative geometry of the Standard Model and we show that the three axioms still hold if the four-dimensional manifold has a Lorentzian metric.

hep-th↗

arXiv2014

The disappearance of causality at small scale in almost-commutative manifolds

This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by so-called \emph{isocones} which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consider the case of isocones inside almost-commutative algebras of the form ${\cal C}(M)\otimes A_f$, with $M$ a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in ${\cal C}(M)\otimes A_f$, with $A_f$ noncommutative, induces an order relation on $M$ with the property that every point in $M$ lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale.

math-ph↗

arXiv2013

Noncommutative ordered spaces : examples and counterexamples

In order to introduce the notion of causality in noncommutative geometry it is necessary to extend Gelfand theory to the context of ordered spaces. In a previous work we have already given an algebraic caracterization of the set of non-decreasing continuous functions on an certain class of topological ordered spaces. Such a set is called an isocone, and there exist at least two versions of them (strong and weak) which coincide in the commutative case. In this paper we introduce yet another breed of isocones, ultraweak isocones, which has a simpler definition with a clear physical meaning. We show that ultraweak and weak isocones are in fact the same, and completely classify those which live in a finite dimensional $C^*$-algebra, hence corresponding to finite noncommutative ordered spaces. We also give some examples in infinite dimension.

math.OA↗