Jean-Christophe Wallet

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• PhD
• Professor (Full) at French National Centre for Scientific Research

Poisson structures of the Poincar\'e group can be linked to deformations of the Minkowski space-time, classified some time ago. We construct the star-products and involutions characterizing the $*$-algebras of various quantum Minkowski space-times with non-centrally extended coordinates Lie algebras. We show that the usual Lebesgue integral defines...

The $\rho$-Minkowski space-time, a Lie-algebraic deformation of the usual Minkowski space-time is considered. A star-product realization of this quantum space-time together with the characterization of the deformed Poincar\'e symmetry acting on it are presented. It is shown that appearance of UV/IR mixing is expected already in scalar field theorie...

The unification of quantum mechanics and general relativity has long been elusive. Only recently have empirical predictions of various possible theories of quantum gravity been put to test, where a clear signal of quantum properties of gravity is still missing. The dawn of multi-messenger high-energy astrophysics has been tremendously beneficial, a...

A bstract We construct a gauge theory model on the 4-dimensional ρ -Minkowski space-time, a particular deformation of the Minkowski space-time recently considered. The corresponding star product results from a combination of Weyl quantization map and properties of the convolution algebra of the special Euclidean group. We use noncommutative differe...

We construct a gauge theory model on the 4-dimensional ρ-Minkowski space-time, a particular deformation of the Minkowski space-time recently considered. The corresponding star product results from a combination of Weyl quantization map and properties of the convolution algebra of the special Euclidean group. We use noncom-mutative differential calc...

We study quantum causal structures in 1+1 κ-Minkowski space-time described by a Lorentzian Spectral Triple whose Dirac operator is built from a natural set of twisted derivations of the κ-Poincaré algebra. We show that the Lorentzian Spectral Triple must be twisted to accommodate the twisted nature of the derivations. We exhibit various interesting...

A bstract We study one-loop perturbative properties of scalar field theories on the ρ -Minkowski space. The corresponding star-product, together with the involution are characterized from a combination of Weyl quantization and defining properties of the convolution algebra of the Euclidean group linked to the coordinate algebra of the ρ -Minkowski...

We study one-loop perturbative properties of scalar field theories on the $\rho$-Minkowski space. The corresponding star-product, together with the involution are characterized from a combination of Weyl quantization and defining properties of the convolution algebra of the Euclidean group linked to the coordinate algebra of the $\rho$-Minkowski sp...

We study quantum causal structures in $1+1$ $\kappa$-Minkowski space-time described by a Lorentzian Spectral Triple whose Dirac operator is built from a natural set of twisted derivations of the $\kappa$-Poincar\'e algebra. We show that the Lorentzian Spectral Triple must be twisted to accommodate the twisted nature of the derivations. We exhibit v...

We review the present status of gauge theories built on various quantum space-times described by noncommutative space-times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with par...

The κ-Minkoswki space-time provides a (quantum) noncommutative-deformation of the usual Minkowski space-time. However, a notion of causality is difficult to be defined in such a space with noncommutative time. In this paper, we define a notion of causality on a (1+1)-dimensional κ-Minkoswki space-time using the more general framework of Lorentzian nonc...

We consider a gauge theory on the 5D κ-Minkowski which can be viewed as the noncommutative analog of a U(1) gauge theory. We show that the Hermiticity condition obeyed by the gauge potential Aμ is necessarily twisted. Performing a Becchi-Rouet-Stora-Tyutin gauge-fixing with a Lorentz-type gauge, we carry out a first exploration of the one loop quan...

Recent results obtained in $\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space are reviewed and commented. A Weyl quantization procedure can be applied to convolution algebras to derive a convenient star product. For such a star product, gauge invariant polynomial action functional depending on the curvature exists only in 5 di...

[Formula: see text]-Poincaré invariant gauge theories on [Formula: see text]-Minkowski space-time, which are noncommutative analogs of the usual [Formula: see text] gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the [Formula: see text]-Minkowsk...

The exploration of the universe has recently entered a new era thanks to the multi-messenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give u...

The $\kappa$-Minkoswki space-time provides a quantum noncommutative-deformation of the usual Minkowski space-time. However, a notion of causality is difficult to be defined in such a space with noncommutative time. In this paper, we define a notion of causality on a (1+1)-dimensional $\kappa$-Minkoswki space-time using the more general framework of...

The exploration of the universe has recently entered a new era thanks to the multi-messenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give u...

\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space-time, which are noncommutative analogs of the usual $U(1)$ gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the $\kappa$-Minkowski space-time. We show that twisting the actio...

We consider a gauge theory on the 5-d $\kappa$-Minkowski which can be viewed as the noncommutative analog of a $U(1)$ gauge theory. We show that the Hermiticity condition obeyed by the gauge potential $A_\mu$ is necessarily twisted. Performing a BRST gauge-fixing, we carry out a first exploration of the one loop quantum properties of this gauge the...

Algebraic properties of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry associated to the twisted gauge symmetry occurring in the κ-Poincaré invariant gauge theories on the κ-Minkowski space are investigated. We find that the BRST operation associated to the gauge invariance of the action functional can be continuously deformed together with its corr...

A bstract We show that κ -Poincaré invariant gauge theories on κ -Minkowski space with physically acceptable commutative (low energy) limit must be 5-d. The gauge invariance requirement of the action fixes the dimension of the κ -Minkowski space to d = 5 and selects the unique twisted differential calculus with which the construction can be achieve...

Algebraic properties of the BRST symmetry associated to the twisted gauge symmetry occurring in the $\kappa$-Poincar\'e invariant gauge theories on the $\kappa$-Minkowski space are investigated. We find that the BRST operation associated to the gauge invariance of the action functional can be continuously deformed together with its corresponding Le...

We show that $\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space with physically acceptable commutative (low energy) limit must be 5-d. The gauge invariance requirement of the action fixes the dimension of the $\kappa$-Minkowski space to $d=5$ and selects the unique twisted differential calculus with which the construction can...

A bstract We discuss the construction of κ -Poincaré invariant actions for gauge theories on κ -Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed t...

We discuss the construction of $\kappa$-Poincar\'e invariant actions for gauge theories on $\kappa$-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deform...

We investigate the vacuum energy in κ-Poincaré invariant field theories. It is shown that for the equivariant Dirac operator one obtains an improvement in UV behavior of the vacuum energy and therefore the cosmological constant problem has to be revised. © 2019 authors. Published by the American Physical Society. Published by the American Physical...

A bstract We consider a family of κ -Poincaré invariant scalar field theories on 4-d κ -Minkowski space with quartic orientable interaction, that is for which ϕ and its conjugate ϕ † alternate in the quartic interaction, and whose kinetic operator is the square of a U κ (iso(4))-equivariant Dirac operator. The formal commutative limit yields the st...

We consider a family of k-Poincaré invariant scalar field theories on 4-d k-Minkowski space with quartic orientable interaction, that is for which $\phi$ and its conjugate $\phi^\dag$ alternate in the quartic interaction, and whose kinetic operator is the square of a $U_\kappa(iso(4))$-equivariant Dirac operator. The formal commutative limit yields...

A natural star product for 4-d κ-Minkowski space is used to investigate various classes of κ-Poincaré invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual ϕ4 theory. κ-Poincaré invariance forces the integral involved in the actions to be a twisted trace, thus defining a Kubo-Martin-Schwinger (KM...

We investigate the vacuum energy in $\kappa$-Poincar\'e invariant field theories. It is shown that for the equivariant Dirac operator one obtains an improvement in UV behavior of the vacuum energy and therefore the cosmological constant problem has to be revised.

A natural star product for 4-d kappa-Minkowski space is used to investigate various classes of kappa-Poincaré invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual phi^4 theory. kappa-Poincaré invariance forces the integral involved in the actions to be a twisted trace, thus defining a KMS weight...

Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Po...

Quantum spaces with $\frak{su}(2)$ noncommutativity can be modelled by using a family of $SO(3)$-equivariant differential $^*$-representations. The quantization maps are determined from the combination of the Wigner theorem for $SU(2)$ with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich pr...

We show that su(2) Lie algebras of coordinate operators related to quantum spaces with su(2) noncommutativity can be conveniently represented by SO(3)-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar dec...

We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to quantum spaces with $\frak{su}(2)$ noncommutativity can be conveniently represented by $SO(3)$-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are det...

Metrics structures stemming from the Connes distance promote Moyal planes to the status of quantum metric spaces. We discuss this aspect in the light of recent developments, emphasizing the role of Moyal planes as representative examples of a recently introduced notion of quantum (noncommutative) locally compact space. We move then to the framework...

The noncommutative space $\mathbb{R}^3_\lambda$, a deformation of $\mathbb{R}^3$, supports a $3$-parameter family of gauge theory models with gauge-invariant harmonic term, stable vacuum and which are perturbatively finite to all orders. Properties of this family are discussed. The partition function factorizes as an infinite product of reduced par...

The noncommutative space Rλ3, a deformation of R3, supports a 3-parameter family of gauge theory models with gauge-invariant harmonic term, stable vacuum and which are perturbatively finite to all orders. Properties of this family are discussed. The partition function factorizes as an infinite product of reduced partition functions, each one corres...

We consider the noncommutative space R^3_\theta, a deformation of R^3 for which the star product is closed for the trace functional. We study one-loop IR and UV properties of the 2-point function for real and complex noncommutative scalar field theories with quartic interactions and Laplacian on R^3 as kinetic operator. We find that the 2-point fun...

We consider the noncommutative space $\mathbb{R}^3_\theta$, a deformation of $\mathbb{R}^3$ for which the star product is closed for the trace functional. We study one-loop IR and UV properties of the 2-point function for real and complex noncommutative scalar field theories with quartic interactions and Laplacian on $\mathbb{R}^3$ as kinetic opera...

The noncommutative space $\mathbb{R}^3_\lambda$, a deformation of $\mathbb{R}^3$, supports a $3$-parameter family of gauge theory models with gauge-invariant harmonic term, stable vacuum and which are perturbatively finite to all orders. Properties of this family are discussed. The partition function factorizes as an infinite product of reduced par...

The conference Conceptual and Technical Challenges for Quantum Gravity at Sapienza University of Rome, from 8 to 12 September 2014, has provided a beautiful opportunity for an encounter between different approaches and different perspectives on the quantum-gravity problem. It contributed to a higher level of shared knowledge among the quantum-gravi...

We show that natural noncommutative gauge theory models on $R^3_\lambda$ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of $R^3_\lambda$ and the components of the gauge invariant 1-form canonical connection. This latter object shows up naturally within the present noncommutative differen...

Reference [59] of the References list, [59] P. Vitale, P. Martinetti and J.-C. Wallet, On noncommutative gauge theories on R 3 λ , in preparation has been published with slightly different title and modified authors list [1] Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permi...

We consider a class of gauge invariant models on the noncommutative space $\mathbb{R}^3_\lambda$, a deformation of $\mathbb{R}^3$. Focusing on massless models with no linear $A_i$ dependance, we obtain noncommutative gauge models for which the computation of the propagator can be done in a convenient gauge. We find that the infrared singularity of...

Jacobi operators appear as kinetic operators of several classes of noncommutative field theories (NCFT) considered recently. This paper deals with the case of bounded Jacobi operators. A set of tools mainly issued from operator and spectral theory is given in a way applicable to the study of NCFT. As an illustration, this is applied to a gauge-fixe...

We study a class of noncommutative gauge theory models on 2-dimensional Moyal space from the viewpoint of matrix models and explore some related properties. Expanding the action around symmetric vacua generates non local matrix models with polynomial interaction terms. For a particular vacuum, we can invert the kinetic operator which is related to...

A bstract In this work we clarify some properties of the one-loop IR divergences in nonAbelian gauge field theories on non-commutative 4-dimensional Moyal space. Additionally, we derive the tree-level Slavnov-Taylor identities relating the two, three and four point functions, and verify their consistency with the divergent one-loop level results. W...

22 pages. References and subsections added. Discussion in Section 4 improved

We consider the noncommutative space R3lambda, a deformation of the algebra of functions on R3 which yields a “foliation” of R3 into fuzzy spheres. We first construct a natural matrix base adapted to R3lambda. We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories wi...

We study metric properties stemming from the Connes spectral distance on three types of non-compact non-commutative spaces which have received attention recently from various viewpoints in the physics literature. These are the non-commutative Moyal plane, a family of harmonic Moyal spectral triples for which the Dirac operator squares to the harmon...

We introduce the new notion of "-graded associative algebras which takes its roots from the notion of commutation factors introduced in the context of Lie algebras ([39]). We define and study the associated notion of "-derivation-based differential calculus, which generalizes the derivation-based calculus on associative algebras. A corresponding no...

We analyze the Maxwell-Chern-Simons theory with minimal and tree level magnetic coupling to a fermion and a scalar. For a unique value of the magnetic moment, this theory allows one to recover an anyon-like behavior which however differs from the one of ideal anyons by an attractive contact term. For this particular value of the magnetic moment, we...

In the framework of finite temperature linear response theory, we analyze to a greater extent the nature of anyonic superconductivity. Using identities among Laguerre polynomials and Bessel functions, we provide simple and useful expressions for the response function in the form of high temperature expansions. The physical penetration depth as well...

We extend a zero temperature non-relativistic anyonic superconductivity model to the finite temperature case. Making use of the finite temperature linear response theory, we calculate temperature corrections to various parameters of zero temperature superconductivity such as the London penetration depth, the acoustic phonon velocity and the coheren...

We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R^2, we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator...

The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding r...

In this paper we consider the Spontaneous Symmetry Breaking Mechanism (SSBM) in the Standard Model of particles in the unitary gauge. We show that the computation usually presented of this mechanism can be conveniently performed in a slightly different manner. As an outcome, the computation we present can change the interpretation of the SSBM in th...

The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding r...

We investigate symmetries of the scalar field theory with harmonic term on the Moyal space with euclidean scalar product and general symplectic form. The classical action is invariant under the orthogonal group if this group acts also on the symplectic structure. We find that the invariance under the orthogonal group can be restored also at the qua...

The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework t...

The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework t...

The long-standing problem of constructing a potential from mixed scattering data is discussed. We first consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schr\"odinger equation, $r_{n}(E)$ which are monotonic functions of the energy, determine a unique potential when the domain of energy is...

Consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schr\"odinger equation, $r_{n}(E)$, which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the $r_{n}(E)$ range from zero to infinity. This suggests that the use of the mixed data...

Consider the fixed-ℓ inverse scattering problem. We show that the zeros of the regular solution of the Schrödinger equation, rn(E), which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the rn(E) range from zero to infinity. This suggests that the use of the mixed data of phase-shifts {...

Candidates for renormalizable gauge theory models on Moyal spaces constructed recently have non-trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the explicit expression in position space for these va...

Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi. We apply this framework to a version of the Moyal algebra ${\cal{M}}$. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ${\cal{M}}$ that can be related to...

Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is the...

The `International Conference on Noncommutative Geometry and Physics' was held on 23–27 April 2007 at the Laboratoire de Physique Théorique d'Orsay located at the Université Paris-Sud 11 campus. It brought together about 70 scientists, either mathematicians or physicists, actively working on the most recent aspects of noncommutative geometry, with...

In this paper we find non-trivial vacuum states for the renormalizable non-commutative φ4 model. An associated linear sigma model is then considered. We further investigate the corresponding spontaneous symmetry breaking.

We compute at the one-loop order the β-functions for a renormalisable non-commutative analog of the Gross–Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The β-function for the non-commuta...

The long-standing problem of constructing a potential from mixed scattering data is discussed. We first consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schr\"odinger equation, $r_{n}(E)$ which are monotonic functions of the energy, determine a unique potential when the domain of energy is...

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field. We find that the gauge-invariant effective action involves, beyond the expected noncommutative vers...

We use a Chern Simons Landau-Ginzburg (CSLG) framework related to hierarchies of composite bosons to describe 2D harmonically trapped fast rotating Bose gases in Fractional Quantum Hall Effect (FQHE) states. The predicted values for $\nu$ (ratio of particle to vortex numbers) are $\nu$$=$${{p}\over{q}}$ ($p$, $q$ are any integers) with even product...

The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional Quantum Hall State is examined. Standard Chern-Simons anyons as well as "non standard" anyons obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non minimal coupling to matter are considered. Their respective abili...

The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional Quantum Hall State is examined. Standard Chern-Simons anyons as well as "non standard" anyons obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non minimal coupling to matter are considered. Their respective abili...

We review the main features of a mathematical framework encompassing some of the salient quantum mechanical and geometrical aspects of Hall systems with finite size and general boundary conditions. Geometrical as well as algebraic structures controlling possibly the integral or fractional quantization of the Hall conductivity are discussed.

We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractionnal quantization of the Hall conductivity depending on the value of $NB/2\pi$ ($N$ is the number...

We consider Maxwell-Chern-Simons models involving different non-minimal coupling terms to a non relativistic massive scalar and further coupled to an external uniform background charge. We study how these models can be constrained to support static radially symmetric vortex configurations saturating the lower bound for the energy. Models involving...

In this paper, we revisit some quantum mechanical aspects related to the Quantum Hall Effect. We consider a Landau type model, paying a special attention to the experimental and geometrical features of Quantum Hall experiments. The resulting formalism is then used to compute explicitely the Hall conductivity from a Kubo formula.

Starting from a framework encoding rather simple symmetry principle based on modular subgroups, we construct a zero temperature global phase diagram for the QHE. This phase diagram is found to involve two insulating phases. One noticeable prediction is the possibility to have direct transitions from an insulting phase to any integer ν as well as ν=...

We consider a class of (2+1)-dimensional nonlocal effective models with a Maxwell–Chern–Simons part for which the Maxwell term involves a suitable nonlocality that permits one to take into account some (3+1)-dimensional features of "real" planar systems. We show that this class of models exhibits a hidden duality symmetry stemming from the Maxwell–...

We performed an experimental study of beta waves occurring in human electroencephalographic signals obtained from six healthy subjects that were monitored during the performance of a task requiring attention to auditory signals. We use wavelet analysis to study whether the fluctuations in the modulation of the beta-wave amplitude related to an indi...