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J. Renaud

Publications and source records attributed to J. Renaud.

At least 19 recordsLinked to original sources
arXiv2025

FLRW embeddings in $\mathbb{R}^{n+2}$, differential geometry and conformal photon propagator

This paper introduces differential-geometric methods to study $n$-dimensional locally conformally flat spaces as submanifolds in $\mathbb{R}^{n+2}$. We derive explicit formulas relating intrinsic and ambient differential-geometric objects, including curvature tensors, the codifferential and laplacian operators. We apply this approach to Friedmann-Lema\^itre-Robertson-Walker (FLRW) spaces using newfound embedding formulas, obtaining new and simplified expressions for the photon propagator in four dimensions.

math-ph↗
arXiv2024

String Shooter's overall shape in ambient air

In this article, we study the behaviour of a looped string launched in ambient air using motorised wheels. We show that the loop, once it reaches its stationary state, is either in the pulley or the air-lifted state. The transition between these two distinct states occurs at the so-called takeoff speed. We prove that this speed differs from one string to another based on its characteristics. However, it is independent from the loop's length and its initial launch angle. This speed indeed corresponds to the threshold where air drag starts compensating for the weight of the string.

physics.class-ph↗
arXiv2022

Restriction of Laplace operator on one-forms: from $\mathbb{R}^{n+2}$ and $\mathbb{R}^{n+1}$ ambient spaces to embedded (A)dS$_n$ submanifolds

The Laplace-de Rham operator acting on a one-form $a$: $\square a$, in $\mathbb{R}^{n+2}$ or $\mathbb{R}^{n+1}$ spaces is restricted to $n$-dimensional pseudo-spheres. This includes, in particular, the $n$-dimensional de Sitter and Anti-de Sitter space-times. The restriction is designed to extract the corresponding $n$-dimensional Laplace-de Rham operator acting on the corresponding $n$-dimensional one-form on pseudo-spheres. Explicit formulas relating these two operators are given in each situation. The converse problem, of extending an $n$-dimensional operator composed of the sum of the Laplace-de Rham operator and additional terms to ambient spaces Laplace-de Rham operator, is also studied. We show that for any additional term this operator on the embedded space is the restriction of Laplace-de Rham operator on the embedding space.These results are translated to the Laplace-Beltrami operator thanks to the Weitzenböck formula, for which a proof is also given.

math-ph↗
arXiv2020

Massive scalar field on (A)dS space from a massless conformal field in $\mathbb{R}^6$

We show how the equations for the scalar field (including the massive, massless, minimally and conformally coupled cases) on de Sitter and Anti-de Sitter spaces can be obtained from both the SO$(2,4)$-invariant equation $\square ϕ= 0$ in $\mathbb{R}^6$ and two geometrical constraints defining the (A)dS space. Apart from the equation in $\mathbb{R}^6$, the results only follow from the geometry.

gr-qc↗
arXiv2017

FLRW spaces as submanifolds of $\mathbb{R}^6$: restriction to the Klein-Gordon operator

The FLRW spacetimes can be realized as submanifolds of $\mathbb{R}^6$. In this paper we relate the Laplace-Beltrami operator for an homogeneous scalar field $ϕ$ of $\mathbb{R}^6$ to its explicit restriction on FLRW spacetimes. We then make the link between the homogeneous solutions of the equation $\square_6 ϕ= 0$ in $\mathbb{R}^6$ and those of the Klein-Gordon equation $(\square_{f} - ξR^f + m^2)ϕ^f=0 $ for the free field $ϕ^f$ in the FLRW spacetime. We obtain as a byproduct a formula for the Ricci scalar of the FRLW spacetime in terms of the function $f$ defining this spacetime in $\mathbb{R}^6$.

math-ph↗
arXiv2014

Conformally invariant wave equation for a symmetric second rank tensor ("spin-2") in d-dimensional curved background

We build the general conformally invariant linear wave operator for a free, symmetric, second-rank tensor field in a d-dimensional ($d\geqslant 2$) metric manifold, and explicit the special case of maximally symmetric spaces. Under the assumptions made, this conformally invariant wave operator is unique. The corresponding conformally invariant wave equation can be obtained from a Lagrangian which is explicitly given. We discuss how our result compares to previous works, in particular we hope to clarify the situation between conflicting results.

gr-qc↗
arXiv2013

Conformally invariant formalism for the electromagnetic field with currents in Robertson-Walker spaces

We show that the Laplace-Beltrami equation $\square_6 a =j$ in $(\setR^6,η)$, $η:= \mathrm{diag}(+----+)$, leads under very moderate assumptions to both the Maxwell equations and the conformal Eastwood-Singer gauge condition on conformally flat spaces including the spaces with a Robertson-Walker metric. This result is obtained through a geometric formalism which gives, as byproduct, simplified calculations. In particular, we build an atlas for all the conformally flat spaces considered which allows us to fully exploit the Weyl rescalling to Minkowski space.

gr-qc↗
arXiv2011

Conformal use of retarded Green's functions for the Maxwell field in de Sitter space

We propose a new propagation formula for the Maxwell field in de Sitter space which exploit the conformal invariance of this field together with a conformal gauge condition. This formula allows to determine the classical electromagnetic field in the de Sitter space from given currents and initial data. It only uses the Green's function of the massless Minkowskian scalar field. This leads to drastic simplifications in practical calculations. We apply this formula to the classical problem of the two charges of opposite signs at rest at the North and South Poles of the de Sitter space.

gr-qc↗
arXiv2010

Conformally covariant quantization of Maxwell field in de Sitter space

In this article, we quantize the Maxwell ("massless spin one") de Sitter field in a conformally invariant gauge. This quantization is invariant under the SO$_0(2,4)$ group and consequently under the de Sitter group. We obtain a new de Sitter invariant two-points function which is very simple. Our method relies on the one hand on a geometrical point of view which uses the realization of Minkowski, de Sitter and anti-de Sitter spaces as intersections of the null cone in $\setR^6$ and a moving plane, and on the other hand on a canonical quantization scheme of the Gupta-Bleuler type.

gr-qc↗
arXiv2008

Revisiting the conformal invariance of the scalar field: from Minkowski space to de Sitter space

In this article, we clarify the link between the conformal (i.e. Weyl) correspondence from the Minkowski space to the de Sitter space and the conformal (i.e. SO(2,$d$)) invariance of the conformal scalar field on both spaces. We exhibit the realization on de Sitter space of the massless scalar representation of SO$(2,d)$. It is obtained from the corresponding representation in Minkowski space through an intertwining operator inherited from the Weyl relation between the two spaces. The de Sitter representation is written in a form which allows one to take the point of view of a Minkowskian observer who sees the effect of curvature through additional terms.

gr-qc↗
arXiv2008

Evaluation and exploitation of knowledge robustness in knowledge-based systems

Industrial knowledge is complex, difficult to formalize and very dynamic in reason of the continuous development of techniques and technologies. The verification of the validity of the knowledge base at the time of its elaboration is not sufficient. To be exploitable, this knowledge must then be able to be used under conditions (slightly) different from the conditions in which it was formalized. So, it becomes vital for the company to permanently evaluate the quality of the industrial knowledge implemented in the system. This evaluation is founded on the concept of robustness of the knowledge formalized by conceptual graphs. The evaluation method is supported by a computerized tool.

cs.OH↗
arXiv2008

Maps of Dipersions for Machining Processes

During the products design, the design office defines dimensional and geometrical parameters according to the use criteria and the product functionality. The manufacturing department must integrate the manufacturing and the workpiece position dispersions during the choice of tools and machines operating modes and parameters values to respect the functional constraints. In this paper, we suggest to model the turning dispersions taking into account not only geometrical specifications of position or orientation but also the experience of method actors. A representation using the principle of know-how maps in two or three dimensions is privileged. The most interesting aspect is that these maps include tacit and explicit knowledge. An experimental study realized on a machine tool (HES 300) allows to elaborate knowledge maps especially for the turning process.

physics.gen-ph↗
arXiv2007

Wick ordering for coherent state quantization in 1+1 de Sitter space

We show that the coherent state quantization of massive particles in 1+1 de Sitter space exhibits an ordering property: There exist some classical observables $A$ and $A^*$ such that $O_{A^{*p}}O_{A^q}=O_{A^{*p} A^q}$ $p, q \in \Z$, where $O_A$ is the quantum observable corresponding to the classical observable $A$.

math-ph↗