Jean-pierre Antoine
UCLouvain (University of Louvain), Physics/IRMP, Faculty Member
We present a general formalism for the construction of coherent states, based on the notion of reproducing triple. It covers the case of continuous frames in Hilbert space, as well as generalized coherent states associated to group... more
We present a general formalism for the construction of coherent states, based on the notion of reproducing triple. It covers the case of continuous frames in Hilbert space, as well as generalized coherent states associated to group representations which are square integrable only on a homogeneous space. Coherent states (CS), originally introduced by Schrodinger in the context of a harmonic oscillator, later popularized by Glauber and Klauder for the description of coherent light, have been generalized to such an extent that they find applications in every single corner of quantum theory. Yet there are cases where the known methods fail to generate CS, for instance, the Galilei or the Poincare groups (in 1+1 or 1+3 dimensions), and other groups of the same type. Our aim here is to treat such situations, and in fact much more general ones, by a suitable extension of the notion of coherent states. The discussion is based on joint work with S.T. Ali and J.-P. Gazeau3−5. First we briefly review the standard method.
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We review the main points in the development of partial *-algebras, at three different levels: (i) The algebraic structure stemming from the partial multiplication; (ii) The locally convex partial *-algebras; (iii) The partial *-algebras... more
We review the main points in the development of partial *-algebras, at three different levels: (i) The algebraic structure stemming from the partial multiplication; (ii) The locally convex partial *-algebras; (iii) The partial *-algebras of closable operators in Hilbert spaces or partial O*-algebras, including the representation theory of the abstract partial *-algebras
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We present a method for the construction of coherent states, based on the notion of square integrability of a group representation on a homogeneous space. This generalized formalism allows to cover cases hitherto inaccessible, such as the... more
We present a method for the construction of coherent states, based on the notion of square integrability of a group representation on a homogeneous space. This generalized formalism allows to cover cases hitherto inaccessible, such as the Poincare group.
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(Docteur en Sciences; groupe Sciences Physiques -- Université catholique de Louvain, 196
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We review the main points in the development of partial *-algebras during the last 15 years, at three different levels. (i) The algebraic structure stemming from the partial multiplication; (ii) The topological partial *-algebras; (iii)... more
We review the main points in the development of partial *-algebras during the last 15 years, at three different levels. (i) The algebraic structure stemming from the partial multiplication; (ii) The topological partial *-algebras; (iii) The partial *-algebras of closable operators in Hilbert spaces or partial O*-algebras, including the representation theory of the abstract partial *-algebras.
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We describe several families of wavelets associated to the Galilei group, extended by space and time dilations. The construction follows a general method based on group representations which are square integrable only on a homogeneous... more
We describe several families of wavelets associated to the Galilei group, extended by space and time dilations. The construction follows a general method based on group representations which are square integrable only on a homogeneous space of the underlying group.
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In this chapter, we will study families of operators acting on a rigged Hilbert space, with a particular interest in their partial algebraic structure. In Section 10.1 the notion of rigged Hilbert space D[t] ↪ H ↪ D × [t ×] is introduced... more
In this chapter, we will study families of operators acting on a rigged Hilbert space, with a particular interest in their partial algebraic structure. In Section 10.1 the notion of rigged Hilbert space D[t] ↪ H ↪ D × [t ×] is introduced and some examples are presented. In Section 10.2, we consider the space.L(D, D ×) of all continuous linear maps from D[t] into D × [t ×] and look for conditions under which (L(D, D ×), L +(D)) is a (topological) quasi *-algebra. Moreover the general problem of introducing in L(D, D ×) a partial multiplication is considered. In Section 10.3 representations of abstract quasi *-algebras into quasi*-algebras of operators are studied and the GNS-construction is revisited for this case. In Section 10.4, we consider the special case where the extreme spaces of the rigged Hilbert space are Hilbert spaces and we construct the maximal CQ*-algebra acting on a triplet of Hilbert spaces.
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The wavelet transform on the two-sphere and related manifolds: a review. [Proceedings of SPIE 7000, 70000B (2008)]. Jean-Pierre Antoine, Daniela Roşca. Abstract. In a first part, we discuss several properties that seem desirable ...
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... Directional features detection; Continuous wavelet transform; Scaleangle representation; Direc tional wavelets; Reproducing kernel; Wavelet calibration; Angular resolving power; Scale resolvingpower * Corresponding author. Fax:... more
... Directional features detection; Continuous wavelet transform; Scaleangle representation; Direc tional wavelets; Reproducing kernel; Wavelet calibration; Angular resolving power; Scale resolvingpower * Corresponding author. Fax: +32.10.47.24.14; email: antoine@fyma.ucl.ac ...
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We study the order structure induced on a rigged Hilbert space ⊂ χ ⪅ ′by a self-dual cone in χ. Under the assumption that is a semi-reflexive locally convex vector lattice, and a solid subset of ′ (we speak then of a rigged Hilbert... more
We study the order structure induced on a rigged Hilbert space ⊂ χ ⪅ ′by a self-dual cone in χ. Under the assumption that is a semi-reflexive locally convex vector lattice, and a solid subset of ′ (we speak then of a rigged Hilbert lattice), we show that ′ is in fact a nondegenerate partial inner product space.
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In the algebraic formulation of quantum theories, a state is often represented by a normal linear functional on some *-algebra 𝔄 of operators on a Hilbert space, i.e., a functional of the form f(X)=Tr TX for some trace class operator X.... more
In the algebraic formulation of quantum theories, a state is often represented by a normal linear functional on some *-algebra 𝔄 of operators on a Hilbert space, i.e., a functional of the form f(X)=Tr TX for some trace class operator X. The question is whether every strongly positive (i.e., positive on positive elements) linear functional is normal. Criteria for this statement to be true are well known in the case where 𝔄 is a *-algebra of bounded operators (C*- or W*-algebra) or a *-algebra of unbounded operators (O*-algebra). Those results are extended to the case where 𝔄 is a (weak) partial *-algebra of closable operators, both for linear functionals and for sesquilinear forms.
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This tutorial text reviews some applications of the Continuous Wavelet Transform (CWT) in Magnetic Resonance Spectroscopy (MRS), focusing on the problems of spectral line estimation, namely, apodization, random noise, baseline, solvent... more
This tutorial text reviews some applications of the Continuous Wavelet Transform (CWT) in Magnetic Resonance Spectroscopy (MRS), focusing on the problems of spectral line estimation, namely, apodization, random noise, baseline, solvent peak, nonstandard lineshapes. First, we use a standard wavelet, namely the Morlet wavelet. Next, we introduce a new type of wavelet, derived from the autocorrelation function of a model signal. Finally, we apply this new technique for constructing adapted wavelets, generated from the metabolite data themselves. A short compendium on the basics of the CWT and some explicit numerical programs are given in two Appendices.
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ABSTRACT A new class of wavelet functions called data-based autocorrelation wavelets is developed for analyzing Magnetic Resonance Spectroscopic (MRS) signals by means of the continuous wavelet transform (CWT), instead of the traditional... more
ABSTRACT A new class of wavelet functions called data-based autocorrelation wavelets is developed for analyzing Magnetic Resonance Spectroscopic (MRS) signals by means of the continuous wavelet transform (CWT), instead of the traditional wavelet like Morlet wavelet. These new wavelets are derived from the normalized autocorrelation function from metabolite data and then used for detecting the presence of a given metabolite in a signal with a presence of many different components and finally for quantifying some of its parameters.
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Before going into details, a word of warning is in order. Mathematical Physics is an ill-defined concept, various interpretations are possible, ranging from the very narrow to the broadest sense – and the boundary with other fields is... more
Before going into details, a word of warning is in order. Mathematical Physics is an ill-defined concept, various interpretations are possible, ranging from the very narrow to the broadest sense – and the boundary with other fields is fuzzy. In this survey, we take the widest possible definition: Mathematical Physics is present whenever a physical problem is studied with mathematically rigorous methods, thus it comprises various forms of applications, to (applied) mathematics, physics, engineering, etc. We feel that these borderline themes are among the liveliest parts of research and should not be left out.
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This chapter is devoted to a fairly detailed examination of the quintessential example of coherent states — the canonical coherent states. It is fair to say that the entire subject of coherent states developed by analogy from this... more
This chapter is devoted to a fairly detailed examination of the quintessential example of coherent states — the canonical coherent states. It is fair to say that the entire subject of coherent states developed by analogy from this example. As mentioned in Chapter 1, this set of states, or rays in the Hilbert space of a quantum mechanical system, was originally discovered by Schrodinger transition from quantum to classical mechanics. They are endowed with a remarkable array of interesting properties, some of which we shall survey in this chapter. Apart from initiating the discussion, this will also help us in motivating the various mathematical directions in which one can try to generalize the notion of a CS.
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... [6] JE Humphreys, Introduction to Lie Algebras and Representation Theory. New York–Heidelberg– Berlin: Springer, 1972. [7] RE Behrends, J. Dreitlein, C. Fronsdal, and W. Lee, Simple groups and strong interaction symmetries, Rev. Mod.... more
... [6] JE Humphreys, Introduction to Lie Algebras and Representation Theory. New York–Heidelberg– Berlin: Springer, 1972. [7] RE Behrends, J. Dreitlein, C. Fronsdal, and W. Lee, Simple groups and strong interaction symmetries, Rev. Mod. Phys., 34 (1962): 1–40. [8] JP. Antoine ...
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We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we... more
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
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We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the (n−1)-sphere Sn−1, based on the construction of general coherent states associated to square integrable group representations. The parameter... more
We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the (n−1)-sphere Sn−1, based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, X∼SO(n)×R*+, is embedded into the generalized Lorentz group SO0(n,1) via the Iwasawa decomposition, so that X≃SO0(n,1)/N, where N≃Rn−1. Then the CWT on Sn−1 is derived from a suitable unitary representation of SO0(n,1) acting in the space L2(Sn−1,dμ) of finite energy signals on Sn−1, which turns out to be square integrable over X. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual filtering properties of the CWT. Next the Euclidean limit of this CWT on Sn−1 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R→∞, from which one recovers the usual CWT on flat Euclidean space. Finally, we discuss the ex...
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A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we... more
A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.
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Magnetic resonance spectroscopy (MRS) is an effective diagnostic technique for monitoring biochemical changes in an organism. The lineshape of MRS signals can deviate from the theoretical Lorentzian lineshape due to inhomogeneities of the... more
Magnetic resonance spectroscopy (MRS) is an effective diagnostic technique for monitoring biochemical changes in an organism. The lineshape of MRS signals can deviate from the theoretical Lorentzian lineshape due to inhomogeneities of the magnetic field applied to patients and to tissue heterogeneity. We call this deviation a distortion and study the self-deconvolution method for automatic estimation of the unknown lineshape
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In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical... more
In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical bra-ket formalism of Dirac, and the more recent rigged Hilbert space approach.
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A powerful analysis method, based on wavelet transforms (WT), was developed and applied to the search of point and extended sources in the EGRET data. The exposure and PSF of EGRET vary strongly over the field of view and the galactic... more
A powerful analysis method, based on wavelet transforms (WT), was developed and applied to the search of point and extended sources in the EGRET data. The exposure and PSF of EGRET vary strongly over the field of view and the galactic gamma-ray background presents many point-like structures, making the source extraction in the wavelet space difficult. A mathematical approach have been studied to successfully estimate the significance level of the wavelet coefficients. The performance of the method will be presented as well as strategy for the future GLAST mission involving an order of magnitude more sources.
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We develop a method of discretization of the continuous theory of coherent states on a general semidirect product Lie group. The group is assumed to have a unitary representation which is square integrable on some homogeneous space. We... more
We develop a method of discretization of the continuous theory of coherent states on a general semidirect product Lie group. The group is assumed to have a unitary representation which is square integrable on some homogeneous space. We show also that the existence of a discrete frame of coherent states in the carrier space of a unitary representation of such a group implies the square integrability of this representation on the label space.
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This chapter is devoted to a detailed development of the theory of square integrable group representations, including the resulting orthogonality relations. Then we study a particular class of semidirect product groups, namely, groups of... more
This chapter is devoted to a detailed development of the theory of square integrable group representations, including the resulting orthogonality relations. Then we study a particular class of semidirect product groups, namely, groups of the form \(G = {\mathbb{R}}^{n} \rtimes H\), where H is an n-dimensional closed subgroup of GL\((n, \mathbb{R})\). Several concrete examples are presented. Finally we generalize the theory to representations that are only square integrable on a homogeneous space. This allows one to treat CS of the Gilmore-Perelomov type and, in particular, CS of the Galilei group.
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Cet article passe en revue les multiples applications de la théorie des groupes aux problèmes de symétrie en physique. En physique classique, il s'agit surtout de la relativité : euclidienne, galiléenne, einsteinienne (relativité... more
Cet article passe en revue les multiples applications de la théorie des groupes aux problèmes de symétrie en physique. En physique classique, il s'agit surtout de la relativité : euclidienne, galiléenne, einsteinienne (relativité restreinte). Passant à la mécanique quantique, on remarque d'abord que les principes de base impliquent que l'espace des états d'un système quantique a une structure intrinsèque d'espace préhilbertien, que l'on complète ensuite en un espace de Hilbert. Dans ce contexte, la description de l'invariance sous un groupe G se base sur une représentation unitaire de G. On parcourt ensuite les différents domaines d'application : physique atomique et moléculaire, matière condensée, optique quantique, ondelettes, symétries internes, symétries approchées. On discute ensuite l'extension aux théories de jauge, en particulier au Modèle Standard des interactions fondamentales. On conclut par quelques indications sur des développements récents.