Jean-Pierre Gazeau

Temporally stable coherent states for infinite well and Poschl-Teller potentials

This article is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Poschl–Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU(1,1) symmetry a la Barut-Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.

Gupta-Bleuler quantization for minimally coupled scalar fields in de Sitter space

We present a fully covariant quantization of the minimally coupled massless field on de Sitter space, thanks to a new representation of the canonical commutation relations. We thus obtain a formalism free of any infrared divergence. Our method is based on a rigorous group-theoretical approach combined with a suitable adaptation (Krein spaces) of the Wightman-Garding axiomatic for massless fields (Gupta-Bleuler scheme). We make explicit the correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter spacetime. The minimally coupled massless field is associated with a representation which is the lowest term of the discrete series of unitary representations of the de Sitter group. In spite of the presence of negative-norm modes in the theory, no negative energy can be measured: expressions such as n k 1 n k 2 ... |T 00 |n k 1 n k 2 ... are always positive.

Beta-integers as natural counting systems for quasicrystals

Recently, discrete sets of numbers, the -integers , have been proposed as numbering tools in quasicrystalline studies. Indeed, there exists a unique numeration system based on the irrational in which the -integers are all real numbers with no fractional part. These -integers appear to be quite appropriate for describing some quasilattices relevant to quasicrystallography when precisely is equal to (golden mean ), to , or to , i.e. when is one of the self-similarity ratios observed in quasicrystalline structures. As a matter of fact, -integers are natural candidates for coordinating quasicrystalline nodes, and also the Bragg peaks beyond a given intensity in corresponding diffraction patterns: they could play the same role as ordinary integers do in crystallography. In this paper, we prove interesting algebraic properties of the sets when is a `quadratic unit PV number, a class of algebraic integers which includes the quasicrystallographic cases. We completely characterize their respective Meyer additive and multiplicative properties where F and G are finite sets, and also their respective Galois conjugate sets . These properties allow one to develop a notion of a quasiring . We hope that in this way we will initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be `module on a quasiring in : . We give also some two-dimensional examples with .

“Massive” spin-2 field in de Sitter space

In this paper we present a covariant quantization of the “massive” spin-2 field on de Sitter (dS) space. By “massive” we mean a field which carries a specific principal series representation of the dS group. The work is in the direct continuation of previous ones concerning the scalar, the spinor, and the vector cases. The quantization procedure, independent of the choice of the coordinate system, is based on the Wightman-Garding axiomatic and on analyticity requirements for the two-point function in the complexified pseudo-Riemanian manifold. Such a construction is necessary in view of preparing and comparing with the dS conformal spin-2 massless case (dS linear quantum gravity) which will be considered in a forthcoming paper and for which specific quantization methods are needed.

'Massless' vector field in de Sitter Universe

We proceed to the quantization of the massless vector field in the de Sitter (dS) space. This work is the natural continuation of a previous article devoted to the quantization of the dS massive vector field [J. P. Gazeau and M. V. Takook, J. Math. Phys. 41, 5920 (2000); T. Garidi et al., ibid. 43, 6379 (2002).] The term “massless” is used by reference to conformal invariance and propagation on the dS lightcone whereas “massive” refers to those dS fields which unambiguously contract to Minkowskian massive fields at zero curvature. Due to the combined occurrences of gauge invariance and indefinite metric, the covariant quantization of the massless vector field requires an indecomposable representation of the de Sitter group. We work with the gauge fixing corresponding to the simplest Gupta–Bleuler structure. The field operator is defined with the help of coordinate-independent de Sitter waves (the modes). The latter are simple to manipulate and most adapted to group theoretical approaches. The physical sta...

Vector coherent states from Plancherel's theorem, Clifford algebras and matrix domains

As a substantial generalization of the technique for constructing canonical and the related nonlinear and q-deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.

Dirac fields and thermal effects in the de Sitter universe

We present a study of Dirac quantum fields in a four-dimensional dexa0Sitter spacetime. The theory is based on the requirement of precise analyticity properties of the waves and the correlation functions in the complexification of the dexa0Sitter manifold. Holomorphic dexa0Sitter spinorial plane waves are introduced in this way and used to construct the two-point functions, whose properties are fully characterized. The physical interpretation of the analyticity properties of Wightmans functions in terms of a KMS-type thermal condition is also given.

Coherent state quantization of a particle in de Sitter space

We present a coherent state quantization of the dynamics of a relativistic test particle on a one-sheet hyperboloid embedded in a three-dimensional Minkowski space. The group SO0(1, 2) is considered to be the symmetry group of the system. Our procedure relies on the choice of coherent states of the motion on a circle. The coherent state realization of the principal series representation of SO0(1, 2) seems to be a new result.

Additive and multiplicative properties of point sets based on beta-integers

To each number β > 1 correspond abelian groups in Rd, of the form Λβ = Σi=1d Zβei, which obey βΛβ ⊂ Λβ. The set Zβ of beta-integers is a countable set of numbers: it is precisely the set of real numbers which are polynomial in β when they are written in basis β, and Zβ = Z when β ∈ N. We prove here a list of arithmetic properties of Zβ: addition, multiplication, relation with integers, when β is a quadratic Pisot-Vijayaraghavan unit (quasicrystallographic inflation factors are particular examples). We also consider the case of a cubic Pisot-Vijayaraghavan unit associated with the seven-fold cyclotomic ring. At the end, we show how the point sets Λβ are vertices of d-dimensional tilings.

Multidimensional generalized coherent states

Generalized coherent states were presented recently for systems with one degree of freedom having discrete and/or continuous spectra. We extend that definition to systems with several degrees of freedom, give some examples and apply the formalism to the model of two-dimensional fermion gas in a constant magnetic field.