We continue the analysis of relativistic phase space, identified with the quotient of the Poincare group in 1+1 dimensions by the time translation subgroup. Proceeding by contraction from the corresponding (Anti) de Sitter group SO 0 (2, 1), we obtain a realisation of the Poincare representation P(m) of mass m in a space of functions defined on phase space. The contraction singles out a privileged section in the Poincare group, with a unique left and right invariant measure. Using that section, we show that the representation P(m) is square integrable over the coset space, i.e. phase space. From this we build a new set of Poincare coherent states, and more generally weighted coherent states, which have all the usual properties, resolution of the identity, overcompleteness, reproducing kernel, orthogonality relations. Finally we derive the corresponding Wigner transform On poursuit l'analyse de l'espace de phase relativiste, identifie au quotient du groupe de Poincare en dimension 1+1 par le sous-groupe des translations temporelles. Par contraction a partir du groupe Anti-de Sitter correspondant SO 0 (2, 1), on obtient une realisation de la representation P(m) de masse m du groupe de Poincare dans un espace de fonctions definies sur l'espace de phase. La contraction selectionne une section privilegiee dans le groupe de Poincare, possedant une unique mesure invariante a la fois a gauche et a droite. Avec ce choix de section, on montre que la representation P(m) est de carre sommable sur l'espace quotient, c'est-a-dire l'espace de phase. A partir de la, on construit une nouvelle famille d'etats coherents de Poincare et, plus generalement, d'etats coherents avec poids, qui ont toutes les proprietes usuelles: resolution de l'identite, surcompletude, noyau reproduisant, relations d'orthogonalite. Enfin, on derive la transformation de Wigner correspondanteWe continue the analysis of relativistic phase space, identified with the quotient of the Poincare group in 1+1 dimensions by the time translation subgroup. Proceeding by contraction from the corresponding (Anti) de Sitter group SO 0 (2, 1), we obtain a realisation of the Poincare representation P(m) of mass m in a space of functions defined on phase space. The contraction singles out a privileged section in the Poincare group, with a unique left and right invariant measure. Using that section, we show that the representation P(m) is square integrable over the coset space, i.e. phase space. From this we build a new set of Poincare coherent states, and more generally weighted coherent states, which have all the usual properties, resolution of the identity, overcompleteness, reproducing kernel, orthogonality relations. Finally we derive the corresponding Wigner transform On poursuit l'analyse de l'espace de phase relativiste, identifie au quotient du groupe de Poincare en dimension 1+1 par le sous-groupe des translations temporelles. Par contraction a partir du groupe Anti-de Sitter correspondant SO 0 (2, 1), on obtient une realisation de la representation P(m) de masse m du groupe de Poincare dans un espace de fonctions definies sur l'espace de phase. La contraction selectionne une section privilegiee dans le groupe de Poincare, possedant une unique mesure invariante a la fois a gauche et a droite. Avec ce choix de section, on montre que la representation P(m) est de carre sommable sur l'espace quotient, c'est-a-dire l'espace de phase. A partir de la, on construit une nouvelle famille d'etats coherents de Poincare et, plus generalement, d'etats coherents avec poids, qui ont toutes les proprietes usuelles: resolution de l'identite, surcompletude, noyau reproduisant, relations d'orthogonalite. Enfin, on derive la transformation de Wigner correspondante

Annales De L Institut Henri Poincare-physique TheoriqueJan 1, 19907 citations

About Coherent states of Relativistic Particles

I. M. Ternov, Vladislav G. Bagrov

MVSFADec 1, 19820 citations

Coherent states of the Poincare group, related frames and transforms

Mohammed Rezaul Karim

We construct here families of coherent states for the full Poincare group, for representations corresponding to mass $m>0$ and arbitrary integral or half-integral spin. Each family of coherent states is defined by an affine section in the group and constitutes a frame. The sections, in their turn, are determined by particular velocity vector fields, the latter always appearing in dual pairs. We discretize the coherent states of Poincare group in 1-space and 1-time dimensions and show that they form a discrete frame, develop a transform, similar to a windowed Fourier transform, which we call the relativistic windowed Fourier transform. We also obtain a reconstruction formula. Finally, we perform numerical computations. We evaluate the discrete frame operator numerically and present it graphically for different sections and windows. We also reconstruct some functions, compare reconstructed functions with the original ones graphically. We compare the reconstruction scheme of the relativistic windowed Fourier transform with that of the standard windowed Fourier transform.We construct here families of coherent states for the full Poincare group, for representations corresponding to mass $m>0$ and arbitrary integral or half-integral spin. Each family of coherent states is defined by an affine section in the group and constitutes a frame. The sections, in their turn, are determined by particular velocity vector fields, the latter always appearing in dual pairs. We discretize the coherent states of Poincare group in 1-space and 1-time dimensions and show that they form a discrete frame, develop a transform, similar to a windowed Fourier transform, which we call the relativistic windowed Fourier transform. We also obtain a reconstruction formula. Finally, we perform numerical computations. We evaluate the discrete frame operator numerically and present it graphically for different sections and windows. We also reconstruct some functions, compare reconstructed functions with the original ones graphically. We compare the reconstruction scheme of the relativistic windowed Fourier transform with that of the standard windowed Fourier transform.

Jan 1, 19961 citation

RELATIVISTIC COHERENT-STATE REPRESENTATIONS

Gerald Kaiser

Group Theoretical Methods in PhysicsJan 1, 19773 citations

We construct here families of coherent states for the full Poincare group, for representations corresponding to mass

m>0

and arbitrary integral or half-integral spin. Each family of coherent states is defined by an affine section in the group and constitutes a frame. The sections, in their turn, are determined by particular velocity vector fields, the latter always appearing in dual pairs. We discretize the coherent states of Poincare group in 1-space and 1-time dimensions and show that they form a discrete frame, develop a transform, similar to a windowed Fourier transform, which we call the relativistic windowed Fourier transform. We also obtain a reconstruction formula. Finally, we perform numerical computations. We evaluate the discrete frame operator numerically and present it graphically for different sections and windows. We also reconstruct some functions, compare reconstructed functions with the original ones graphically. We compare the reconstruction scheme of the relativistic windowed Fourier transform with that of the standard windowed Fourier transform.