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Coherent states of the Poincare group, related frames and transforms

Mohammed Rezaul Karim

Published Jan 1, 1996• 1 citation
OpenAlex

Abstract

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.

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We construct here families of coherent states for the full Poincare group, for representations corresponding to mass m>0m>0m>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.
Drawing upon some earlier work on coherent states of the Poincare' group in l-space and l-time dimensions, we use these states to define a relativistic windowed Fourier transform, as relativistic extension of the usual windowed Fourier transform. We discretize the resulting transform and obtain conditions under which the discretized transform can be used to reconstruct arbitrary square integrable functions. We present some numerical and graphical exercises to illustrate the theory, as well as to compare the relativistic windowed Fourier transform with the orthodox windowed Fourier transform.
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 correspondante
We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number NNN of phase samples {θk=2πk/N}k=0N−1\{θ_k=2πk/N\}_{k=0}^{N-1}{θk​=2πk/N}k=0N−1​ for a given mean number ppp of particles. The resulting Coherent State (CS) subsystem S={∣zk=p1/2eiθk>}{\cal S}=\{|z_k=p^{1/2}e^{iθ_k}>\}S={∣zk​=p1/2eiθk​>} is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, S{\cal S}S is "almost complete" (a \textit{pseudo-frame}) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when $p
We show that there exists a V -admissible-type subspace in the carrier space of the Wigner representation of the Poincaré group in 1+3 dimensions. This subspace is built up from an orthonormal set of Sturmian-type functions which verify naturally the assumption of rotational invariance under which relativistic coherent states frames were previously obtained. Then, we propose an extension of the concept of V -admissibility for the class of semi-direct product groups, which takes into account the relativity groups which were not covered by the existing approaches. In the light of this extension, the square-integrability modulo an appropriate subgroup of the Wigner representation of the group of Poincaré is revisited and some physical implications are discussed.
We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number N of phase samples {θ k = 2πk/N } N -1 k=0 for a given mean number p of particles.The resulting Coherent State (CS) subsystem S = {|z k = p 1 2 e iθ k } is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact.For an unbounded number of particles, S is "almost complete" (a pseudo-frame) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when p < N and/or N → ∞.
: n-dimensional coherent states systems generated by translations, modulations, rotations and dilations are described. Starting from unitary irreducible representations of the n-dimen- sional affine Weyl-Heisenberg group, which are not square-integrable, one is led to consider systems of coherent states labeled by the elements of quotients of the original group. Such systems can yield a resolution of the identity, and then be used as alternatives to usual wavelet or windowed Fourier analysis. When the quotient space is the phase space of the representation, different embeddings of it into the group provide different descriptions of the phase space. R'esum'e: On d'ecrit des syst`emes d"etats coh'erents engendr'es par des translations, dilatations et rotations en n-dimensions. Partant de repr'esentations unitaires irr'eductibles du groupe de Weyl-Heisenberg affine en dimension n, qui ne sont pas de carr'e integrable, on est naturellement conduit `a consid'erer des syst`emes d"etats coh...