Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

16th Congress of the International Commission for Optics: Optics as a Key to High Technology

An extension of Glauber's definition of coherence for a given lie algebra

E. Ledinegg

Published Jul 26, 1993• 0 citations
Publisher

Abstract

We show that fermionic particles emitted by a stochastic source develop a behaviour which is different from usual Fermi statistics. The particles are described by creation and anihilation operators which span a sl(2, R) algebra. The corresponding Glauber-coherent states,1 defined in an appropriate representation are identical to coherent states found earlier by Barut and Girardello2 in solving the eigenvalue equation of the step operators of the su(1, 1) algebra. This establishes a case in which the mathematically extended notion of coherent states as eigenstates of step operators of an arbitrary Lie algebra finds a physical interpretation.

DOI
10.1117/12.2308458

Related papers

Coherent neutron fields and the Lie algebra sl(2,R)

E. Ledinegg, E. Schachinger

The application of Glauber's definition (1963) of quantum coherence allows the introduction of neutron fields of partial coherence under the assumption that the complete occupation number space is a direct product of Fermi subspaces. Excitations are described by 'collective' creation and annihilation operators which span an algebra isomorphic to the sl(2,R) algebra. The associated coherent states are of partial coherence and the finite dimensional representation of the su(2,R) algebra is not regularly coherent. In contrast, the transition to an infinite dimensional representation space results in regular coherent properties of the field. This is demonstrated using the representation space of the su(1,1) algebra which has a real isomorphism to the sl(2,R) algebra. The coherent states calculated from Glauber's condition for coherence are completely coherent, and are, moreover, identical to those found by Barut and Girardello (1971) in starting from a far more abstract argument.

Journal of Physics A Mathematical and General•Jun 21, 1993•1 citation

Bosonic andk-fermionic coherent states for a class of polynomial Weyl–Heisenberg algebras

Mohammed Daoud A, M. Kibler

The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.

arXiv:1110.4799•Journal of Physics A Mathematical and Theoretical•May 30, 2012•22 citations

SU(2) and SU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states

Brif, C

We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in the both cases, by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.

•Jan 1, 1997•44 citations

Fermions, Bosons, Anyons, Boltzmanions and Lie-Hopf Algebras

E. Celeghini

Usual quantum statistics is written in Fock space but it is not an algebraic theory. We show that at a deeper level it can be algebraically formalized defining the different statistics as (multi-mode) coherent states of the appropriate (but different from the usual ones) Lie-Hopf groups. The traditional connection between groups and statistics, established in vacuum, is indeed subverted by the interaction with the thermal bath. We show indeed that h(1), related in quantum field theory to bosons, must be used to define in presence of a bath the Boltzmann statistics while, to build the Bose statistics, we have to take into account su(1,1). Astonishing to describe fermions we are forced to use not the superalgebra h(1|1) but su(2) in the fundamental representation. Higher representations of su(2) allow also to give a possible definition of anyon statistics with generalized Pauli principle. Physical implications are discussed; the results is more general then the usual on the discrete spectrum, but everything collapses to standard theory when the continuum limit is performed.

arXiv:cond-mat/9803321•Mar 26, 1998•0 citations

Coherent states and Lie algebras

F. T. Hioe

The Bloch coherent states for a spin or a system of spins and the Glauber coherent states for bosons are examined from the viewpoint of Lie algebras. It is pointed out that the Bloch coherent states are vectors in the space spanned by the basis functions for an irreducible representation of the unitary unimodular group SU(2), and that the Glauber coherent states are vectors in the space spanned by the basis functions for the infinite-dimensional irreducible representation of a contracted group of SU(2). A deeper understanding of many of the useful properties of these coherent states is gained.

Journal of Mathematical Physics•Aug 1, 1974•16 citations

SU (N) coherent states

Manu Mathur, Hamdi Mani

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

arXiv:quant-ph/0206005•Journal of Mathematical Physics•Oct 24, 2002•26 citations
We show that fermionic particles emitted by a stochastic source develop a behaviour which is different from usual Fermi statistics. The particles are described by creation and anihilation operators which span a sl(2, R) algebra. The corresponding Glauber-coherent states,1 defined in an appropriate representation are identical to coherent states found earlier by Barut and Girardello2 in solving the eigenvalue equation of the step operators of the su(1, 1) algebra. This establishes a case in which the mathematically extended notion of coherent states as eigenstates of step operators of an arbitrary Lie algebra finds a physical interpretation.
The application of Glauber's definition (1963) of quantum coherence allows the introduction of neutron fields of partial coherence under the assumption that the complete occupation number space is a direct product of Fermi subspaces. Excitations are described by 'collective' creation and annihilation operators which span an algebra isomorphic to the sl(2,R) algebra. The associated coherent states are of partial coherence and the finite dimensional representation of the su(2,R) algebra is not regularly coherent. In contrast, the transition to an infinite dimensional representation space results in regular coherent properties of the field. This is demonstrated using the representation space of the su(1,1) algebra which has a real isomorphism to the sl(2,R) algebra. The coherent states calculated from Glauber's condition for coherence are completely coherent, and are, moreover, identical to those found by Barut and Girardello (1971) in starting from a far more abstract argument.
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in the both cases, by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.
Usual quantum statistics is written in Fock space but it is not an algebraic theory. We show that at a deeper level it can be algebraically formalized defining the different statistics as (multi-mode) coherent states of the appropriate (but different from the usual ones) Lie-Hopf groups. The traditional connection between groups and statistics, established in vacuum, is indeed subverted by the interaction with the thermal bath. We show indeed that h(1), related in quantum field theory to bosons, must be used to define in presence of a bath the Boltzmann statistics while, to build the Bose statistics, we have to take into account su(1,1). Astonishing to describe fermions we are forced to use not the superalgebra h(1|1) but su(2) in the fundamental representation. Higher representations of su(2) allow also to give a possible definition of anyon statistics with generalized Pauli principle. Physical implications are discussed; the results is more general then the usual on the discrete spectrum, but everything collapses to standard theory when the continuum limit is performed.
The Bloch coherent states for a spin or a system of spins and the Glauber coherent states for bosons are examined from the viewpoint of Lie algebras. It is pointed out that the Bloch coherent states are vectors in the space spanned by the basis functions for an irreducible representation of the unitary unimodular group SU(2), and that the Glauber coherent states are vectors in the space spanned by the basis functions for the infinite-dimensional irreducible representation of a contracted group of SU(2). A deeper understanding of many of the useful properties of these coherent states is gained.
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.