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

AIP conference proceedings

Finite Hamiltonian Systems: Linear Transformations and Aberrations

Kurt Bernardo Wolf, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov

Published Jan 1, 2008• 0 citations
Publisher

Abstract

In finite Hamiltonian systems, the operators of position, momentum, and energy have a finite number N of eigenvalues. These operators can be naturally realized as generators of the Lie algebra su(2), in a representation of spin j, of dimension N = 2j+1. Time evolution is rotation of a phase space sphere, whose projections perform the harmonic motion of an oscillator. The (centrally extended) group of rigid—linear—motions of this phase space is then U(2). On the other hand, N‐point wavefunctions—signals—can be subject to a U(N) group of unitary matrices, containing the linear U(2); aberrations are transformations outside that subgroup. As in geometric optics, we classify the aberration multiplets by order and weight. Their action on phase space is displayed by means of a Wigner function on the sphere, to be compared with the corresponding geometric canonical transformations.

DOI
10.1063/1.3043865

Related papers

Finite Hamiltonian systems: Linear transformations and aberrations

Kurt Bernardo Wolf

Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su(2), and their phase space is a sphere. Rigid motions of this phase space form the group SU(2); overall phases complete this to U(2). But since N-point states can be subject to U(N) ⊃U(2) transformations, the rest of the generators will provide all N 2 unitary transformations of the states, which appear as nonlinear transformations—aberrations—of the system phase space. They are built through the “finite quantization” of a classical optical system.

Physics of Atomic Nuclei•Mar 1, 2010•0 citations

Linear transformations and aberrations in continuous and finite systems

Kurt Bernardo Wolf

In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space. In the Lie-theoretical presentation, linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space, while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates. In a way parallel but distinct from the Schrödinger quantization of continuous classical systems, we quantize the geometric optical model into discrete, finite-dimensional systems based on the Lie algebra , whose wavefunctions are N -point signals, phase space is a sphere and transformations are represented by the N × N unitary matrices that form the group . We factor this group into -linear and nonlinear unitary transformations of phase space and classify all its N 2 − 4 aberrations. This offers a new parametrization of based on a chosen subgroup.

Journal of Physics A Mathematical and Theoretical•Jul 15, 2008•10 citations

Finite optical Hamiltonian systems

Kurt Bernardo Wolf, Natig M. Atakishiyev, Luis Edgar Vincent +2 more

In this essay we finitely quantize the Hamiltonian system of geometric optics to a finite system that is also Hamiltonian, but where signals are described by complex N-vectors, which are subject to unitary transformations that form the group U(N). This group can be decomposed into U(2)-paraxial and aberration transformations. Proper irreducible representation bases are thus provided by quantum angular momentum theory. For one-dimensional systems we have waveguide models. For two-dimensional systems we can have Cartesian or polar sensor arrays, where digital images are subject to unitary rotation, gyration or asymmetric Fourier transformations, as well as a unitary map between the two arrays.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 10, 2011•1 citation

Geometry and dynamics of squeezing in finite systems

Kurt Bernardo Wolf, Guillermo Krötzsch

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.

Journal of the Optical Society of America A•Jan 1, 2007•7 citations

Quantization in terms of symplectic groups: The harmonic oscillator as a generic example

H.A. Kastrup

The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.

Journal of Physics Conference Series•Mar 1, 2011•1 citation

The Group Theoretical Structure of Fermion Many-Body Systems Arising from the Canonical Anticommutation Relation. I: Lie Algebras of Fermion Operators and Exact Generator Coordinate Representations of State Vectors

Hideo Fukutome

We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.

Progress of Theoretical Physics•Mar 1, 1981•46 citations
In finite Hamiltonian systems, the operators of position, momentum, and energy have a finite number N of eigenvalues. These operators can be naturally realized as generators of the Lie algebra su(2), in a representation of spin j, of dimension N = 2j+1. Time evolution is rotation of a phase space sphere, whose projections perform the harmonic motion of an oscillator. The (centrally extended) group of rigid—linear—motions of this phase space is then U(2). On the other hand, N‐point wavefunctions—signals—can be subject to a U(N) group of unitary matrices, containing the linear U(2); aberrations are transformations outside that subgroup. As in geometric optics, we classify the aberration multiplets by order and weight. Their action on phase space is displayed by means of a Wigner function on the sphere, to be compared with the corresponding geometric canonical transformations.
Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su(2), and their phase space is a sphere. Rigid motions of this phase space form the group SU(2); overall phases complete this to U(2). But since N-point states can be subject to U(N) ⊃U(2) transformations, the rest of the generators will provide all N 2 unitary transformations of the states, which appear as nonlinear transformations—aberrations—of the system phase space. They are built through the “finite quantization” of a classical optical system.
In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space. In the Lie-theoretical presentation, linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space, while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates. In a way parallel but distinct from the Schrödinger quantization of continuous classical systems, we quantize the geometric optical model into discrete, finite-dimensional systems based on the Lie algebra , whose wavefunctions are N -point signals, phase space is a sphere and transformations are represented by the N × N unitary matrices that form the group . We factor this group into -linear and nonlinear unitary transformations of phase space and classify all its N 2 − 4 aberrations. This offers a new parametrization of based on a chosen subgroup.
In this essay we finitely quantize the Hamiltonian system of geometric optics to a finite system that is also Hamiltonian, but where signals are described by complex N-vectors, which are subject to unitary transformations that form the group U(N). This group can be decomposed into U(2)-paraxial and aberration transformations. Proper irreducible representation bases are thus provided by quantum angular momentum theory. For one-dimensional systems we have waveguide models. For two-dimensional systems we can have Cartesian or polar sensor arrays, where digital images are subject to unitary rotation, gyration or asymmetric Fourier transformations, as well as a unitary map between the two arrays.
Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.