Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice ${\\ee Z}_{D} \\times {\\ee Z}_{D}$ with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in ${\\ee Z}_{D} \\times q-oscillator subalgebra, an {\\it algebraic} approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower- Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice ${\\ee Z}_{D} \\times {\\ee Z}_{D}$ with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in ${\\ee Z}_{D} \\times q-oscillator subalgebra, an {\\it algebraic} approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower- Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.

Jan 1, 199826 citations

Finite dimensional Hilbert space

A. C. de la Torre, Dardo Goyeneche

American Journal of PhysicsJun 14, 20040 citations

Finite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase

T Hakioglu

Schwinger's finite ( D ) dimensional periodic Hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in is studied and, based on the admissibility of the underlying q -oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.Schwinger's finite ( D ) dimensional periodic Hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in is studied and, based on the admissibility of the underlying q -oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.

arXiv:quant-ph/9809074Journal of Physics A Mathematical and GeneralAug 21, 199836 citations

Finite Dimensional Hilbert Spaces

WORLD SCIENTIFIC eBooksSep 1, 20220 citations

Finite-Dimensional Hilbert Spaces

Paul Richard Halmos

Springer eBooksJan 1, 19831 citation

Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Clifford Algebra

Ashmeet Singh, Sean M. Carroll

The finite entropy of black holes suggests that local regions of spacetime \nare described by finite-dimensional factors of Hilbert space, in contrast with \nthe infinite-dimensional Hilbert spaces of quantum field theory. With this in \nmind, we explore how to cast finite-dimensional quantum mechanics in a form \nthat matches naturally onto the smooth case, especially the recovery of \nconjugate position/momentum variables, in the limit of large Hilbert-space \ndimension. A natural tool for this task is the generalized Clifford algebra \n(GCA). Based on an exponential form of Heisenberg's canonical commutation \nrelation, the GCA offers a finite-dimensional generalization of conjugate \nvariables without relying on any a priori structure on Hilbert space. We \nhighlight some features of the GCA, its importance in studying concepts such as \nlocality of operators, and point out departures from infinite-dimensional \nresults (possibly with a cutoff) that might play a crucial role in our \nunderstanding of quantum gravity. We introduce the concept of "Schwinger \nlocality," which characterizes how the action of an operator spreads a quantum \nstate along conjugate directions. We illustrate these concepts with a worked \nexample of a finite-dimensional harmonic oscillator, demonstrating how the \nenergy spectrum deviates from the familiar infinite-dimensional case.The finite entropy of black holes suggests that local regions of spacetime \nare described by finite-dimensional factors of Hilbert space, in contrast with \nthe infinite-dimensional Hilbert spaces of quantum field theory. With this in \nmind, we explore how to cast finite-dimensional quantum mechanics in a form \nthat matches naturally onto the smooth case, especially the recovery of \nconjugate position/momentum variables, in the limit of large Hilbert-space \ndimension. A natural tool for this task is the generalized Clifford algebra \n(GCA). Based on an exponential form of Heisenberg's canonical commutation \nrelation, the GCA offers a finite-dimensional generalization of conjugate \nvariables without relying on any a priori structure on Hilbert space. We \nhighlight some features of the GCA, its importance in studying concepts such as \nlocality of operators, and point out departures from infinite-dimensional \nresults (possibly with a cutoff) that might play a crucial role in our \nunderstanding of quantum gravity. We introduce the concept of "Schwinger \nlocality," which characterizes how the action of an operator spreads a quantum \nstate along conjugate directions. We illustrate these concepts with a worked \nexample of a finite-dimensional harmonic oscillator, demonstrating how the \nenergy spectrum deviates from the familiar infinite-dimensional case.

Jun 25, 20186 citations

Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice

{\\ee Z}_{D} \\times {\\ee Z}_{D}

with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in ${\\ee Z}_{D} \\times q-oscillator subalgebra, an {\\it algebraic} approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower- Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.