dorsal/arxiv
View SchemaFinite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase
| Authors | T. Hakioglu |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9809074 |
| URL | https://arxiv.org/abs/quant-ph/9809074 |
| DOI | 10.1088/0305-4470/31/33/008 |
| Journal | J.Phys.A31:6975-6994,1998 |
Abstract
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 ${\ee Z}_{D} \times {\ee Z}_{D}$ is studied and, based on the admissibility of the underlying 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.
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"abstract": "Schwinger\u0027s finite (D) dimensional periodic Hilbert space representations are\nstudied on the toroidal lattice ${\\ee Z}_{D} \\times {\\ee Z}_{D}$ with specific\nemphasis on the deformed oscillator subalgebras and the generalized\nrepresentations of the Wigner function. These subalgebras are shown to be\nadmissible endowed with the non-negative norm of Hilbert space vectors. Hence,\nthey provide the desired canonical basis for the algebraic formulation of the\nquantum phase problem. Certain equivalence classes in the space of labels are\nidentified within each subalgebra, and connections with area-preserving\ncanonical transformations are examined. The generalized representations of the\nWigner function are examined in the finite-dimensional cyclic Schwinger basis.\nThese representations are shown to conform to all fundamental conditions of the\ngeneralized phase space Wigner distribution. As a specific application of the\nSchwinger basis, the number-phase unitary operator pair in ${\\ee Z}_{D} \\times\n{\\ee Z}_{D}$ is studied and, based on the admissibility of the underlying\nq-oscillator subalgebra, an {\\it algebraic} approach to the unitary quantum\nphase operator is established. This being the focus of this work, connections\nwith the Susskind-Glogower- Carruthers-Nieto phase operator formalism as well\nas standard action-angle Wigner function formalisms are examined in the\ninfinite-period limit. The concept of continuously shifted Fock basis is\nintroduced to facilitate the Fock space representations of the Wigner function.",
"arxiv_id": "quant-ph/9809074",
"authors": [
"T. Hakioglu"
],
"categories": [
"quant-ph",
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],
"doi": "10.1088/0305-4470/31/33/008",
"journal_ref": "J.Phys.A31:6975-6994,1998",
"title": "Finite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase",
"url": "https://arxiv.org/abs/quant-ph/9809074"
},
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