TY - JOUR

T1 - Finite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase

AU - Hakioǧlu, T.

PY - 1998/8/21

Y1 - 1998/8/21

N2 - Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice ℤD × ℤD 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 ℤD × ℤD 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.

AB - Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice ℤD × ℤD 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 ℤD × ℤD 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.

UR - http://www.scopus.com/inward/record.url?scp=0032555392&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/31/33/008

DO - 10.1088/0305-4470/31/33/008

M3 - Article

AN - SCOPUS:0032555392

SN - 0305-4470

VL - 31

SP - 6975

EP - 6994

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 33

ER -