FINITE-DIMENSIONAL SCHWINGER BASIS, DEFORMED SYMMETRIES, WIGNER FUNCTION, AND AN ALGEBRAIC APPROACH TO QUANTUM PHASE

Schwinger's finite (D) dimensional periodic Hilbert space representati ons are studied on the toroidal lattice Z(D) x Z(D) with specific emph asis on the deformed oscillator subalgebras and the generalized repres entations of the Wigner function. These subalgebras are shown to be ad missible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic form ulation of the quantum phase problem. Certain equivalence classes in t he space of labels are identified within each subalgebra, and connecti ons with area-preserving canonical transformations are examined.The ge neralized representations of the Wigner function are examined in the f inite-dimensional cyclic Schwinger basis. These representations are sh own 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 Z(D) x Z(D) is studie d and, based on the admissibility of the underlying q-oscillator subal gebra, an algebraic approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the S usskind-Glogower-Carmthers-Nieto phase operator formalism as well as s tandard action-angle Wigner function formalisms are examined in the in finite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.