A consistent Lie algebraic representation of quantum phase and number operators (original) (raw)
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A fully consistent Lie algebraic representation of quantum phase and number operators
Journal of Physics A: Mathematical and General, 2004
A consistent realization of the quantum operators corresponding to the canonically conjugate phase and number variables is proposed, resorting to the κ = 1 2 positive discrete series of the irreducible unitary representation of the Lie algebra su(1, 1) of the double covering group of SO ↑ (1, 2).
Journal of Mathematical Physics, 1975
The purpose of this article is to present a detailed analysis on the quantum mechnical level of the canonical transformation between coordinate-momentum and number-phase descriptions for systems possessing an s i (2,R) dynamical algebra, specifically, the radial harmonic oscillator and pseudo-Coulomb systems. The former one includes the attractive and repulsive oscillators and the free particle, each with an additional "centrifugal" force. while the latter includes the bound, free and threshold states with an added "centrifugal" force. This is implemented as a unitary mappingcanonical transform-between the usual Hilbert space L 2 of quantum mechanics and a new set of Hilbert spaces on the circle whose coordinate has the meaning of a phase variable. Moreover, the UIR's D t of the universal covering group of S L (2,R) realized on the former space are mapped unitarily onto the latter.
Linear canonical transformations and quantum phase: a unified canonical and algebraic approach
Journal of Physics A: Mathematical and General, 1999
The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwinger's unitary-canonical operator basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with dynamical symmetry is examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.
Admissible cyclic representations and an algebraic approach to quantum phase
Journal of Physics A: Mathematical and General, 1998
Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscillator (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototype cyclic oscillator endowed with a non-negative (admissible) spectrum. By changing the deformation parameter, the cyclic spectrum can be tuned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the proper algebraic formulation of the quantum-phase operator. Using a general scheme, it is shown that admissible cyclic algebras permit a well-defined Hermitian phase operator of which properties are studied in detail at finite periods as well as at the infinite-period limit. Fujikawa's index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the specific example of GHO it is confirmed that the infiniteperiod limit is distinctively singular. The connection with the Pegg-Barnett phase formalism is established in this singular limit as the period of the cyclic representations tends to infinity. The singular behaviour at this limit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase operator.
On generalized phase operators for the quantum harmonic oscillator
Il Nuovo Cimento B Series 10, 1970
Following the demonstration by Lerner that the phase (cosine and sine) operators for a harmonic oscillator can be much more general than the specific set considered earlier in the literature, we make a systematic study of the properties of such generalized phase operators. We determine their eigenstates and show that the coefficients of transformation from the number eigenstates to these are, quite generally, orthogonal polynomials in the eigenvalue parameter. Using these generalized C and S operators the number-phase minimum uncertainty state is given. Finally we point out that though the commutator [C, S] cannot be made to vanish, nevertheless one can construct C and S in such a way that their commutator has a vanishing expectation value with respect to all coherent states having a given mean occupation number exceeding unity. 1.-Introduction.
Journal of Physics A: Mathematical and General, 1998
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 studied. The generalised 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 is studied and, based on the admissibility of the underlying q-oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. Connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as the 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.
Refined Algebraic Quantization with the Triangular Subgroup of SL(2, ℝ)
International Journal of Modern Physics D, 2005
We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL(2, R). The unreduced phase space is T * R p+q with p ≥ 1 and q ≥ 1, and the system has a distinguished classical o(p, q) observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O(p, q). The representation is nontrivial iff (p, q) = (1, 1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantisation that imposes the constraints in the senseĤ a Ψ = 0 and postulates self-adjointness of the o(p, q) observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantisation gives no quantum theory when (p, q) = (1, 2) or (2, 1), or when p ≥ 2, q ≥ 2 and p + q ≡ 1 (mod 2).