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International Journal of Theoretical Physics

Unitary Operator of su q (n)-Covariant Oscillator Algebra

Won Sang Chung

Published Oct 1, 2002• 0 citations
Publisher
DOI
10.1023/a:1021013109038

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Modern Physics Letters A•Aug 20, 2009•4 citations
The polar decomposition of the su(2) algebra leads to unitary phase operators which do not close to an algebra with the number operator. It is shown that phase and number operators can be embedded into a larger su(2j + 1) algebra with trigonometric structure constants. In the contraction limit where we pass from the su(2) to the oscillator algebra, the embedding algebra for the phase operators becomes su(∞). The coherent states realization of the su(∞) algebra and its relation to the q-deformed oscillator algebra is briefly discussed.
We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. The unitary operator U, which is found to be able to transform the Fock space of a three–dimensional isotropic harmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum–mechanical rotation in three–mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.
We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with [Formula: see text] and [Formula: see text] symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are [Formula: see text]-pseudo-Hermiticity and [Formula: see text]-anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.