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arXiv:1206.2507 · Journal of Physics A Mathematical and Theoretical

Complementarity and phases inSU(3)

Hubert de Guise, Apostol Vourdas, Luis Lorenzo Sánchez-Soto

Published May 30, 2012• 4 citations
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Abstract

Phase operators and phase states are introduced for irreducible representations of the Lie algebra su(3) using a polar decomposition of ladder operators. In contradistinction with su(2), it is found that the su(3) polar decomposition does not uniquely determine a Hermitian phase operator. We describe two possible ways of proceeding: one based in imposing SU(2) invariance and the other based on the idea of complementarity. The generalization of these results to SU(n) is sketched.

DOI
10.1088/1751-8113/45/24/244030

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Phase operators and phase states are introduced for irreducible representations of the Lie algebra su(3) using a polar decomposition of ladder operators. In contradistinction with su(2), it is found that the su(3) polar decomposition does not uniquely determine a Hermitian phase operator. We describe two possible ways of proceeding: one based in imposing SU(2) invariance and the other based on the idea of complementarity. The generalization of these results to SU(n) is sketched.
Phase operators and phase states are introduced in the Hilbert space H2j+1{H}_{2j+1}H2j+1​ associated with the SU(2) group. The phase operators obey the SU(2) algebra and play a dual role to the standard angular-momentum operators. A finite Weyl group plays a fundamental role in those ideas. In the SU(1,1) case the exponential of the phase operators is nonunitary, and the phase states form an overcomplete set which is used to formulate an analytic representation.
The main aim of this work is to build unitary phase operators and the corresponding temporally stable phase states for the [Formula: see text] Lie algebra. We first introduce an irreducible finite-dimensional Hilbertian representation of the [Formula: see text] Lie algebra which is suitable for our purpose. The phase operators obtained from the [Formula: see text] generators are defined and the phase states are derived as eigenstates associated to these unitary phase operators. The special cases of [Formula: see text] and [Formula: see text] Lie algebras are also explicitly investigated.
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.
The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the generators J- and J+ of the SU(2) group, with J+ = J-† = HUr where H is Hermitean and Ur unitary, and (ii) an alternative to the {J2,Jz} quantization scheme, viz., the {J2,Ur} quantization scheme. The representation theory of the SU(2) group can be developed in this nonstandard scheme. The key ideas for developing the Wigner-Racah algebra of the SU(2) group in the {J2,Ur} scheme are given. In particular, some properties of the coupling and recoupling coefficients as well as the Wigner-Eckart theorem in the {J2,Ur} scheme are examined in great detail.