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arXiv:quant-ph/0504025 · Collection of Czechoslovak Chemical Communications

Representation Theory and Wigner-Racah Algebra of the SU(2) Group in a Noncanonical Basis

M. Kibler

Published Jan 1, 2005• 9 citations
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Abstract

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.

DOI
10.1135/cccc20050771

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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.
The Lie algebra of the group SU 2 is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J2, Ur} scheme, an alternative to the familiar {J2, Jz} quantization scheme corresponding to common eigenvectors of the Casimir operator J2 and the Cartan operator Jz. A connection is established between the eigenvectors of the complete set of commuting operators {J2, Ur} and mutually unbiased bases in spaces of constant angular momentum.
This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU(2) corresponding to an irreducible representation of SU(2). The representation theory of SU(2) is reconsidered via the use of two truncated deformed oscillators. This leads to replace the familiar scheme {j^2, j_z} by a scheme {j^2, v(ra)}, where the two-parameter operator v(ra) is defined in the enveloping algebra of the Lie algebra su(2). The eigenvectors of the commuting set of operators {j^2, v(ra)} are adapted to a tower of chains SO(3) > C(2j+1), 2j integer, where C(2j+1) is the cyclic group of order 2j+1. In the case where 2j+1 is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices.
This is a challenging paper that includes some reviews and new results. Since the non-commutative version of the classical system based on the compact group SU(2) has been constructed in (quant-ph/0502174) by making use of Jaynes–Commings model and so-called quantum diagonalization method in (quant-ph/0502147), we construct a non-commutative version of the classical system based on the non-compact group SU(1,1) by modifying the compact case. In this model the Hamiltonian is not hermite but pseudo hermite, which causes a big difference between the two models. For example, in the classical representation theory of SU(1,1), unitary representations are infinite dimensional from the starting point. Therefore, to develop a unitary theory of non-commutative system of SU(1,1) we need an infinite number of non-commutative systems, which means a kind of second non-commutativization. This is a very hard and interesting problem. We develop a corresponding theory though it is not always enough, and present some challenging problems concerning how classical properties can be extended to the non-commutative case. This paper is arranged for the convenience of readers as the first subsection is based on the standard model (SU(2) system) and the next one is based on the non-standard model (SU(1,1) system). This contrast may make the similarities and differences between the standard and non-standard models clearer.
A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.
Noncanonical quantization schemes have been proposed in quantum physics long time ago. Recalling some of them I associate these investigations to the quantum mechanical im-plications of the canonically inequivalent Hamilton formula-tions of classical Newtonian dynamics. As the simplest illus-tration of such alternative Hamiltonian descriptions I consider one dimensional system of two particles interacting harmoni-cally. Description of the motion of their center of mass is stan-dard and guarantees the Galilean covariance but the internal motion is described as a noncanonical oscillator for which the Hamiltonian and the total energy are not related by a simple proportionality. Alternative Hamiltonians lead to alternative Poisson structures which may be considered as classical ana-logues of the noncanonical quantum commutation rules. Gen-Concepts of Physics, Vol. I (2004) 11 erated alternative quantizations are consistent with the same Heisenberg equations of motion, i.e., they satisfy the Wigner principle of quantization. For the one dimensional harmonic oscillator the simplest noncanonical algebraic description is given in terms of the Lie algebra so(p, q), p+ q = 3. In such a case the Hamiltonian is still a linear function while the energy is a quadratic polynomial of the main quantum number. This leads to nonstandard termodynamical properties obeyed by such a noncanonical oscillator. The 3-dimensional harmonic oscillator also has a noncanonical description given in terms of the simple Lie algebra, so(p, q), p+ q = 5 in the case under consideration. The most important physical consequence of such a description is the noncommutativity of the coordinate operators, interpreted to be an implication of the noncommu-tative geometry of the underlying spacetime.
This paper deals with bases in a finite-dimensional Hilbert space.Such a space can be realized as a subspace of the representation space of SU 2 corresponding to an irreducible representation of SU 2 .The representation theory of SU 2 is reconsidered via the use of two truncated deformed oscillators.This leads to replacement of the familiar scheme {j 2 , j z } by a scheme {j 2 , v ra }, where the two-parameter operator v ra is defined in the universal enveloping algebra of the Lie algebra su 2 .The eigenvectors of the commuting set of operators {j 2 , v ra } are adapted to a tower of chains SO 3 ⊃ C 2j+1 (2j ∈ N * ), where C 2j+1 is the cyclic group of order 2j + 1.In the case where 2j + 1 is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases.Some useful relations on generalized quadratic Gauss sums are exposed in three appendices.