Moyal-like form of the star product for generalized \(SU(2)\) Stratonovich-Weyl symbols. (English) Zbl 1052.81060

E. P. Wigner in his seminal paper [Phys. Rev., II. Ser. 40, 749–759 (1932; JFM 58.0948.07 and Zbl 0004.38201)] presented the phase space method which has been successfully applied in quantum mechanics. The method provides a useful insight into quantum classical correspondence in non-relativistic quantum mechanics. Stratonovich gave an axiomatic approach which is known as “Stratonovich-Weyl correspondence”. The pivot of the Stratonovich-Weyl correspondence is the star (or twisted) product which associates the product of two operators \(\widehat{f}_1\) and \(\widehat{f}_2\) with an associative star product \(f_1(\Omega) * f_2(\Omega)\) by their symbols \(f_i(\Omega)\) (the star \(*\) is associative, non-commutative). J. E. Moyal’s paper [Proc. Camb. Philos. Soc. 45, 99–124 (1949; Zbl 0031.33601)] explored both integral and differential representations of the star product for the Heisenberg-Weyl group, where states and observables are viewed as functions on a given phase-space.
The main result of the paper under review is to give an exact differential form representation of the star product for spin-like system. By using \((2s+1)\)-dimensional representation of the universal enveloping algebra of \(su(2)\), operators \(\widehat{f}_i\) (\(i=1, 2\)) can be expressed as \(\widehat{f}_i=\sum_{l=0}^{2s}\sum_{-l}^l f_{lk}^i \widehat{T}_{lk}^{(s)}\), where \(\widehat{T}_{lk}^{(s)}\) is the irreducible tensor operator and their product \(\widehat{T}_{l_1k_1}^{(s)}\cdot \widehat{T}_{l_2k_2}^{(s)}\) is a linear form on irreducible tensor operator of \(6j\)-symbols in terms of the expansion on Clebsch-Gordon coefficients. Thus, using the relation \(\widehat{f}_1 \widehat{f}_2=\frac{2s+1}{4\pi} \int d\Omega \widehat{\omega}_{-s} W_{f_1f_2}^{(s)}\) with \(W_{f_1f_2}^{(s)} = \widehat{L}_{f_1f_2}^{s}( W_{f_1}^{(s_1)}W_{f_2}^{(s_2)})\) and expressions of \(\widehat{f}_i\) and \(W_{f_i}^{(s_i)}\), the author obtain the differential operator \(\widehat{L}_{f_1f_2}^s\) of the star product.
The derived procedure is given in section 4 with an example in section 5. Section 2 provides mathematical ideas, and reviews Stratonovich-Weyl kernel for \(SU(2)\) (spin-like) system in section 3. The large spin limit is reduced from section 4 in order to achieve a simpler form of the differential form representation of the star-product in section 6. The last section 7 includes a discussion to replace the operator algebra by differential calculus in the classical phase-space. The evolution equation becomes the quantum Liouville equation for quasi-distributions on the sphere.


81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
53D55 Deformation quantization, star products
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