## 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.

### MSC:

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

### Citations:

Zbl 0004.38201; Zbl 0031.33601; JFM 58.0948.07
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