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Star product and star function

Annotation

We give a brief review on star products and star functions [8, 9]. We introduce a star product on polynomials. Extending the product to functions on complex space, we introduce exponential element in the star product algebra. By means of the star exponential functions we can define several functions called star functions in the algebra.We show certain examples.

Keywords

Moyal product; star product; star product algebra; star exponential functions

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DOI

10.20310/2686-9667-2019-24-127-281-292

UDC

517.9

Pages

281-292

References

[1] G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I. Mapping Theorems and ordering of functions of noncommuting operators”, Physical Review D, 2:10 (1970), 2161-2186. [2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, “Deformation theory and quantization. I. Deformations of symplectic structures”, Annals of Physics, 111:1 (1978), 61-110. [3] J. E. Moyal, “Quantum mechanics as a statistical theory”, Proceedings of the Cambridge Philosophical Society, 45 (1949), 99-124. [4] H. Omori, “Toward geometric quantum theory”, Progress in Mathematics. V. 252: From Geometry to Quantum Mechanics, eds. Y. Maeda, T. Ochiai, P. Michor, A. Yoshioka, Birkhauser, Boston, 2007, 213-251. [5] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, “Strange phenomena related to ordering problems in quantizations”, Journal Lie Theory, 13:2 (2003), 481-510. [6] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, “Orderings and non-formal deformation quantization”, Letters in Mathematical Physics, 82 (2007), 153-175. [7] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, “Geometric objects in an approach to quantum geometry”, Progress in Mathematics. V. 252: From Geometry to Quantum Mechanics, eds. Y. Maeda, T. Ochiai, P. Michor, A. Yoshioka, Birkhauser, Boston, 2007, 303-324. [8] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, “Deformation Expression for Elements of Algebra”, arXiv: math.ph/1104.1708v1. [9] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, “Deformation Expression for Elements of Algebras (II)”, arXiv: math.ph/1105.1218v2.

Received

2019-06-14

Section of issue

Scientific articles

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