Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 084, 7 pages      arXiv:2105.12593      https://doi.org/10.3842/SIGMA.2021.084

Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra

Stjepan Meljanac a and Rina Štrajn b
a) Division of Theoretical Physics, Ruder Bošković Institute, Bijenička cesta 54, 10002 Zagreb, Croatia
b) Department of Electrical Engineering and Computing, University of Dubrovnik, Ćira Carića 4, 20000 Dubrovnik, Croatia

Received May 27, 2021, in final form September 09, 2021; Published online September 15, 2021

Abstract
We consider a class of exponentials in the Weyl-Heisenberg algebra with exponents of type at most linear in coordinates and arbitrary functions of momenta. They are expressed in terms of normal ordering where coordinates stand to the left from momenta. Exponents appearing in normal ordered form satisfy differential equations with boundary conditions that could be solved perturbatively order by order. Two propositions are presented for the Weyl-Heisenberg algebra in 2 dimensions and their generalizations in higher dimensions. These results can be applied to arbitrary noncommutative spaces for construction of star products, coproducts of momenta and twist operators. They can also be related to the BCH formula.

Key words: exponential operators; normal ordering; Weyl-Heisenberg algebra; noncommutative geometry.

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