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arXiv · 2405.03001

Old and new powerful tools for the normal ordering problem and noncommutative binomials

Kei Beauduin
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Abstract

In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. These tools are subsequently applied to derive counterparts for the exponential identity $e^{A+B} = e^A e^B$ and the binomial theorem $(A+B)^n = \sum \binom{n}{k} A^k B^{n-k}$ when the commutator $[B, A]$ is either an arbitrary quadratic polynomial or a monomial in $A$ or $B$. Analogous formulas are found when the commutator is bivariate. Furthermore, we introduce a novel operator bridging between the normal and antinormal ordered forms.

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Kei Beauduin. 2024-05-05. Old and new powerful tools for the normal ordering problem and noncommutative binomials. https://doi.org/10.54550/eca2025v5s1r4

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