Related papers: An advance in the arithmetic of the Lie groups as an alternative to the forms of the Campbell-Baker-Hausdorff-Dynkin theorem

An advance in the arithmetic of the Lie groups as an alternative to the forms of the Campbell-Baker-Hausdorff-Dynkin theorem

URL: http://arxiv.org/abs/2401.15732v1
Date: Sun, 28 Jan 2024 19:20:02 GMT
Title: An advance in the arithmetic of the Lie groups as an alternative to the forms of the Campbell-Baker-Hausdorff-Dynkin theorem
Authors: Sunghyun Kim, Zhichen Liu, and Richard A. Klemm
Abstract summary: The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. Here it is proven that $rm eabf X+bbf Y$ is equivalent to $rm epbf Zrm eqbf Xrm e-pbf Z$ for scalar $p$ and $q$.
Score: 0.7373617024876725
License: http://creativecommons.org/licenses/by/4.0/
Abstract: The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators ${\bf X}$ and ${\bf Y}$, according to the Campbell-Baker-Hausdorff-Dynkin theorem, ${\rm e}^{{\bf X}+{\bf Y}}$ is not equivalent to ${\rm e}^{\bf X}{\rm e}^{\bf Y}$, but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators $\{{\bf X,Y,Z}\}$, such that $[{\bf X}, {\bf Y}] = \kappa{\bf Z}$ for scalar $\kappa$ and cyclic permutations, here it is proven that ${\rm e}^{a{\bf X}+b{\bf Y}}$ is equivalent to ${\rm e}^{p{\bf Z}}{\rm e}^{q{\bf X}}{\rm e}^{-p{\bf Z}}$ for scalar $p$ and $q$. Extensions for ${\rm e}^{a{\bf X}+b{\bf Y}+c{\bf Z}}$ are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.

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