베이커-캠벨-하우스도르프 공식

수학노트
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개요

  • 리대수에 정의된 bracket을 이용하여, $\exp$에 의한 리군의 원소의 곱셈을 정의

$$ e^x e^y = e^{H(x,y)} $$ 여기서 $$H(x,y)=x+y+\frac{1}{2}[x,y]+\frac{1}{12}([x,[x,y]]+[y,[y,x]])+\cdots$$


보조정리

  • $n\times n$ 행렬 $X, Y$에 대하여, 다음이 성립한다

$$ e^{X}Y e^{-X} = e^{\operatorname{ad}X} Y =Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots $$


1

$$U Q U^{-1}=Q+\alpha\hbar I$$

  • 다항식 $f(Q)$에 대하여, 다음이 성립한다

$$U f(Q) U^{-1}=f(Q+\alpha\hbar I)$$ $$UVU^{-1}=e^{i\hbar \alpha \beta}V$$


2

$$q^h x q^{-h}=q^{\lambda} x$$


메모


매스매티카 파일 및 계산 리소스


사전 형태의 자료


관련도서

  • Bonfiglioli, Andrea, and Roberta Fulci. Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin. Springer Science & Business Media, 2011.


리뷰, 에세이, 강의노트


관련논문

  • J. Mostovoy, J. M. Perez-Izquierdo, I. P. Shestakov, A Non-associative Baker-Campbell-Hausdorff formula, arXiv:1605.00953 [math.RA], May 03 2016, http://arxiv.org/abs/1605.00953
  • Nishimura, Hieokazu, and Hirowaki Takamiya. ‘A Note on the Infinitesimal Baker-Campbell-Hausdorff Formula’. arXiv:1507.01453 [math], 25 June 2015. http://arxiv.org/abs/1507.01453.
  • Van-Brunt, Alexander, and Matt Visser. ‘Explicit Baker-Campbell-Hausdorff Formulae for Some Specific Lie Algebras’. arXiv:1505.04505 [hep-Th, Physics:math-Ph, Physics:quant-Ph], 18 May 2015. http://arxiv.org/abs/1505.04505.
  • Matone, Marco. “Closed Form of the Baker-Campbell-Hausdorff Formula for Semisimple Complex Lie Algebras.” arXiv:1504.05174 [hep-Ph, Physics:hep-Th, Physics:math-Ph, Physics:quant-Ph], April 20, 2015. http://arxiv.org/abs/1504.05174.
  • Matone, Marco. ‘Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas’. arXiv:1503.08198 [hep-Th, Physics:math-Ph, Physics:quant-Ph], 27 March 2015. http://arxiv.org/abs/1503.08198.
  • Matone, Marco. ‘An Algorithm for the Baker-Campbell-Hausdorff Formula’. arXiv:1502.06589 [hep-Ph, Physics:hep-Th, Physics:math-Ph, Physics:quant-Ph], 23 February 2015. http://arxiv.org/abs/1502.06589.
  • Van-Brunt, Alexander, and Matt Visser. “Simplifying the Reinsch Algorithm for the Baker-Campbell-Hausdorff Series.” arXiv:1501.05034 [hep-Th, Physics:math-Ph, Physics:quant-Ph], January 20, 2015. http://arxiv.org/abs/1501.05034.
  • Van-Brunt, Alexander, and Matt Visser. “Special-Case Closed Form of the Baker-Campbell-Hausdorff Formula.” arXiv:1501.02506 [math-Ph, Physics:quant-Ph], January 11, 2015. http://arxiv.org/abs/1501.02506.
  • Casas, Fernando, and Ander Murua. “An Efficient Algorithm for Computing the Baker–Campbell–Hausdorff Series and Some of Its Applications.” Journal of Mathematical Physics 50, no. 3 (March 1, 2009): 033513. doi:10.1063/1.3078418.
  • Alekseev, Anton, and Charles Torossian. ‘The Kashiwara-Vergne Conjecture and Drinfeld’s Associators’. arXiv:0802.4300 [math], 28 February 2008. http://arxiv.org/abs/0802.4300.
  • Newman, Morris, and Robert C. Thompson. “Numerical Values of Goldberg’s Coefficients in the Series for $\log e^xe^y$” Mathematics of Computation 48, no. 177 (1987): 265–71. doi:10.1090/S0025-5718-1987-0866114-9.
  • Kashiwara, Masaki, and Michèle Vergne. ‘The Campbell-Hausdorff Formula and Invariant Hyperfunctions’. Inventiones Mathematicae 47, no. 3 (1 October 1978): 249–72. doi:10.1007/BF01579213.