베이커-캠벨-하우스도르프 공식

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

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

예

1

하이젠베르크 교환관계식 \([P,Q] = -i \hbar I\)
\(U=e^{i \alpha P},V=e^{i\beta Q}\)이면

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

다항식 \(f(Q)\)에 대하여, 다음이 성립한다

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

양자 바일 대수와 양자평면의 관계식을 얻는다

2

Quantized universal enveloping algebra
\([h,x]=\lambda x\) 이면, (리대수 \(\mathfrak{sl}(2)\) 등에서 나타나는 관계식. sl(2)의 유한차원 표현론 참조)

\[q^h x q^{-h}=q^{\lambda} x\]

메모

Baker-Campbell-Hausdorff formula
http://terrytao.wordpress.com/2011/09/01/254a-notes-1-lie-groups-lie-algebras-and-the-baker-campbell-hausdorff-formula/

매스매티카 파일 및 계산 리소스

https://docs.google.com/file/d/0B8XXo8Tve1cxSm1mQXNlTENneUk/edit

사전 형태의 자료

http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula

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.

메타데이터

위키데이터

ID : Q160131

Spacy 패턴 목록

[{'LEMMA': 'baker'}]