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Pythagoras0 (토론 | 기여) (→관련논문) |
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* 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. “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. | * 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. | ||
* 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. | * 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. | ||
[[분류:리군과 리대수]] | [[분류:리군과 리대수]] | ||
2015년 1월 23일 (금) 17:14 판
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
사전 형태의 자료
관련논문
- 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.
- 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.