"베이커-캠벨-하우스도르프 공식"의 두 판 사이의 차이

수학노트
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(같은 사용자의 중간 판 6개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
−
* 리대수에 정의된 bracket을 이용하여, $\exp$에 의한 리군의 원소의 곱셈을 정의
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* 리대수에 정의된 bracket을 이용하여, <math>\exp</math>에 의한 리군의 원소의 곱셈을 정의
−
$$
+
:<math>
 
e^x e^y = e^{H(x,y)}
 
e^x e^y = e^{H(x,y)}
−
$$
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</math>
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여기서 $$H(x,y)=x+y+\frac{1}{2}[x,y]+\frac{1}{12}([x,[x,y]]+[y,[y,x]])+\cdots$$
+
여기서 :<math>H(x,y)=x+y+\frac{1}{2}[x,y]+\frac{1}{12}([x,[x,y]]+[y,[y,x]])+\cdots</math>
  
  
 
==보조정리==
 
==보조정리==
−
* $n\times n$ 행렬 $X, Y$에 대하여, 다음이 성립한다
+
* <math>n\times n</math> 행렬 <math>X, Y</math>에 대하여, 다음이 성립한다
−
$$
+
:<math>
 
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
 
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
−
$$
+
</math>
  
  
 
==예==
 
==예==
 
===1===
 
===1===
−
* [[하이젠베르크 군과 대수|하이젠베르크 교환관계식]] $[P,Q] = -i \hbar I$
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* [[하이젠베르크 군과 대수|하이젠베르크 교환관계식]] <math>[P,Q] = -i \hbar I</math>
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* $U=e^{i \alpha P},V=e^{i\beta Q}$이면
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* <math>U=e^{i \alpha P},V=e^{i\beta Q}</math>이면
−
$$U Q U^{-1}=Q+\alpha\hbar I$$
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:<math>U Q U^{-1}=Q+\alpha\hbar I</math>
−
* 다항식 $f(Q)$에 대하여, 다음이 성립한다
+
* 다항식 <math>f(Q)</math>에 대하여, 다음이 성립한다
−
$$U f(Q) U^{-1}=f(Q+\alpha\hbar I)$$
+
:<math>U f(Q) U^{-1}=f(Q+\alpha\hbar I)</math>
−
$$UVU^{-1}=e^{i\hbar \alpha \beta}V$$
+
:<math>UVU^{-1}=e^{i\hbar \alpha \beta}V</math>
 
* [[양자 바일 대수와 양자평면]]의 관계식을 얻는다
 
* [[양자 바일 대수와 양자평면]]의 관계식을 얻는다
  
27번째 줄: 27번째 줄:
 
===2===
 
===2===
 
* [[Quantized universal enveloping algebra]]
 
* [[Quantized universal enveloping algebra]]
−
* $[h,x]=\lambda x$ 이면, (리대수 <math>\mathfrak{sl}(2)</math> 등에서 나타나는 관계식. [[sl(2)의 유한차원 표현론]] 참조)
+
* <math>[h,x]=\lambda x</math> 이면, (리대수 <math>\mathfrak{sl}(2)</math> 등에서 나타나는 관계식. [[sl(2)의 유한차원 표현론]] 참조)
−
$$q^h x q^{-h}=q^{\lambda} x$$
+
:<math>q^h x q^{-h}=q^{\lambda} x</math>
  
  
42번째 줄: 42번째 줄:
 
==사전 형태의 자료==
 
==사전 형태의 자료==
 
* http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
 
* http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
 +
 +
 +
==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==
 +
* Alekseev, [http://www.6ecm.pl/docs/Alekseev.pdf Bernoulli numbers, Drinfeld associators, and the Kashiwara–Vergne problem]
 +
* Kurlin, [http://hamilton.nuigalway.ie/DeBrunCentre/SecondWorkshop/metaBCH.pdf The metabelian BCH formula and compressed Drinfeld associators]
  
  
 
==관련논문==
 
==관련논문==
−
* http://arxiv.org/abs/1505.04505
+
* 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. “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. ‘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.
52번째 줄: 64번째 줄:
 
* 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.
 
* 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.
+
* 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 <math>\log e^xe^y</math>” 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.
 
* 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.
  
−
==관련도서==
 
−
* Bonfiglioli, Andrea, and Roberta Fulci. Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin. Springer Science & Business Media, 2011.
 
  
  
 
[[분류:리군과 리대수]]
 
[[분류:리군과 리대수]]
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q160131 Q160131]
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===Spacy 패턴 목록===
 +
* [{'LEMMA': 'baker'}]

2021년 2월 17일 (수) 02:26 기준 최신판

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

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

  • 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.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LEMMA': 'baker'}]
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