About: Baker–Campbell–Hausdorff formula

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In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in and and iterated commutators thereof. The first few terms of this series are:where "" indicates terms involving higher commutators of and . If and are sufficiently small elements of the Lie algebra of a Lie group , the series is convergent. Meanwhile, every element sufficiently close to the identity in can be expressed as for a small in . Thus, we can say that near the identity the group multiplication in —written as —can be expressed in purely Lie a

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rdf:type	software yago:WikicatLieAlgebras yago:WikicatLieGroups yago:Abstraction100002137 yago:Algebra106012726 yago:Cognition100023271 yago:Content105809192 yago:Discipline105996646 yago:Group100031264 yago:KnowledgeDomain105999266 yago:Mathematics106000644 yago:PsychologicalFeature100023100 yago:PureMathematics106003682 yago:Science105999797
rdfs:label	Baker-Campbell-Hausdorff-Formel (de) Φόρμουλα Μπέικερ – Κάμπελ – Χάουσντορφ (el) Baker–Campbell–Hausdorff formula (en) Fórmula de Baker-Campbell-Hausdorff (es) Formule de Baker-Campbell-Hausdorff (fr) Formula di Baker-Campbell-Hausdorff (it) Fórmula de Baker-Campbell-Hausdorff (pt) Формула Бейкера — Кэмпбелла — Хаусдорфа (ru) 贝克-坎贝尔-豪斯多夫公式 (zh)
rdfs:comment	In der Mathematik ist die Baker-Campbell-Hausdorff-Formel eine nach den Mathematikern Henry Frederick Baker, John Edward Campbell und Felix Hausdorff benannte Gleichung, die ein Vertauschungsgesetz für bestimmte lineare Operatoren angibt. (de) En mathématiques, la formule de Baker- (en)-Hausdorff est la solution Z de l'équation : où , et sont des matrices, ou plus généralement des éléments d'une algèbre de Lie d'un groupe de Lie. Expression Avec les crochets de Lie, elle s'écrit : Une formule reliée est la formule de Zassenhaus: En particulier lorsque et commutent nous avons Lorsque et commutent avec leur commutateur (c'est-à-dire ) le résultat se restreint à la formule dite de Glauber : . Elle est souvent appliquée en physique quantique avec les opérateurs position et impulsion , . (fr) Em Teoria de Lie, teoria de operadores e teoria matricial, a fórmula de Baker-Campbell-Hausdorff descreve a exponenciação de elementos de uma álgebra de Lie que não necessariamente comutam: onde é o comutador da álgebra, e os termos posteriores são todos comutadores de comutadores. (pt) Формула Бейкера — Кэмпбелла — Хаусдорфа определяет выражение для из следующего равенства здесь , и — элементы алгебры Ли близкие к нулю.Выражение на является довольно сложным рядом с членами составленными из скобок Ли от , . Существование этой формулы играет ключевую роль в доказательстве того, что алгебра Ли полностью определяет локальную структуру своей группы Ли.Частный случай этой формулы применяется в квантовой механике и особенно в квантовой оптике. (ru) 在数学中，贝克-坎贝尔-豪斯多夫公式（英語：Baker–Campbell–Hausdorff formula）指的是下列方程中的解：其中，和是李群李代数中的非对易元素。贝克-坎贝尔-豪斯多夫公式有很多种写法，下列是最常见的一种：这里的表示还应有高阶项。 (zh) Στα μαθηματικά, η φόρμουλα Μπέικερ – Κάμπελ – Χάουσντορφ είναι η λύση για στην εξίσωση για πιθανώς μη αντιμεταθετικό X και Y στην μιας . Υπάρχουν διάφοροι τρόποι γραφής του τύπου, αλλά όλοι τελικά δίνουν μια έκφραση για με Λι αλγεβρικούς όρους, δηλαδή ως σειρά (όχι απαραίτητα συγκλίνουσα) στο και και επαναλαμβάνοντας τους μεταγωγείς τους. Οι πρώτοι όροι αυτής της σειράς είναι: Αν και είναι αρκετά μικρά πίνακες, τότε μπορεί να υπολογιστεί ως ο λογάριθμος του , όπου τα εκθετικά και ο λογάριθμος μπορούν να υπολογιστούν ως σειρές ισχύος. (el) In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in and and iterated commutators thereof. The first few terms of this series are:where "" indicates terms involving higher commutators of and . If and are sufficiently small elements of the Lie algebra of a Lie group , the series is convergent. Meanwhile, every element sufficiently close to the identity in can be expressed as for a small in . Thus, we can say that near the identity the group multiplication in —written as —can be expressed in purely Lie a (en) En matemáticas, la fórmula de Baker-Campbell-Hausdorff permite hallar la solución de para la ecuación con e que pueden ser no conmutativos en el álgebra de Lie de un grupo de Lie. Hay varias formas de escribir la fórmula, pero todas finalmente producen una expresión para en términos algebraicos de Lie, es decir, como una serie formal (no necesariamente convergente) en e y conmutadores iterados de los mismos. Los primeros términos de esta serie son: Las exposiciones modernas de la fórmula se pueden encontrar, entre otros lugares, en los libros de Rossmann y Hall. (es) In matematica, la formula di Baker–Campbell–Hausdorff è la soluzione dell'equazione: per due grandezze e non commutanti (ad esempio matrici quadrate). Questa formula collega i gruppi di Lie con le algebre di Lie esprimendo il logaritmo del prodotto di due elementi del gruppo di Lie come un elemento dell'algebra di Lie in coordinate canoniche. La soluzione coinvolge le parentesi di Lie degli elementi e ; la sua scrittura, interrotta al terzo ordine, è: Questa formula prende il nome da Henry Frederick Baker, John Edward Campbell e Felix Hausdorff. (it)
dcterms:subject	Exponentials Combinatorics Lie groups Mathematical physics
Wikipage page ID	285422 (xsd:integer)
Wikipage revision ID	1115922051 (xsd:integer)
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