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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Non-abelian gauge transformation ： ウィキペディア英語版 Non-abelian gauge transformation In theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group ''G'', the elements of which do not obey the commutative law when they are multiplied. By contrast, the original choice of gauge group in the physics of electromagnetism had been U(1), which is commutative. For a non-abelian Lie group ''G'', its elements do not commute, i.e. they in general do ''not'' satisfy :$a$ *b=b *a \,. The quaternions marked the introduction of non-abelian structures in mathematics. In particular, its generators $t^a$, which form a basis for the vector space of infinitesimal transformations (the Lie algebra), have a commutation rule: :$\backslash left()\; =\; t^a\; t^b\; -\; t^b\; t^a\; =\; C^\; t^c.$ The structure constants $C^$ quantify the lack of commutativity, and do not vanish. We can deduce that the structure constants are antisymmetric in the first two indices and real. The normalization is usually chosen (using the Kronecker delta) as :$Tr(t^at^b)\; =\; \backslash frac\backslash delta^.$ Within this orthonormal basis, the structure constants are then antisymmetric with respect to all three indices. An element $\backslash omega$ of the group can be expressed near the identity element in the form :$\backslash omega\; =\; exp(\backslash theta^at^a)$, where $\backslash theta^a$ are the parameters of the transformation. Let $\backslash varphi(x)$ be a field that transforms covariantly in a given representation $T(\backslash omega)$. This means that under a transformation we get :$\backslash varphi(x)\; \backslash to\; \backslash varphi\text{'}(x)\; =\; T(\backslash omega)\backslash varphi(x).$ Since any representation of a compact group is equivalent to a unitary representation, we take :$T(\backslash omega)$ to be a unitary matrix without loss of generality. We assume that the Lagrangian $\backslash mathcal$ depends only on the field $\backslash varphi(x)$ and the derivative $\backslash partial\_\backslash mu\backslash varphi(x)$: :$\backslash mathcal\; =\; \backslash mathcal\backslash big(\backslash varphi(x),\backslash partial\_\backslash mu\backslash varphi(x)\backslash big).$ If the group element $\backslash omega$ is independent of the spacetime coordinates (global symmetry), the derivative of the transformed field is equivalent to the transformation of the field derivatives: :$\backslash partial\_\backslash mu\; T(\backslash omega)\backslash varphi(x)\; =\; T(\backslash omega)\backslash partial\_\backslash mu\backslash varphi(x).$ Thus the field $\backslash varphi$ and its derivative transform in the same way. By the unitarity of the representation, scalar products like $(\backslash varphi,\backslash varphi)$, $(\backslash partial\_\backslash mu\backslash varphi,\backslash partial\_\backslash mu\backslash varphi)$ or $(\backslash varphi,\backslash partial\_\backslash mu\backslash varphi)$ are invariant under global transformation of the non-abelian group. Any Lagrangian constructed out of such scalar products is globally invariant: :$\backslash mathcal\backslash big(\backslash varphi(x),\backslash partial\_\backslash mu\backslash varphi(x)\backslash big)\; =\; \backslash mathcal\backslash big(T(\backslash omega)\backslash varphi,T(\backslash omega)\backslash partial\_\backslash mu\; \backslash varphi\backslash big).$ ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Non-abelian gauge transformation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.