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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 Background field method ： ウィキペディア英語版 Background field method In theoretical physics, background field method is a useful procedure to calculate the effective action of a quantum field theory by expanding a quantum field around a classical "background" value ''B'': : $\backslash phi(x)\; =\; B(x)\; +\; \backslash eta\; (x)$. After this is done, the Green's functions are evaluated as a function of the background. This approach has the advantage that the gauge invariance is manifestly preserved if the approach is applied to gauge theory. ==Method== We typically want to calculate expressions like :$Z()\; =\; \backslash int\; \backslash mathcal\; D\; \backslash phi\; e^$ where ''J''(''x'') is a source, $\backslash mathcal\; L(x)$ is the Lagrangian density of the system, ''d'' is the number of dimensions and φ(''x'') is a field. In the background field method, one starts by splitting this field into a classical background field ''B''(''x'') and a field η(''x'') containing additional quantum fluctuations: :$\backslash phi(x)\; =\; B(x)\; +\; \backslash eta(x)\; \backslash ,.$ Typically, ''B''(''x'') will be a solution of the classical equations of motion :$\backslash left.\backslash frac\backslash right|\_\; =\; 0$ where ''S'' is the action, i.e. the space integral of the Lagrangian density. Fields obeying these equations typically yield the greatest contribution in a path integral, so it is natural to expand around them. Switching on a source ''J''(''x'') will change the equations into δ''S''/δφ|φ = ''B'' + ''J'' = 0. Then the action is expanded around the background ''B''(''x''): :$$ \begin \int d^d x (\mathcal L () + J(x) \phi(x)) & = \int d^d x (\mathcal L () + J(x) B(x)) \\ & + \int d^d x \left(\frac () + J(x)\right) \eta(x) \\ & + \frac12 \int d^d x d^d y \frac () \eta(x) \eta(y) + \cdots \end The second term in this expansion is zero by the equations of motion. The first term does not depend on any fluctuating fields, so that it can be brought out of the path integral. The result is :$Z()\; =\; e^\; \backslash int\; \backslash mathcal\; D\; \backslash eta\; e^\; ()\; \backslash eta(x)\; \backslash eta(y)\; +\; \backslash cdots\}.$ The path integral which now remains is (neglecting the corrections in the dots) of Gaussian form and can be integrated exactly: :$Z()\; =\; C\; e^\; \backslash left(\backslash det\; \backslash frac\; ()\backslash right)^\; +\; \backslash cdots$ where "det" signifies a functional determinant and ''C'' is a constant. The power of minus one half will naturally be plus one for Grassmann fields. The above derivation gives the Gaussian approximation to the functional integral. Corrections to this can be computed, producing a diagrammatic expansion. 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「Background field method」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.