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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 Fujikawa's method ： ウィキペディア英語版 Fujikawa method Fujikawa's method is a way of deriving the chiral anomaly in quantum field theory. Suppose given a Dirac field ψ which transforms according to a ρ representation of the compact Lie group ''G''; and we have a background connection form of taking values in the Lie algebra $\backslash mathfrak\backslash ,.$ The Dirac operator (in Feynman slash notation) is :$D\backslash !\backslash !\backslash !\backslash !/\backslash \; \backslash stackrel\backslash \; \backslash partial\backslash !\backslash !\backslash !/\; +\; i\; A\backslash !\backslash !\backslash !/$ and the fermionic action is given by :$\backslash int\; d^dx\backslash ,\; \backslash overlineiD\backslash !\backslash !\backslash !\backslash !/\; \backslash psi$ The partition function is :$Z()=\backslash int\; \backslash mathcal\backslash overline\backslash mathcal\backslash psi\; e^.$ The axial symmetry transformation goes as :$\backslash psi\backslash to\; e^\backslash psi\backslash ,$ :$\backslash overline\backslash to\; \backslash overlinee^$ :$S\backslash to\; S\; +\; \backslash int\; d^dx\; \backslash ,\backslash alpha(x)\backslash partial\_\backslash mu\backslash left(\backslash overline\backslash gamma^\backslash mu\backslash gamma^5\backslash psi\backslash right)$ Classically, this implies that the chiral current, $j\_^\backslash mu\; \backslash equiv\; \backslash overline\backslash gamma^\backslash mu\backslash gamma^5\backslash psi$ is conserved, $0\; =\; \backslash partial\_\backslash mu\; j\_^\backslash mu$. Quantum mechanically, the chiral current is not conserved: Jackiw discovered this due to the non-vanishing of a triangle diagram. Fujikawa reinterpreted this as a change in the partition function measure under a chiral transformation. To calculate a change in the measure under a chiral transformation, first consider the dirac fermions in a basis of eigenvectors of the Dirac operator: :$\backslash psi\; =\; \backslash sum\backslash limits\_\backslash psi\_ia^i,$ :$\backslash overline\backslash psi\; =\; \backslash sum\backslash limits\_\backslash psi\_ib^i,$ where $\backslash$ are Grassmann valued coefficients, and $\backslash$ are eigenvectors of the Dirac operator: :$D\backslash !\backslash !\backslash !\backslash !/\; \backslash psi\_i\; =\; -\backslash lambda\_i\backslash psi\_i.$ The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space, :$\backslash delta\_i^j\; =\; \backslash int\backslash frac\backslash psi^(x)\backslash psi\_i(x).$ The measure of the path integral is then defined to be: :$\backslash mathcal\backslash psi\backslash mathcal\backslash overline\; =\; \backslash prod\backslash limits\_i\; da^idb^i$ Under an infinitesimal chiral transformation, write :$\backslash psi\; \backslash to\; \backslash psi^\backslash prime\; =\; (1+i\backslash alpha\backslash gamma\_)\backslash psi\; =\; \backslash sum\backslash limits\_i\; \backslash psi\_ia^,$ :$\backslash overline\backslash psi\; \backslash to\; \backslash overline^\backslash prime\; =\; \backslash overline(1+i\backslash alpha\backslash gamma\_)\; =\; \backslash sum\backslash limits\_i\; \backslash psi\_ib^.$ The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors :$C^i\_j\; \backslash equiv\; \backslash left(\backslash frac\backslash right)^i\_j\; =\; \backslash int\; d^dx\; \backslash ,\backslash psi^(x)()\backslash psi\_j(x)\; =\; \backslash delta^i\_j\backslash ,\; -\; i\backslash int\; d^dx\; \backslash ,\backslash alpha(x)\backslash psi^(x)\backslash gamma\_\backslash psi\_j(x).$ The transformation of the coefficients $\backslash$ are calculated in the same manner. Finally, the quantum measure changes as :$\backslash mathcal\backslash psi\backslash mathcal\backslash overline\; =\; \backslash prod\backslash limits\_i\; da^i\; db^i\; =\; \backslash prod\backslash limits\_i\; da^db^^(C^i\_j),$ where the Jacobian is the reciprocal of the determinant because the integration variables are Grassmannian, and the 2 appears because the a's and b's contribute equally. We can calculate the determinant by standard techniques: : to first order in α(x). Specialising to the case where α is a constant, the Jacobian must be regularised because the integral is ill-defined as written. Fujikawa employed heat-kernel regularization, such that : &= 2i\lim\limits_\alpha\int d^dx\, \psi^(x)\gamma_ e^e^\psi_i(k) :$=\; -\backslash frac)!\}(\backslash tfracF)^,$ after applying the completeness relation for the eigenvectors, performing the trace over γ-matrices, and taking the limit in M. The result is expressed in terms of the field strength 2-form, $F\; \backslash equiv\; F\_\backslash ,dx^\backslash mu\backslash wedge\; dx^\backslash nu\backslash ,.$ This result is equivalent to $(\backslash tfrac)^$ Chern class of the $\backslash mathfrak$-bundle over the d-dimensional base space, and gives the chiral anomaly, responsible for the non-conservation of the chiral current. 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「Fujikawa 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.