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In quantum field theory, a Ward–Takahashi identity is an identity between correlation functions that follows from the global or gauge symmetries of the theory, and which remains valid after renormalization. The Ward–Takahashi identity of quantum electrodynamics was originally used by John Clive Ward and Yasushi Takahashi to relate the wave function renormalization of the electron to its vertex renormalization factor F1(0), guaranteeing the cancellation of the ultraviolet divergence to all orders of perturbation theory. Later uses include the extension of the proof of Goldstone's theorem to all orders of perturbation theory. The Ward–Takahashi identity is a quantum version of the classical Noether's theorem, and any symmetries in a quantum field theory can lead to an equation of motion for correlation functions. This generalized sense should be distinguished when reading literature, such as Michael Peskin and Daniel Schroeder's textbook, ''An Introduction to Quantum Field Theory'' (see references), from the original sense of the Ward identity. == The Ward–Takahashi identity == The Ward–Takahashi identity applies to correlation functions in momentum space, which do not necessarily have all their external momenta on-shell. Let :: be a QED correlation function involving an external photon with momentum k (where is the polarization vector of the photon and summation over =0,...,3 is implied), ''n'' initial-state electrons with momenta , and ''n'' final-state electrons with momenta . Also define to be the simpler amplitude that is obtained by removing the photon with momentum ''k'' from our original amplitude. Then the Ward–Takahashi identity reads :: :::::::::::::::::: where ''−e'' is the charge of the electron. Note that if has its external electrons on-shell, then the amplitudes on the right-hand side of this identity each have one external particle off-shell, and therefore they do not contribute to S-matrix elements. 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「Ward–Takahashi identity」の詳細全文を読む スポンサード リンク
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