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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 Källén–Lehmann spectral representation ： ウィキペディア英語版 Källén–Lehmann spectral representation The Källén–Lehmann spectral representation gives a general expression for the two-point function of an interacting quantum field theory as a sum of free propagators. It was discovered by Gunnar Källén and Harry Lehmann independently. This can be written as :$\backslash Delta(p)=\backslash int\_0^\backslash infty\; d\backslash mu^2\backslash rho(\backslash mu^2)\backslash frac$ being $\backslash rho(\backslash mu^2)$ the spectral density function that should be positive definite. In a gauge theory, this latter condition cannot be granted but nevertheless a spectral representation can be provided. This belongs to non-perturbative techniques of quantum field theory. ==Mathematical derivation== In order to derive a spectral representation for the propagator of a field $\backslash Phi(x)$, one consider a complete set of states $\backslash$ so that, for the two-point function one can write :$\backslash langle\; 0|\backslash Phi(x)\backslash Phi^\backslash dagger(y)|0\backslash rangle=\backslash sum\_n\backslash langle\; 0|\backslash Phi(x)|n\backslash rangle\backslash langle\; n|\backslash Phi^\backslash dagger(y)|0\backslash rangle.$ We can now use Poincaré invariance of the vacuum to write down :$\backslash langle\; 0|\backslash Phi(x)\backslash Phi^\backslash dagger(y)|0\backslash rangle=\backslash sum\_n\; e^|\backslash langle\; 0|\backslash Phi(0)|n\backslash rangle|^2.$ Let us introduce the spectral density function :$\backslash rho(p^2)\backslash theta(p\_0)(2\backslash pi)^=\backslash sum\_n\backslash delta^4(p-p\_n)|\backslash langle\; 0|\backslash Phi(0)|n\backslash rangle|^2$. We have used the fact that our two-point function, being a function of $p\_\backslash mu$, can only depend on $p^2$. Besides, all the intermediate states have $p^2\backslash ge\; 0$ and $p\_0>0$. It is immediate to realize that the spectral density function is real and positive. So, one can write :$\backslash langle\; 0|\backslash Phi(x)\backslash Phi^\backslash dagger(y)|0\backslash rangle\; =\; \backslash int\backslash frac\backslash int\_0^\backslash infty\; d\backslash mu^2e^\backslash rho(\backslash mu^2)\backslash theta(p\_0)\backslash delta(p^2-\backslash mu^2)$ and we freely interchange the integration, this should be done carefully from a mathematical standpoint but here we ignore this, and write this expression as :$\backslash langle\; 0|\backslash Phi(x)\backslash Phi^\backslash dagger(y)|0\backslash rangle\; =\; \backslash int\_0^\backslash infty\; d\backslash mu^2\backslash rho(\backslash mu^2)\backslash Delta\text{'}(x-y;\backslash mu^2)$ being :$\backslash Delta\text{'}(x-y;\backslash mu^2)=\backslash int\backslash frace^\backslash theta(p\_0)\backslash delta(p^2-\backslash mu^2)$. From CPT theorem we also know that holds an identical expression for $\backslash langle\; 0|\backslash Phi^\backslash dagger(x)\backslash Phi(y)|0\backslash rangle$ and so we arrive at the expression for the chronologically ordered product of fields :$\backslash langle\; 0|T\backslash Phi(x)\backslash Phi^\backslash dagger(y)|0\backslash rangle\; =\; \backslash int\_0^\backslash infty\; d\backslash mu^2\backslash rho(\backslash mu^2)\backslash Delta(x-y;\backslash mu^2)$ being now :$\backslash Delta(p;\backslash mu^2)=\backslash frac$ a free particle propagator. Now, as we have the exact propagator given by the chronologically ordered two-point function, we have obtained the spectral decomposition. 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「Källén–Lehmann spectral representation」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.