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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 Schwinger–Dyson equation ： ウィキペディア英語版 Schwinger–Dyson equation The Schwinger–Dyson equations (SDEs), also known as the Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between Green functions in quantum field theories (QFTs). They are also referred to as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion of the corresponding Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs. In his paper "The S-Matrix in Quantum electrodynamics", Dyson derived relations between different S-matrix elements, or more specific "one-particle Green's functions", in quantum electrodynamics, by summing up infinitely many Feynman diagrams, thus working in a perturbative approach. Starting from his variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively, which generalize Dyson's equations to the Schwinger–Dyson equations for the Green functions of quantum field theories. Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields of theoretical physics, such as solid-state physics and elementary particle physics. Schwinger also derived an equation for the two-particle irreducible Green functions,〔 which is nowadays referred to as the inhomogeneous Bethe–Salpeter equation. == Derivation == Given a polynomially bounded functional ''F'' over the field configurations, then, for any state vector (which is a solution of the QFT), $|\backslash psi\backslash rangle$, we have :$\backslash left\backslash langle\backslash psi\backslash left|\backslash mathcal\backslash left\backslash F()\backslash right\backslash \}\backslash right|\backslash psi\backslash right\backslash rangle\; =\; -i\backslash left\backslash langle\backslash psi\backslash left|\backslash mathcal\backslash left\backslash S()\backslash right\backslash \}\backslash right|\backslash psi\backslash right\backslash rangle$ where ''S'' is the action functional and $\backslash mathcal$ is the time ordering operation. Equivalently, in the density state formulation, for any (valid) density state ρ, we have :$\backslash rho\backslash left(\backslash mathcal\backslash left\backslash F()\backslash right\backslash \}\backslash right)\; =\; -i\backslash rho\backslash left(\backslash mathcal\backslash left\backslash S()\backslash right\backslash \}\backslash right).$ This infinite set of equations can be used to solve for the correlation functions nonperturbatively. To make the connection to diagrammatic techniques (like Feynman diagrams) clearer, it's often convenient to split the action S as S()=1/2 D−1ij φi φj+Sint() where the first term is the quadratic part and D−1 is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the deWitt notation whose inverse, D is called the bare propagator and Sint is the "interaction action". Then, we can rewrite the SD equations as :$\backslash langle\backslash psi|\backslash mathcal\backslash |\backslash psi\backslash rangle=\backslash langle\backslash psi|\backslash mathcal\backslash -FS\_D^\backslash \}|\backslash psi\backslash rangle.$ If ''F'' is a functional of φ, then for an operator ''K'', ''F''() is defined to be the operator which substitutes ''K ''for φ. For example, if :$F()=\backslash frac\}\backslash phi(x\_1)\backslash cdots\; \backslash frac\}\backslash phi(x\_n)$ and ''G'' is a functional of ''J'', then : If we have an "analytic" (whatever that means for functionals) functional ''Z'' (called the generating functional) of ''J'' (called the source field) satisfying :$\backslash frac()=i^n\; Z()\; \backslash langle\backslash phi(x\_1)\backslash cdots\; \backslash phi(x\_n)\backslash rangle$, then, from the properties of the functional integrals :$\backslash left(\backslash right)+J(x)\backslash right\backslash rangle\}\_J=0$, the Schwinger–Dyson equation for the generating functional is :$\backslash frac\backslash left(\backslash frac\backslash right)Z()+J(x)Z()=0.$ If we expand this equation as a Taylor series about ''J'' = 0, we get the entire set of Schwinger–Dyson equations. 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「Schwinger–Dyson equation」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.