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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 Bethe–Salpeter equation ： ウィキペディア英語版 Bethe–Salpeter equation The Bethe–Salpeter equation,〔 〕 named after Hans Bethe and Edwin Salpeter, describes the bound states of a two-body (particles) quantum field theoretical system in a relativistically covariant formalism. The equation was actually first published in 1950 at the end of a paper by Yoichiro Nambu, but without derivation.〔 〕 Due to its generality and its application in many branches of theoretical physics, the Bethe–Salpeter equation appears in many different forms. One form, that is quite often used in high energy physics is :$\backslash Gamma(P,p)\; =\backslash int\backslash !\backslash frac\; \backslash ;\; K(P,p,k)\backslash ,\; S(k-\backslash tfrac)\; \backslash ,\backslash Gamma(P,k)\backslash ,\; S(k+\backslash tfrac)$ where ''Γ'' is the Bethe–Salpeter amplitude, ''K'' the interaction and ''S'' the propagators of the two participating particles. In quantum theory, bound states are objects that live for an infinite time (otherwise they are called resonances), thus the constituents interact infinitely many times. By summing up all possible interactions, that can occur between the two constituents, infinitely many times, the Bethe–Salpeter equation is a tool to calculate properties of bound states and its solution, the Bethe–Salpeter amplitude, is a description of the bound state under consideration. As it can be derived via identifying bound-states with poles in the S-matrix, it can be connected to the quantum theoretical description of scattering processes and Green's functions. The Bethe–Salpeter equation is a general quantum field theoretical tool, thus applications for it can be found in any quantum field theory. Some examples are positronium, bound state of an electron–positron pair, excitons (bound state of an electron–hole pair〔 〕), and meson as quark-antiquark bound-state.〔 〕 Even for simple systems such as the positronium, the equation cannot be solved exactly although the equation can in principle be formulated exactly. Fortunately, a classification of the states can be achieved without the need for an exact solution. If one of the particles is significantly more massive than the other, the problem is considerably simplified as one solves the Dirac equation for the lighter particle under the external potential of the heavier particle. == Derivation == The starting point for the derivation of the Bethe–Salpeter equation is the two-particle (or four point) Dyson equation :$G\; =\; S\_1\backslash ,S\_2\; +\; S\_1\backslash ,S\_2\backslash ,\; K\_\backslash ,\; G$ in momentum space, where "G" is the two-particle Green function $\backslash langle\backslash Omega|\; \backslash phi\_1\; \backslash ,\backslash phi\_2\backslash ,\; \backslash phi\_3\backslash ,\; \backslash phi\_4\; |\backslash Omega\backslash rangle$, "S" are the free propagators and "K" is an interaction kernel, which contains all possible interaction between the two particles. The crucial step is now, to assume that bound states appear as poles in the Green function. One assumes, that two particles come together and form a bound state with mass "M", this bound state propagates freely, and then the bound state splits in its two constituents again. Therefore, one introduces the Bethe–Salpeter wave function $\backslash Psi\; =\; \backslash langle\backslash Omega|\; \backslash phi\_1\; \backslash ,\backslash phi\_2|\backslash psi\backslash rangle$, which is a transition amplitude of two constituents $\backslash phi\_i$ into a bound state $\backslash psi$, and then makes an ansatz for the Green function in the vicinity of the pole as :$G\; \backslash approx\; \backslash frac,$ where ''P'' is the total momentum of the system. One sees, that if for this momentum the equation $P^2\; =\; M^2$ holds, what is exactly the Einstein Einstein energy-momentum relation (with the Four-momentum $P\_\backslash mu\; =\; \backslash left(E/c,\backslash vec\; p\; \backslash right)$ and $P^2\; =\; P\_\backslash mu\backslash ,P^\backslash mu$ ) the four-point Green function contains a pole. If one plugs that ansatz into the Dyson equation above, and sets the total momentum "P" such the energy-momentum relation holds, on both sides of the term a pole appears. :$\backslash frac\; =\; S\_1\backslash ,S\_2\; +S\_1\backslash ,S\_2\backslash ,\; K\_\backslash frac$ Comparing the residues yields :$\backslash Psi=S\_1\backslash ,S\_2\backslash ,\; K\_\backslash Psi,\; \backslash ,$ This is already the Bethe–Salpeter equation, written in terms of the Bethe–Salpeter wave functions. To obtain the above form one introduces the Bethe–Salpeter amplitudes "Γ" :$\backslash Psi\; =\; S\_1\backslash ,S\_2\backslash ,\backslash Gamma$ and gets finally :$\backslash Gamma=\; K\_\backslash ,S\_1\backslash ,S\_2\backslash ,\backslash Gamma$ which is written down above, with the explicit momentum dependence. 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「Bethe–Salpeter 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.