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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 LSZ reduction formula ： ウィキペディア英語版 LSZ reduction formula In quantum field theory, the LSZ reduction formula is a method to calculate S-matrix elements (the scattering amplitudes) from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann. Although the LSZ reduction formula cannot handle bound states, massless particles and topological solitons, it can be generalized to cover bound states, by use of composite fields which are often nonlocal. Furthermore, the method, or variants thereof, have turned out to be also fruitful in other fields of theoretical physics. For example in statistical physics they can be used to get a particularly general formulation of the fluctuation-dissipation theorem. ==In and out fields== S-matrix elements are amplitudes of transitions between ''in'' states and ''out'' states. An ''in'' state $|\backslash \backslash \; \backslash mathrm\backslash rangle$ describes the state of a system of particles which, in a far away past, before interacting, were moving freely with definite momenta and, conversely, an ''out'' state $|\backslash \backslash \; \backslash mathrm\backslash rangle$ describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta ''In'' and ''out'' states are states in Heisenberg picture so they should not be thought to describe particles at a definite time, but rather to describe the system of particles in its entire evolution, so that the S-matrix element: :$S\_=\backslash langle\; \backslash \backslash \; \backslash mathrm|\; \backslash \backslash \; \backslash mathrm\backslash rangle$ is the probability amplitude for a set of particles which were prepared with definite momenta to interact and be measured later as a new set of particles with momenta The easy way to build ''in'' and ''out'' states is to seek appropriate field operators that provide the right creation and annihilation operators. These fields are called respectively ''in'' and ''out'' fields. Just to fix ideas, suppose we deal with a Klein–Gordon field that interacts in some way which doesn't concern us: :$\backslash mathcal\; L=\; \backslash frac\; 1\; 2\; \backslash part\_\backslash mu\; \backslash varphi\backslash part^\backslash mu\; \backslash varphi\; -\; \backslash frac\; 1\; 2\; m\_0^2\; \backslash varphi^2\; +\backslash mathcal\; L\_\}$ may contain a self interaction or interaction with other fields, like a Yukawa interaction $g\backslash \; \backslash varphi\backslash bar\backslash psi\backslash psi$. From this Lagrangian, using Euler–Lagrange equations, the equation of motion follows: :$\backslash left(\backslash part^2+m\_0^2\backslash right)\backslash varphi(x)=j\_0(x)$ where, if $\backslash mathcal\; L\_\}\}$ We may expect the ''in'' field to resemble the asymptotic behaviour of the free field as , making the assumption that in the far away past interaction described by the current is negligible, as particles are far from each other. This hypothesis is named the adiabatic hypothesis. However self interaction never fades away and, besides many other effects, it causes a difference between the Lagrangian mass and the physical mass of the boson. This fact must be taken into account by rewriting the equation of motion as follows: :$\backslash left(\backslash part^2+m^2\backslash right)\backslash varphi(x)=j\_0(x)+\backslash left(m^2-m\_0^2\backslash right)\backslash varphi(x)=j(x)$ This equation can be solved formally using the retarded Green's function of the Klein–Gordon operator $\backslash partial^2+m^2$: :$\backslash Delta\_^3k\}\; \backslash left(e^-e^\backslash right)\_\backslash qquad\; \backslash omega\_k=\backslash sqrt$ allowing us to split interaction from asymptotic behaviour. The solution is: :$\backslash varphi(x)=\backslash sqrt\; Z\; \backslash varphi\_^4y\; \backslash Delta\_\}(x)=0,$ and hence is a free field which describes an incoming unperturbed wave, while the last term of the solution gives the perturbation of the wave due to interaction. The field is indeed the ''in'' field we were seeking, as it describes the asymptotic behaviour of the interacting field as , though this statement will be made more precise later. It is a free scalar field so it can be expanded in flat waves: :$\backslash varphi\_^3k\; \backslash left\backslash )+f^$ *_k(x) a^\dagger_)\right\} where: :$f\_k(x)=\backslash left.\backslash frac\}(2\backslash omega\_k)^\}\}\backslash right|\_$ The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form: :$a\_)=i\backslash int\; \backslash mathrm^3x\; f^$ *_k(x)\overleftrightarrow\partial_0\varphi_}_0 f = \mathrm\partial_0 f -f\partial_0 \mathrm. The Fourier coefficients satisfy the algebra of creation and annihilation operators: :$()=0;\backslash quad\; ()=\backslash delta^3(\backslash mathbf-\backslash mathbf);$ and they can be used to build ''in'' states in the usual way: :$\backslash left|k\_1,\backslash ldots,k\_n\backslash \; \backslash mathrm\backslash right\backslash rangle=\backslash sqrt\}^\backslash dagger(\backslash mathbf\_1)\backslash ldots\; \backslash sqrt\}^\backslash dagger(\backslash mathbf\_n)|0\backslash rangle$ The relation between the interacting field and the ''in'' field is not very simple to use, and the presence of the retarded Green's function tempts us to write something like: :$\backslash varphi(x)\backslash sim\backslash sqrt\; Z\backslash varphi\_\backslash quad\; x^0\backslash to-\backslash infty$ implicitly making the assumption that all interactions become negligible when particles are far away from each other. Yet the current contains also self interactions like those producing the mass shift from to . These interactions do not fade away as particles drift apart, so much care must be used in establishing asymptotic relations between the interacting field and the ''in'' field. The correct prescription, as developed by Lehmann, Symanzik and Zimmermann, requires two normalizable states $|\backslash alpha\backslash rangle$ and $|\backslash beta\backslash rangle$, and a normalizable solution of the Klein–Gordon equation $(\backslash part^2+m^2)f(x)=0$. With these pieces one can state a correct and useful but very weak asymptotic relation: :$\backslash lim\_\; \backslash int\; \backslash mathrm^3x\; \backslash langle\backslash alpha|f(x)\backslash overleftrightarrow\backslash part\_0\backslash varphi(x)|\backslash beta\backslash rangle=\; \backslash sqrt\; Z\; \backslash int\; \backslash mathrm^3x\; \backslash langle\backslash alpha|f(x)\backslash overleftrightarrow\backslash part\_0\backslash varphi\_\}(x)\; +\backslash int\; \backslash mathrm^4y\; \backslash Delta\_\; \backslash int\; \backslash mathrm^3x\; \backslash langle\backslash alpha|f(x)\backslash overleftrightarrow\backslash part\_0\backslash varphi(x)|\backslash beta\backslash rangle=\; \backslash sqrt\; Z\; \backslash int\; \backslash mathrm^3x$ \langle\alpha|f(x)\overleftrightarrow\part_0\varphi_{\mathrm{out}}(x)|\beta\rangle 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「LSZ reduction formula」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. 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