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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 Feynman's path integral ： ウィキペディア英語版 Path integral formulation The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude. The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 paper.〔; also see 〕 The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier, in the course of his doctoral thesis work by John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point. This formulation has proven crucial to the subsequent development of theoretical physics, because it is manifestly symmetric between time and space. Unlike previous methods, the path-integral allows a physicist to easily change coordinates between very different canonical descriptions of the same quantum system. The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks. For this reason path integrals were used in the study of Brownian motion and diffusion a while before they were introduced in quantum mechanics. == Quantum action principle == In quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time-translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative imaginary unit, −''i''). For states with a definite energy, this is a statement of the De Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle. But the Hamiltonian in classical mechanics is derived from a Lagrangian, which is a more fundamental quantity relative to special relativity. The Hamiltonian tells you how to march forward in time, but the time is different in different reference frames. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics. The Hamiltonian is a function of the position and momentum at one time, and it tells you the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a Legendre transform, and the condition that determines the classical equations of motion (the Euler–Lagrange equations) is that the action is an extremum. In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. So what does the Legendre transform mean? In classical mechanics, with discretization in time, :$\backslash epsilon\; H\; =\; p(t)(q(t+\backslash epsilon)\; -\; q(t))\; -\; \backslash epsilon\; L\; \backslash ,$ and :$p\; =\; \backslash ,$ where the partial derivative with respect to $\backslash dot$ holds ''q''(''t'' + ε) fixed. The inverse Legendre transform is: :$\backslash epsilon\; L\; =\; \backslash epsilon\; p\; \backslash dot\; -\; \backslash epsilon\; H\; \backslash ,$ where :$\backslash dot\; q\; =\; \backslash ,$ and the partial derivative now is with respect to ''p'' at fixed ''q''. In quantum mechanics, the state is a superposition of different states with different values of ''q'', or different values of ''p'', and the quantities ''p'' and ''q'' can be interpreted as noncommuting operators. The operator ''p'' is only definite on states that are indefinite with respect to ''q''. So consider two states separated in time and act with the operator corresponding to the Lagrangian: :$e^\backslash ,$ If the multiplications implicit in this formula are reinterpreted as matrix multiplications, what does this mean? It can be given a meaning as follows: The first factor is :$e^\; \backslash ,$ If this is interpreted as doing a ''matrix'' multiplication, the sum over all states integrates over all ''q''(''t''), and so it takes the Fourier transform in ''q''(''t''), to change basis to ''p''(''t''). That is the action on the Hilbert space – change basis to p at time t. Next comes: :$e^\; \backslash ,$ or evolve an infinitesimal time into the future. Finally, the last factor in this interpretation is :$e^\; \backslash ,$ which means change basis back to q at a later time. This is not very different from just ordinary time evolution: the ''H'' factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just doing Fourier transforms to change to a pure ''q'' basis from an intermediate ''p'' basis. Another way of saying this is that since the Hamiltonian is naturally a function of ''p'' and ''q'', exponentiating this quantity and changing basis from ''p'' to ''q'' at each step allows the matrix element of ''H'' to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to Paul Dirac. , which is just the action function which classical mechanics requires to be stationary for small variations in all the intermediate ''q''s. This shows the way in which equation (11) goes over into classical results when ''h'' becomes extremely small." |source=''Dirac (1932) op. cit., p. 69''}} Dirac further noted that one could square the time-evolution operator in the S representation :$e^\; \backslash ,$ and this gives the time evolution operator between time ''t'' and time ''t'' + 2ε. While in the ''H'' representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the S representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of ''q''(0) and the later one with a fixed value of ''q''(''t''). The result is a sum over paths with a phase which is the quantum action. Crucially, Dirac identified in this paper the deep quantum mechanical reason for the principle of least action controlling the classical limit (see quotation box). 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「Path integral formulation」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.