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The Feynman checkerboard or relativistic chessboard model was Richard Feynman’s sum-over-paths formulation of the kernel for a free spin ½ particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equation in (1+1)-dimensional spacetime as discrete sums. The model can be visualised by considering relativistic random walks on a two-dimensional spacetime checkerboard. At each discrete timestep the particle of mass moves a distance ( being the speed of light) to the left or right. For such a discrete motion the Feynman path integral reduces to a sum over the possible paths. Feynman demonstrated that if each 'turn' (change of moving from left to right or vice versa) of the spacetime path is weighted by (with denoting the reduced Planck's constant), in the limit of vanishing checkerboard squares the sum of all weighted paths yields a propagator that satisfies the one-dimensional Dirac equation. As a result, helicity (the one-dimensional equivalent of spin) is obtained from a simple cellular-automata type rule. The Checkerboard model is important because it connects aspects of spin and chirality with propagation in spacetime〔 Silvan S. Schweber, ''QED and the men who made it'' , Princeton University Press, 1994 〕 and is the only sum-over-path formulation in which quantum phase is discrete at the level of the paths, taking only values corresponding to the 4th roots of unity. ==History== Feynman invented the model in the 1940s while developing his spacetime approach to quantum mechanics.〔 R. P. Feynman, ''Space-time approach to non-relativistic quantum mechanics'', Rev. Mod. Physics, 20, 1948 〕 He did not publish the result until it appeared in a text on path-integrals coauthored by Albert Hibbs in the mid 1960s.〔 Feynman and Hibbs, ''Quantum Mechanics and Path Integrals'', New York: McGraw-Hill, Problem 2-6, pp. 34-36, 1965 〕 The model was not included with the original path-integral paper〔 because a suitable generalization to a four dimensional spacetime had not been found.〔 R. P. Feynman, ''(The Development of the Space-Time View of Quantum Electrodynamics )'', Science, 153, pp. 699-708, 1966 (Reprint of the Nobel Prize lecture) 〕 One of the first connections between the amplitudes prescribed by Feynman for the Dirac particle in 1+1 dimensions, and the standard interpretation of amplitudes in terms of the Kernel or propagator, was established by Jayant Narlikar in a detailed analysis.〔 J. Narlikar, ''Path Amplitudes for Dirac particles'', Journal of the Indian Mathematical Society, 36, pp. 9-32, 1972 〕 The name 'Feynman Chessboard Model' was coined by Gersch when he demonstrated its relationship to the one-dimensional Ising model.〔 H. A. Gersch, ''(Feynman's Relativistic Chessboard as an Ising Model )'', 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「Feynman checkerboard」の詳細全文を読む スポンサード リンク
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