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Feynman Checkers: Number theory methods in quantum theory

By Mikhail Skopenkov, Alexey Ustinov

Appears in collections : Combinatorics, Number Theory, Mathematical Physics, Jean-Morlet Chair 2020 - Workshop: Discrepancy Theory and Applications - Part 1 / Chaire Jean-Morlet 2020 - Workshop : Théorie de la discrépance et applications - Part 1

In the 40s R. Feynman invented a simple model of electron motion, which is now known as Feynman's checkers. This model is also known as the one-dimensional quantum walk or the imaginary temperature Ising model. In Feynman's checkers, a checker moves on a checkerboard by simple rules, and the result describes the quantum-mechanical behavior of an electron. We solve mathematically a problem by R. Feynman from 1965, which was to prove that the model reproduces the usual quantum-mechanical free-particle kernel for large time, small average velocity, and small lattice step. We compute the small-lattice-step and the large-time limits, justifying heuristic derivations by J. Narlikar from 1972 and by A.Ambainis et al. from 2001. The main tools are the Fourier transform and the stationary phase method. A more detailed description of the model can be found in Skopenkov M.& Ustinov A. Feynman checkers: towards algorithmic quantum theory. (2020) https://arxiv.org/abs/2007.12879

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Citation data

  • DOI 10.24350/CIRM.V.19682203
  • Cite this video Skopenkov Mikhail (11/30/20). Feynman Checkers: Number theory methods in quantum theory. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19682203
  • URL https://dx.doi.org/10.24350/CIRM.V.19682203

Bibliography

  • Feynman, R. P.; Hibbs, A. R.; Quantum mechanics and path integrals (International Series in Pure and Applied Physics). Maidenhead, Berksh.: McGraw-Hill Publishing Company, Ltd., 365 p. (1965).
  • KEMPE, Julia. Quantum random walks: an introductory overview. Contemporary Physics, 2009, vol. 50, no 1, p. 339-359. - https://doi.org/10.1080/00107151031000110776
  • SKOPENKOV, Mikhail et USTINOV, Alexey. Feynman checkers: towards algorithmic quantum theory. arXiv preprint arXiv:2007.12879, 2020. - https://arxiv.org/abs/2007.12879
  • VENEGAS-ANDRACA, Salvador Elías. Quantum walks: a comprehensive review. Quantum Information Processing, 2012, vol. 11, no 5, p. 1015-1106. - https://doi.org/10.1007/s11128-012-0432-5

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