Virtualconference
H Feynman Checkers: Number theory methods in quantum theory
Auteurs : Ustinov, Alexey (Auteur de la Conférence) ; Skopenkov, Mikhail (Auteur de la Conférence)
CIRM (Editeur )
CIRM (Editeur )
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- 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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Résumé :
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
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
Keywords : Feynman checkerboard; quantum walk; Ising model; Young diagram; Dirac equation; stationary phase method
Codes MSC :
05A17
- Partitions of integers (combinatorics)
11L03
- Trigonometric and exponential sums, general
11P82
- Analytic theory of partitions
33C45
- Orthogonal polynomials and functions (Chebyshev, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite, Hahn, etc.)
81P68
- Quantum computation
81T25
- Quantum field theory on lattices
81T40
- Two-dimensional field theories, conformal field theories, etc.
82B20
- Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
68Q12
- Quantum algorithms and complexity
Ressources complémentaires :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/presentation-checkers-cirm.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 30/11/2020 Date de captation : 30/11/2020 Collection : Research talks ; Combinatorics ; Number Theory ; Mathematical Physics Durée : 00:46:33 Domaine : Combinatorics ; Number Theory ; Mathematical Physics Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2020-11-17_Skopenkov_Ustinov.mp4 
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Informations sur la Rencontre Virtuelle
Nom de la rencontre : Jean-Morlet Chair 2020 - Workshop: Discrepancy Theory and Applications - Part 1 / Chaire Jean-Morlet 2020 - Workshop : Théorie de la discrépance et applications - Part 1Organisateurs de la rencontre : Madritsch, Manfred ; Rivat, Joël ; Tichy, Robert Dates : 30/11/2020 - 01/12/2020 Année de la rencontre : 2020 URL Congrès : https://www.cirm-math.com/2257virtual.html
Citation Data
DOI : 10.24350/CIRM.V.19682203Cite this video as: Ustinov, Alexey ;Skopenkov, Mikhail (2020). Feynman Checkers: Number theory methods in quantum theory.CIRM . Audiovisual resource. doi:10.24350/CIRM.V.19682203 URI : http://dx.doi.org/10.24350/CIRM.V.19682203 |
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