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Feynman Checkers: Number theory methods in quantum theory
Virtualconference
Authors : Ustinov, Alexey (Author of the conference) ; Skopenkov, Mikhail (Author of the conference)
CIRM (Publisher )
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
Additional resources :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/presentation-checkers-cirm.pdf
CIRM (Publisher )
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Abstract :
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
Keywords : Feynman checkerboard; quantum walk; Ising model; Young diagram; Dirac equation; stationary phase method
MSC Codes : 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
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
Additional resources :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/presentation-checkers-cirm.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 30/11/2020 Conference Date : 30/11/2020 Subseries : Research talks arXiv category : Combinatorics ; Number Theory ; Mathematical Physics Mathematical Area(s) : Combinatorics ; Number Theory ; Mathematical Physics Format : MP4 (.mp4) - HD Video Time : 00:46:33 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-11-17_Skopenkov_Ustinov.mp4 
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Information on the Event
Event Title : Jean-Morlet Chair 2020 - Workshop: Discrepancy Theory and Applications - Part 1 / Chaire Jean-Morlet 2020 - Workshop : Théorie de la discrépance et applications - Part 1Event Organizers : Madritsch, Manfred ; Rivat, Joël ; Tichy, Robert Dates : 30/11/2020 - 01/12/2020 Event Year : 2020 Event URL : 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 |
See Also
- [Virtualconference] Greedy energy minimization and the van der Corput sequence (part 3) / Author of the conference Pausinger, Florian.
- [Virtualconference] Discrepancy of stratified samples from partitions of the unit cube (part 2) / Author of the conference Pausinger, Florian.
- [Virtualconference] Billiard orbits and geodesics in non-integrable flat dynamical systems (part 2) / Author of the conference Chen, William.
- [Virtualconference] Pick's theorem and Riemann sums: a Fourier analytic tale / Author of the conference Travaglini, Giancarlo.
- [Virtualconference] On two different topics in discrepancy theory: From the discrepancy of stratified samples to greedy energy minimization (part 1) / Author of the conference Pausinger, Florian.
- [Virtualconference] Billiard orbits and geodesics in non-integrable flat dynamical systems (part 1) / Author of the conference Chen, William.
- [Virtualconference] On the interplay between uniform distribution, discrepancy, and energy minimization / Author of the conference Bilyk, Dmitriy.
- [Virtualconference] Local and global statistics for point sequences / Author of the conference Aistleitner, Christoph.
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
