Feynman checkers: towards algorithmic quantum theory

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Abstract

We survey and develop the most elementary model of electron motion introduced by Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a Feynman problem from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results due to Narlikar from 1972 and to Sunada and Tate from 2012.For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model.We also present a survey of known results on Feynman checkers.Bibliography: 53 titles.

About the authors

Mihail Borisovich Skopenkov

HSE University; Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute); King Abdullah University of Science and Technology

Email: skopenkov@rambler.ru

Alexey Vladimirovich Ustinov

HSE University; Khabarovsk Division of the Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences

Email: ustinov.alexey@gmail.com
Doctor of physico-mathematical sciences, Associate professor

References

  1. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, J. Watrous, “One-dimensional quantum walks”, Proceedings of the thirty-third annual ACM symposium on theory of computing, ACM, New York, 2001, 37–49
  2. C. M. Bender, L. R. Mead, K. A. Milton, “Discrete time quantum mechanics”, Comput. Math. Appl., 28:10-12 (1994), 279–317
  3. C. M. Bender, K. A. Milton, D. H. Sharp, “Gauge invariance and the finite-element solution of the Schwinger model”, Phys. Rev. D (3), 31:2 (1985), 383–388
  4. I. Bialynicki-Birula, “Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata”, Phys. Rev. D (3), 49:12 (1994), 6920–6927
  5. P. Billingsley, Probability and measure, Wiley Ser. Probab. Math. Statist., 3rd ed., John Wiley & Sons, Inc., New York, 1995, xiv+593 pp.
  6. I. Bogdanov, Feynman checkers: the probability of direction reversal, 2020, 8 pp.
  7. M. J. Cantero, F. A. Grünbaum, L. Moral, L. Velazquez, “The CGMV method for quantum walks”, Quantum Inf. Process., 11:5 (2012), 1149–1192
  8. D. Chelkak, S. Smirnov, “Discrete complex analysis on isoradial graphs”, Adv. Math., 228:3 (2011), 1590–1630
  9. Li-Chen Chen, M. E. H. Ismail, “On asymptotics of Jacobi polynomials”, SIAM J. Math. Anal., 22:5 (1991), 1442–1449
  10. М. Дмитриев, Модель {“}шашки Фейнмана{rm”} с поглощением, 2022, 10 с.
  11. Р. Фейнман, КЭД: Странная теория света и вещества, Наука, М., 1988, 144 с.
  12. Р. П. Фейнман, А. Хиббс, Квантовая механика и интегралы по траекториям, Мир, М., 1968, 384 с.
  13. S. R. Finch, Mathematical constants, Encyclopedia Math. Appl., 94, Cambridge Univ. Press, Cambridge, 2003, xx+602 pp.
  14. G. B. Folland, Quantum field theory. A tourist guide for mathematicians, Math. Surveys Monogr., 149, Amer. Math. Soc., Providence, RI, 2008, xii+325 pp.
  15. B. Z. Foster, T. Jacobson, “Spin on a 4D Feynman checkerboard”, Internat. J. Theoret. Phys., 56:1 (2017), 129–144
  16. B. Gaveau, L. S. Schulman, “Dirac equation path integral: interpreting the Grassmann variables”, Nuovo Cimento D (1), 11:1-2 (1989), 31–51
  17. K. Georgopoulos, C. Emary, P. Zuliani, “Comparison of quantum-walk implementations on noisy intermediate-scale quantum computers”, Phys. Rev. A, 103:2 (2021), 022408, 10 pp.
  18. H. A. Gersch, “Feynman's relativistic chessboard as an Ising model”, Internat. J. Theoret. Phys., 20:7 (1981), 491–501
  19. И. С. Градштейн, И. М. Рыжик, Таблицы интегралов, сумм, рядов и произведений, 4-е изд., Физматгиз, М., 1963, 1100 с.
  20. G. Grimmett, S. Janson, P. F. Scudo, “Weak limits for quantum random walks”, Phys. Rev. E, 69:2 (2004), 026119
  21. M. N. Huxley, Area, lattice points, and exponential sums, London Math. Soc. Monogr. (N. S.), 13, The Clarendon Press, Oxford Univ. Press, New York, 1996, xii+494 pp.
  22. T. Jacobson, “Feynman's checkerboard and other games”, Non-linear equations in classical and quantum field theory (Meudon/Paris, 1983/1984), Lecture Notes in Phys., 226, Springer, Berlin, 1985, 386–395
  23. T. Jacobson, L. S. Schulman, “Quantum stochastics: the passage from a relativistic to a non-relativistic path integral”, J. Phys. A, 17:2 (1984), 375–383
  24. P. Jizba, “Feynman checkerboard picture and neutrino oscillations”, J. Phys. Conf. Ser., 626 (2015), 012048, 11 pp.
  25. G. L. Jones, “Complex temperatures and phase transitions”, J. Math. Phys., 7:11 (1966), 2000–2005
  26. А. А. Карацуба, Основы аналитической теории чисел, 2-е изд., Наука, М., 1983, 240 с.
  27. J. Kempe, “Quantum random walks: an introductory overview”, Contemp. Phys., 50:1 (2009), 339–359
  28. R. Kenyon, “The Laplacian and Dirac operators on critical planar graphs”, Invent. Math., 150:2 (2002), 409–439
  29. N. Konno, “A new type of limit theorems for the one-dimensional quantum random walk”, J. Math. Soc. Japan, 57:4 (2005), 1179–1195
  30. N. Konno, “Quantum walks”, Quantum potential theory, Lecture Notes in Math., 1954, Springer, Berlin, 2008, 309–452
  31. N. Konno, “Quantum walks”, Sugaku Expositions, 33:2 (2020), 135–158
  32. A. B. J. Kuijlaars, A. Martinez-Finkelshtein, “Strong asymptotics for Jacobi polynomials with varying nonstandard parameters”, J. Anal. Math., 94 (2004), 195–234
  33. M. Maeda, H. Sasaki, E. Segawa, A. Suzuki, K. Suzuki, “Scattering and inverse scattering for nonlinear quantum walks”, Discrete Contin. Dyn. Syst., 38:7 (2018), 3687–3703
  34. J. Maldacena, “The symmetry and simplicity of the laws of physics and the Higgs boson”, European J. Phys., 37:1 (2016), 015802, 25 pp.
  35. V. Matveev, R. Shrock, “A connection between complex-temperature properties of the 1D and 2D spin $s$ Ising model”, Phys. Lett. A, 204:5-6 (1995), 353–358
  36. J. V. Narlikar, “Path amplitudes for Dirac particles”, J. Indian Math. Soc. (N. S.), 36 (1972), 9–32
  37. I. Novikov, “Feynman checkers: the probability to find an electron vanishes nowhere inside the light cone”, Rev. Math. Phys., 2020 (to appear)
  38. G. N. Ord, “Classical particles and the Dirac equation with an electromagnetic field”, Chaos Solitons Fractals, 8:5, Special issue (1997), 727–741
  39. G. N. Ord, J. A. Gualtieri, “The Feynman propagator from a single path”, Phys. Rev. Lett., 89:25 (2002), 250403
  40. F. Ozhegov, Feynman checkers: external electromagnetic field and asymptotic properties, preprint, 2021
  41. М. Е. Пескин, Д. В. Шрeдер, Введение в квантовую теорию поля, НИЦ “Регулярная и хаотическая динамика”, М.–Ижевск, 2001, 784 с.
  42. S. S. Schweber, “Feynman and the visualization of space-time processes”, Rev. Modern Phys., 58:2 (1986), 449–508
  43. М. Б. Скопенков, А. А. Пахарев, А. В. Устинов, “Сквозь сеть сопротивлений”, Матем. просв., серия 3, 18, МЦНМО, М., 2014, 33–65
  44. M. Skopenkov, A. Ustinov, Feynman checkers: towards algorithmic quantum theory, 2020, 55 pp.
  45. M. Skopenkov, A. Ustinov, Feynman checkers: auxiliary computations,par, Last accessed 31.12.2021
  46. M. Skopenkov, A. Ustinov, Feynman checkers: Minkowskian lattice quantum field theory, preprint, 2022
  47. N. J. A. Sloane (ed.), The on-line encyclopedia of integer sequences
  48. R. P. Stanley, “Irreducible symmetric group characters of rectangular shape”, Sem. Lothar. Combin., 50 (2003/04), Art. B50d, 11 pp.
  49. T. Sunada, T. Tate, “Asymptotic behavior of quantum walks on the line”, J. Funct. Anal., 262:6 (2012), 2608–2645
  50. Г. Сегe, Ортогональные многочлены, Физматлит, М., 1962, 500 с.
  51. S. E. Venegas-Andraca, “Quantum walks: a comprehensive review”, Quantum Inf. Process., 11:5 (2012), 1015–1106
  52. J. Yepez, “Relativistic path integral as a lattice-based quantum algorithm”, Quantum Inf. Process., 4:6 (2005), 471–509
  53. P. Zakorko, Feynman checkers: a uniform approximation of the wave function by Airy function, preprint, 2021

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