# Feynman checkerboard

Feynman checkerboard with two paths contributing to the sum for the propagator from ($x/\epsilon c$, $t/\epsilon$) = (0, 0) to (3, 7).

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 $\epsilon\,$ the particle of mass $m\,$ moves a distance $\epsilon c\,$ ($c\,$ 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 $-i \epsilon mc^2/\hbar$ (with $\hbar\,$ 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[1] 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.[2] He did not publish the result until it appeared in a text on path-integrals coauthored by Albert Hibbs in the mid 1960s.[3] The model was not included with the original path-integral paper[2] because a suitable generalization to a four dimensional spacetime had not been found.[4]

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.[5] The name 'Feynman Chessboard Model' was coined by Gersch when he demonstrated its relationship to the one-dimensional Ising model.[6] Gaveau et al. discovered a relationship between the model and a stochastic model of the Telegraph equations due to Mark Kac through analytic continuation.[7] Jacobson and Schulman examined the passage from the relativistic to the non-relativistic path integral.[8] Subsequently Ord showed that the Chessboard model was embedded in correlations in Kac’s original stochastic model[9] and so had a purely classical context, free of formal analytic continuation.[10] In the same year, Kauffman and Noyes[11] produced a fully discrete version related to bit-string physics, that has recently been developed into a general approach to discrete physics.[12]

## Extensions

Although Feynman did not live to publish extensions to the Chessboard model, it is evident from his archived notes that he was interested in establishing a link between the 4th roots of unity (used as statistical weights in chessboard paths) and his discovery, with J. A. Wheeler, that antiparticles are equivalent to particles moving backwards in time.[1] His notes contain several sketches of chessboard paths with added spacetime loops.[13] The first extension of the model to explicitly contain such loops was the ‘Spiral Model' in which chessboard paths were allowed to spiral in spacetime. Unlike the Chessboard case, causality had to be implemented explicitly to avoid divergences, however with this restriction the Dirac equation emerged as a continuum limit.[14] Subsequently the roles of Zitterbewegung, antiparticles and the Dirac Sea in the Chessboard model have been elucidated[15] and the implications for the Schrödinger equation considered through the non-relativistic limit.[16]

Further extensions of the original 2-dimensional spacetime model include features such as improved summation rules[17] and generalized lattices.[18] There has been no consensus on an optimal extension of the Chessboard model to a fully four-dimensional space-time. Two distinct classes of extensions exist, those working with a fixed underlying lattice[19][20] and those that embed the two dimensional case in higher dimension.[21][22] The advantage of the former is that the sum-over-paths is closer to the non-relativistic case, however the simple picture of a single directionally independent speed of light is lost. In the latter extensions the fixed speed property is maintained at the expense of variable directions at each step.

## References

1. ^ a b Silvan S. Schweber, QED and the men who made it , Princeton University Press, 1994
2. ^ a b R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Physics, 20, 1948
3. ^ Feynman and Hibbs, Quantum Mechanics and Path Integrals, New York: McGraw-Hill, Problem 2-6, pp. 34-36, 1965
4. ^ 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)
5. ^ J. Narlikar, Path Amplitudes for Dirac particles, Journal of the Indian Mathematical Society, 36, pp. 9-32, 1972
6. ^ H. A. Gersch, Feynman's Relativistic Chessboard as an Ising Model, Int. J. Theor. Physics, 20, pp. 491-501, 1981
7. ^ B. Gaveau and T. Jacobson and M. Kac and L. S. Schulman, Relativistic extension of the analogy between quantum mechanics and Brownian motion, Physical Review Letters, 53, pp. 419-422, 1984
8. ^ T. Jacobson and L. S. Schulman, Quantum Stochastics: the passage from a relativistic to a non-relativistic path integral, J. Phys. A, 17, pp. 375-383, 1984
9. ^ M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain Journal of Mathematics, 4, 497-509, 1974
10. ^ G. N. Ord, The Schroedinger and Dirac Free-Particle Equations without Quantum Mechanics, Annals of Physics, 250, pp. 51-62, 1996
11. ^ L. H. Kauffman and H. P. Noyes, Discrete Physics and the Dirac Equation, Phys. Lett. A, 218, 1996, arXiv:hep-th/9603202
12. ^ Louis H. Kauffman, Non-Commutative Worlds --A Summary, 2005, arXiv:quant-ph/0503198
13. ^ S. S. Schweber, Feynman and the visualization of space-time processes, Rev. Mod. Phys., 58, pp 449-508, 1986
14. ^ G. N. Ord, A Classical Analog of Quantum Phase, Int., J. Theor. Physics, 31, pp. 1177-1195, 1992
15. ^ G. N. Ord, and J. A.Gualtieri, The Feynman Propagator from a Single Path, Phys. Rev. Lett., 89, pp. 250403-250407, 2002 arXiv:quant-ph/0109092
16. ^ G. N. Ord and R. B. Mann, Entwined pairs and Schrödinger's equation, Annals of Physics, 308, pp. 478-492, 2003 arXiv:quant-ph/0206095
17. ^ A. Kull and R.A. Treumann, On the path integral of the relativistic electron, 1999, arXiv:quant-ph/9901058
18. ^ A. Kull, Quantum mechanical motion of relativistic particles in non-continuous spacetime, 2002 arXiv:quant-ph/0212053
19. ^ T. Jacobson, Feynman’s checkerboard and other games, in Non-Linear Equations in Classical and Quantum Field Theory, Lecture Notes in Physics, Springer/Heidelberg, 226, pp. 386-395, 1985
20. ^ Frank D. Smith, HyperDiamond Feynman Checkerboard in 4-dimensional Spacetime, 1995, arXiv:quant-ph/9503015
21. ^ D. G. C. McKeon and G. N. Ord, On the Dirac Equation in 3+1 space, Annals of Physics, 222, pp. 244-253, 1993
22. ^ G. Rosen, Feynman path summation for the Dirac equation: An underlying one-dimensional aspect of relativistic particle motion, Phys. Rev. A, 28, pp1139-1140, 1983