The Feynman Checkerboard is a discrete representation of the Dirac equation in (1+1)-dimensional spacetime. It is named after Richard Feynman who presented this model in his 1965 book on path integrals.[1] The model can be visualised by considering relativistic random walks on a two-dimensional spacetime checkerboard. At each discrete timesteps <math>\epsilon\,</math> the particle of mass <math>m\,</math> moves a distance <math>\epsilon c\,</math> (<math>c\,</math> 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 vise-versa) of the spacetime path is weighted by <math>-i \epsilon mc^2/\hbar</math> (with <math>\hbar\,</math> denoting the reduced Planck's constant), 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.
Extensions
Following the introduction of the Feynman checkerboard model, other discrete models for the quantum mechanical motion of relativistic particles have been introduced. Most of these models remain limited to (1+1)-dimensional spacetime, but introduce modifications such as impoved summation rules[2], or generalised lattices[3]. A fully discrete (3+1)-dimensional model for the Dirac equation remains to be constructed.
References
- ^ Feynman and Hibbs, Quantum Mechanics and Path Integrals, Problem 2-6, pages 34-36
- ^ A. Kull and R.A. Treumann, On the path integral of the relativistic electron
- ^ A. Kull, Quantum mechanical motion of relativistic particles in non-continuous spacetime
