# FeynmanCheckerboard

The Feynman checkerboard is a discrete model of the Dirac equation in 1 space dimension and 1 time dimension.

This page will link to different variants of the method.

In 1 space and 1 time dimension the Dirac equation does not express spin but only helicity. So only 2 components of the Dirac spinor are needed.

The essential idea is that **Left and Right handed versions of the fermion are to be counted differently** (two components of the Dirac wave function in the Weyl representation). The particle, represented as a square on a checkerboard, moves diagonally Left or Right (the forward dimension being time towards the future), in the direction of its helicity.

It changes direction according to random chance with a fixed probability {$\epsilon$}. That probability represents the particle's mass; a greater {$\epsilon$} means a greater mass.

As the particle turns, its amplitude gets multiplied by {$ i \epsilon$} and it changes its contribution between {$\psi_{\rm L}$} and {$\psi_{\rm R}$}.

Here as elsewhere I'll take {$z$} to be the spin direction. In the {$R$} helicity, we assume the particle moves in the spin direction, to the Right, and in the {$L$} helicity it moves opposite the spin direction, to the Left.

So here we eliminate what I'll call the "spatial derivatives", that is the {$\partial \psi/\partial x$} and {$\partial \psi / \partial y$} terms, so that the Feynman Checkerboard expresses only:

{$\frac{\partial \psi_{\rm R}}{\partial t} + \frac{\partial \psi_{\rm R}}{\partial z} + i m \psi_{\rm L} = 0 $}

and

{$ \frac{\partial \psi_{\rm L}}{\partial t} - \frac{\partial \psi_{\rm L}}{\partial z} + i m \psi_{\rm R} = 0 $}

The first two terms of each equation are seen in the Feynman Checkerboard to express a kind of "advection" or flow.

Imagine a superposition of Checkerboards with {$R$} checkers at all positions {$z<z_0$}, with equal positive amplitude. At {$z_0$} there is a negative {$\partial \psi_{\rm R} / \partial z$}, and because as the train of checkers marches to the right, the site at {$z=z_0$} will have a positive proportional {$\partial \psi_{\rm R} / \partial t$}, fulfilling the first equation above. Similarly for an infinite train of superposed checkers moving Left for the second equation.

The "spatial derivatives" that are ignored in 1+1 dimensions connect the "flow" in the z direction to the derivatives in the *opposite spin* in the x and y direction.

This has proved not as easy to visualize as the result of a "flow"

The "flow" that randomly changes directions is similar to the Zitterbewegung effect, or "trembling motion" that can be discerned in solutions to the Dirac equation. (I haven't thought much about the fact that, as the Wikipedia entry brings up, that this effect is not clearly seen in quantum field theory but only in the single particle situation described by the Dirac equation.)

Skopenkov and Ustinov (2022) show rigorously under what conditions the Feynman checkerboard reduces to the Dirac equation in 1 space dimension. Surprisingly many treatments have neglected this. On the other hand taking the limit as lattice spacing goes to zero eliminates the hope that a cutoff distance removes ultraviolet divergences. This paper also extends the checkerboard model to the many-particle situation and claims to explore how antiparticles can be represented. *I need to read further in this paper.*

**Variants:**

The stipulation that the amplitude be multiplied by {$i \epsilon$} upon turning can be eliminated because the Dirac equation in 1+1 dimensions can be expressed entirely with Real numbers. A more direct interpretation of {$\psi$} as expressing a current follows if backwards in time moves are allowed as well. This can explain how a current can be negative.

Ord and McKeon conjectured a model with backwards in time moves that also expressed an electromagnetic field. This was through recognition that the Checkerboard allows an "orthogonal sublattice." That is, if the squares are alternating red and black, then diagonal moves keep the Checker on either Red or Black squares. The electromagnetic field acts to move the Checkers from one color to the other, that is, non-diagonally.

In summary, Ord and McKeon had moves going Forward in time both in the positive and negative spatial direction (called {$F_+$} and {$F_-$}), and also Backwards in time moves {$B_+$} and {$B_-$}. The authors require "consistency conditions" relating the fields so that arrows forward in time match those coming back, for example {$F_+(x,t)=B_-(x+1,t+1)$}. Currents are defined from for example {$F_+(x,t)-B_-(x,t)$}, and a 1+1 Dirac equation derived only upon re-definition of the variables.

I've just found this related approach, in which an analogy is made to Brownian motion and the methods of "stochastic control theory" are applied. As with some variants of the Feynman Checkerboard, it allows random diversion of the particle's path both in space and time. **I need to read this!** This paper is the tip of the iceberg of another approach. It appears however *not* to have a discrete spacetime setting.