Feynman checkerboard with two paths contributing to the sum for the propagator
from (x / εc, t / ε) = (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.
It can be puzzling as to why a rotation of 720 degrees or two turns is necessary to return to the original state. This comes about because in quantum theory the state of a particle or system is represented by a complex probability amplitude and then when a measurement is made on the system the probability of it coming out some way is given by the square of absolute value of the appropriate amplitude.
Say you send a particle into a system with a detector that can be rotated where the probabilities of it detecting some state are affected by the rotation. When the system is rotated through 360 degrees the observed output and physics are the same as at the start but the amplitudes are changed for a spin ½ particle by a factor of -1 or a phase shift of half of 360 degrees. When the probabilities are calculated the -1 is squared and equals a factor of one so the predicted physics is same as in the starting position. Also in a spin 1/2 particle there are only two spin stated and the amplitudes for both change by the same -1 factor so the interference effects are identical unlike the case for higher spins. The complex probability amplitudes are something of a theoretical construct and cannot be directly observed.
If the probability amplitudes changed by the same amount as the rotation of the equipment then they would have changed by a factor of -1 when the equipment was rotated by 180 degrees which when squared would predict the same output as at the start but this is wrong experimentally. If you rotate the detector 180 degrees the output with spin ½ particles can be different to what it would be if you did not hence the factor of a half is necessary to make the predictions of the theory match reality.
The Checkerboard model is important because it connects aspects of spin and chirality with propagation in spacetime 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.
