The HyperDiamond Random Walk
To generalize the Quantum random walk to four dimensions, enumerate the paths between two space-time locations, and weigh each path with the quaternion that connects each path change. That’s the most obvious generalization of the two dimensional case where the imaginary unit i connects the 2D checkerboard basis. To get the difference equation of the process ,consider the flow contribution from the four nearest neighbors:
To make this more clear consider only the lattice point one red basis vector away
To get an equation relating the quaternion KB at the point (nR,nG,nB,nY) , multiply each flow
quaternion at the adjacent lattice point by the quaternion that rotates it into KB. We get that quaternion from the rotation relations described previously
:
(1)
The flow equations at the other 3 nearest neighbors are:
(2)
(3)
(4)
The Continuum Limit
Let the lattice spacing, 
be very fine so that
distances in the respective lattice basis directions
(5)
Also denote the continuous versions of the flow vectors with a prime symbol:
(6)
(likewise for the others).
At nearest neighbor lattice point
(7)
Furthermore, to first order in 
:
(8)
In the continuum limit, the difference equation for K`R becomes the following differential equation (As in the 1+1 dimensional case, switch to the forward difference version)
(Similarly, the other three equations for the quantum random walk are:
(10)
(11)
(Equations ( 9)-(12) describe the Quantum Random Walk in four dimensions.
A solution to the QRW equations
Let
(13)
Then a set of solutions for the QRW
equations ( 9)-(12)
(14)
A solution for the sum of the K’s:
Let:
(15)
The function
(16)
(Which is the same equation you get from adding 9-12 together.
Equation (17) resembles the Dirac equation and the solutions resemble Dirac plane waves. Big difference however is that probability density is not conserved. .