Classical and Quantum Random Walk 

Classical Random Walk

First, start with the classical random walk in one dimension. This has been used for modeling Brownian motion, diffusion process. Here one models the motion of a particle, e.g. a speck of dust floating in viscous fluid water e.g.. The random impacts of the water molecules impart an irregular drifting motion to the particle. The velocities of the water molecules are distributed over a wide range and likewise so are the velocity fluctuations of the particle. In practice one makes the following simplifications in order to analyze the motion. First, restrict the motion to one dimension: the particle is constrained to move along a line. Second, assume the particle moves at only one speed at any time. It moves only to the left or the right at a speed related to the average thermal energy of the fluid. Third, assume the impacts only occur at discrete intervals. So, we have a particle that moves along a line at a speed we call c. At time intervals 0,î,2 î... the particle may or may not change direction with equal probability. Likewise, the initial speed is c and the initial direction is equally likely to be positive or negative. These assumptions seem drastic but thanks to the law of large numbers they provide a good description of the particle motion after many time intervals. To find the probability that particle will at a location a after n steps, count the number of distinct paths N₁(n,a) from the origin n,a=0 to the location in question

                                                                                                    (1)

The total number of paths is simply 2ⁿ so we find that the probability P₁(n,a) that after n steps the particle is at location a is: 

                                                                                              (2)

Difference Equation for the Classical Random Walk

Another solution is by way of the difference equation: Together with boundary/initial conditions, the probability is determined from the difference equation: 

                                                                                              (3)

Differential Equation for the Classical Random Walk

Let the spacing of the grid be very small with values and in the t and x directions, so this grid appears like a continuum, then the probability may treated as a continuous probability density, and we can evaluate points near the origin with help of a Taylor series: 

                                   (4)

Substitute these expressions into the difference equation (3) and ignore the high order terms to get a differential equation for P(x,t):

                                                                                                                               (5)

(Einstein 1905). 

Quantum Random Walk

In 1965? Richard Feynman proposed a random walk process that seems to emulate the dirac equation least for the one dimensional case. :This time each path is weighed by the following rule: count the number of corners , R, in the path, then assign a weight of

to each path. Add these complex numbers together to get complex probability amplitude. The probability of the particle traversing the locations is the square of the amplitude. is assumably a dimensionless constant related to phase but most author ascribe a time to epsilon. I should point out here that this theory remains in a half-baked state-- a few notes by Feynman and a paper here and there. The book Quantum Mechanics and Path Integrals by Feynman and Hibbs briefly mention this theory in problem 2-6. To get the path summation then one enumerates all paths between two space-time points, then classify these paths according to the number of corners.(actually the number modulo 4 is all that's necessary). We have already counted the number of paths--that's the function N₁(n,a), but counting the corners is more complicated. Here is an exact solution of the 1+1 dimensiona quantum random walk based on work of  Bill Hammel  and Louis Kauffman

Difference Equation for the Quantum Random Walk

(Also see the following newsgroup posting Discrete Dirac by Charles Bloom.(or type the words “discrete dirac” into Google groups if the link does not work)

Let K(n,a) be the sum of weighed paths from the origin [0,0] to the space-time point [n,a] Express K(n,a) as the sum of two flows K+(n,a) and  K-(n,a) that originate from the two light-like directions.

The probability of finding the particle at the lattice point [n,a] is 

                                                                                                                 (6)

   Note: if you take the Feynman’s original description literally, the complex amplitude of the process is K(n,a): the sum of two complex valued flows going into the lattice point:

                                                                                                                           (7)

the probability then is. This expression however, will not lead to the Dirac equation.

Because of the weighing rule, the  flows are determined from:

                                                                                        (8)

Transform to the following light cone coordinates:

                                                                                                                          (9)

Then

                                                                                       (10)

Shift the lattice coordinates:

                                                                                              (11)

Again use a first order Taylor expansion for the K’s

                                                                                    (12)

                                                                                                                      (13)

Where  are continuous light cone coordinates:

                                                                                  (14)

The usual form of the one-dimensional Dirac equation is:

                                                                                                   (15)

with

                                                                                            (16)

eq. 13 reduces to eq 15 with:

                                                                              (17)

The transition from the matrices in 16 to those in 17 amount to a similarity transform with transform matrix:

                                                                                                                                                      (18)

Note that the transform W is the same as the transform from Cartesian coordinates to the light cone coordinates (14).Also note that the wave function  transforms as a vector.

Next(to 4 dimensional  generalization)
To: Exact Solution of  2D Quantum Random Walk
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