I was thinking, how a random walk in a 1D chain could be realized in a wolfram model (on a quantum level). I started making something, but it doesn't seem to work as expected.
Suppose, we have a string substitution system with 3 rules: ABA->AAB, ABA->ABA, ABA->BAA (on each step B can move left, right, or not move at all). The starting string is a sequence with one B in a row of As: AAAABAAAA (could be an infinite sequence as well). This model represents a random walk, where B can move to the left, to the right, or remain still.
From here I am starting to struggle with the basic formalism. According to Wolfram, the transition between two states is supposed to be expressed by a path integral. First question:
Is state AAAABAAAA at the time t=0 is the same state as AAAABAAAA at time t=10? In other words, does the "transition amplitude" depend on the time we consider the states at? (in quantum mechanics there is universal time, so all transition amplitudes are functions of time)
I initially thought that an answer to the questions must be "states at different update iterations are different states for the purpose of transition amplitude calculations" (after all, we can enforce each time step to be different by modifying the rules a little bit), but I am not sure anymore, because if it is true, the complex phases of all paths will be the same. The last statement follows from the length of all path leading from one point to another being the same, and Lagrangian density at each node of the causal graph also being the same (so all paths, leading from one point to the other have the same action). If phases for all paths are the same, the random walk will be classical, and not quantum.
Can someone help me to clarify the situation? Was my example wrong? What would be the right way to do a quantum random walk?