March 9, 2020

Answered by: Jonathan Gorard

How does the uncertainty principle work in your models?

One particularly exciting feature of the Wolfram model is that its basic structure allows us to prove many deep quantum mechanical results, such as the uncertainty principle, as pure theorems about abstract term rewriting systems.

One begins by noting that a pair of abstract rewrite relations, R1 and R2, are said to “commute” if the state obtained by applying R1 and then R2 is identical to the state obtained by applying R2 and then R1. If a multiway Wolfram model evolution is not confluent, in the sense that there exist bifurcations in the multiway evolution graph that never re-converge, then this immediately implies the existence of non-commutative rewrite relations (since an abstract rewriting system is confluent if and only if it commutes with itself). Since each updating event in the multiway system can be thought of as being the application of an abstract rewrite relation, it follows that there must exist pairs of updating events that do not commute, in the sense that the final hypergraph obtained will depend upon the timelike-ordering of the application of those events.

If we now interpret the multiway system as being the discrete analog of a (complex) projective Hilbert space, with the rewrite relations being linear operators acting on this space, then this statement immediately reduces to the statement of the standard uncertainty principle regarding the timelike-orderings of measurement operations for pairs of non-commuting observables in quantum mechanics.