Discrete quantum mechanics

While there are many issues in finding a complete underlying discrete model for quantum phenomena, it is quite straightforward to set up continuous cellular automata whose limiting behavior reproduces the evolution of probability amplitudes in standard quantum mechanics. One starts by assigning a continuous complex number value to each cell. Then given the list of such values the crucial constraint imposed by the standard formalism of quantum mechanics is unitarity: that the quantity Tr[Abs[list]²] representing total probability should be conserved. This is in a sense analogous to conservation of total density in diffusion processes. From the discussion of page 1024 one can reproduce the 1D diffusion equation with a continuous block cellular automaton in which the new value of each block is given by {{1 - ξ, ξ}, {ξ, 1 - ξ}} . {a₁, a₂}. So in the case of quantum mechanics one can consider having each new block be given by {{Cos[θ],  Sin[θ]}, { Sin[θ], Cos[θ]}} .{a₁, a₂}. The pictures below show examples of behavior obtained with this rule. (Gray levels represent magnitude for each cell, and arrows phase.) And it turns out that in suitable limits one generally gets essentially the behavior expected from either the Dirac or Klein–Gordon equations for relativistic particles, or the Schrödinger equation for non-relativistic particles. (Versions of this were noticed by Richard Feynman in the 1940s in connection with his development of path integrals, and were pointed out again several times in the 1980s and 1990s.)

One might hope to be able to get an ordinary cellular automaton with a limited set of possible values by choosing a suitable θ. But in fact in non-trivial cases most of the cells generated at each step end up having distinct values. One can generalize the setup to more dimensions or to allow n × n matrices that are elements of SU(n). Such matrices can be viewed in the context of ordinary quantum formalism as S matrices for elementary evolution events—and can in general represent interactions. (Note that all rules based on matrices are additive, reflecting the usual assumption of linearity at the level of amplitudes in quantum mechanics. Non-additive unitary rules can also be found. The analog of an external potential can be introduced by progressively changing values of certain cells at each step. Despite their basic setup the systems discussed here are not direct analogs of standard quantum spin systems, since these normally have local Hamiltonians and non-local evolution functions, while the systems here have local evolution functions but seem always to require non-local Hamiltonians.)