Putting DM to the Test

Fredkin points out that even if a preferred frame, or underlying lattice, is found, its implications are in one sense only philosophical the integrity of the theory of relativity remains intact, it is only our philosophical perspective that changes. Similarly, if a deterministic RUCA-like rule is the basis of the real physics, it does not mean that we should all throw away our quantum mechanics texts. On the other, if the finite nature hypothesis is correct and a RUCA-like rule exists and can be found, it should in principle be able to supply us with values of all of the fundamental constants of physics. 

Lorentz-Invariance on a Lattice One of the most obvious shortcomings of a CA-based microphysics has to do with the lack of conventional symmetries. A lattice, by definition, has preferred directions and so is structurally anisotropic. How can we hope to generate symmetries where none fundamentally exist A strong hint comes from our discussion of lattice gases in chapter 9, where we saw that symmetries that do not exist on the microscopic lattice level often emerge on the macroscopic dyneimical level. For example, an appropriate set of microscopic LG rules can spawn circular wavefronts on anisotropic lattices. 

Another simple example is the traiditional two-dimensional random-walk on a four-neighbor Euclidean lattice [toff89]. Despite the fact that the underlying lattice is symmetric only with respect to rotations that are multiples of 90 deg, the probability distribution p(s, y) for a particle that begins its random walk at the origin becomes circularly symmetric in the limit as time t — oo p x,y,t) — (see figure 12.12). 

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

Before outlining Toffoli s model of a deterministic relativistic diffusion C A model, we motivate the discussion by recalling a simple formal analogy that holds between a circular rotation by an angle 6 in the x,y) plane and a Lorentz transformation with velocity 

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