Field theory, curdling, limit cycles, and cellular automata

Abstract

It is suggested that the process of curdling is an important question for the science of fractals. A field equation which displays nucleation (curdling) of particles out of a pure radiation field is discussed. The particle formation arises naturally from the nonlinear character of the equation rather than from imposed quantization conditions. The relativistically invariant equation is

$$div(\rho ^\mu (r,t,\Omega _1 )) = \int {[\rho _\mu (r,t,\Omega ),\rho ^\mu (r,t,\Omega _2 )]d} \Omega _2 $$

where ¦, ¦ denotes commutator.ρ ^μ(r,t,Ω) is both a 4-vector and a 2×2 matrix. It represents substance atr, t traveling with the velocity of light in direction Ω. A unique feature is that the scattering ofρ(Ω ₁) byρ(Ω ₂) as determined by the right-hand side of the above equation results in fields that persist at a given place even thoughρ itself represents substance traveling always at the speed of light. Explicit solutions are given for the case of one dimension. Fields representing particles are obtained and shown to have specially oscillatory structure with incipient fractal character.