Complex wave functions, CPT and quantum field theory for classical generalized Ising models

The quantum or quantum field theory concept of a complex wave function is useful for understanding the information transport in classical statistical generalized Ising models. We relate complex conjugation to the discrete transformations charge conjugation (C), parity (P) and time reversal (T). A subclass of generalized Ising models are probabilistic cellular automata (PCA) with deterministic updating and probabilistic initial conditions. Simple two-dimensional PCA correspond to discretized quantum field theories for Majorana–Weyl, Weyl or Dirac fermions. Momentum and energy are conserved statistical observables. For PCA describing free massless fermions we investigate the vacuum and field operators for particle excitations. For the correlation function one finds the Lorentz-invariant Feynman propagator of quantum field theory. Furthermore, these automata admit probabilistic boundary conditions that correspond to thermal equilibrium with the quantum Fermi–Dirac distribution. PCA with updating sequences of propagation and interaction steps can realize a rich variety of discrete quantum field theories for fermions with interactions. For information theory the quantum formalism for PCA sheds new light on deterministic computing or signal processing with probabilistic input.