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

TL;DR

This work shows how complex wave functions and a linear evolution framework can be embedded in classical probabilistic Ising systems, enabling a quantum-like description of information transport and a lattice-based quantum field theory for fermions in two dimensions. By mapping particle–hole symmetry to complex conjugation and implementing discrete C, P, T (and CPT) transformations, the authors construct probabilistic cellular automata with deterministic updates that reproduce, in the continuum limit, the Dirac/Weyl equations and Feynman propagators for massless fermions, including a particle-physics vacuum and one-particle states. The paper develops a full framework: a complex wave function derived from real PCA data, a fermionic representation with creation/annihilation operators, field operators, density matrices, thermal states with Fermi–Dirac statistics, and a scheme for local interactions that preserves key symmetries. It also connects the discrete model to the complex functional integral, showing how Grassmann path integrals arise from PCA updates. Overall, the results establish PCA as a concrete, unitary, lattice-regularized realization of discrete quantum field theories for fermions, with potential implications for information processing and foundational questions about quantum-classical correspondence.

Abstract

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.