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arXiv:2505.24392 · Nuclear Physics B

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

CHRISTOF WETTERICH

Published Nov 20, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.1016/j.nuclphysb.2025.117204

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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.
A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.
A simple probabilistic cellular automaton is shown to be equivalent to a relativistic fermionic quantum field theory with interactions. Occupation numbers for fermions are classical bits or Ising spins. The automaton acts deterministically on bit configurations. The genuinely probabilistic character of quantum physics is realized by probabilistic initial conditions. In turn, the probabilistic automaton is equivalent to the classical statistical system of a generalized Ising model. For a description of the probabilistic information at any given time quantum concepts as wave functions and non-commuting operators for observables emerge naturally. Quantum mechanics can be understood as a particular case of classical statistics. This offers prospects to realize aspects of quantum computing in the form of probabilistic classical computing. This article is part of the theme issue 'Quantum technologies in particle physics'.
Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.
A classical local cellular automaton can describe an interacting quantum field theory for fermions. We construct a simple classical automaton for a particular version of the Thirring model with imaginary coupling. This interacting fermionic quantum field theory obeys a unitary time evolution and shows all properties of quantum mechanics. Classical cellular automata with probabilistic initial conditions admit a description in the formalism of quantum mechanics. Our model exhibits interesting features as spontaneous symmetry breaking or solitons. The same model can be formulated as a generalized Ising model. This euclidean lattice model can be investigated by standard techniques of statistical physics as Monte Carlo simulations. Our model is an example how quantum mechanics emerges from classical statistics.
An Ising-type classical statistical model is shown to describe quantum fermions. For a suitable time-evolution law for the probability distribution of the Ising-spins our model describes a quantum field theory for Dirac spinors in external electromagnetic fields, corresponding to a mean field approximation to quantum electrodynamics. All quantum features for the motion of an arbitrary number of electrons and positrons, including the characteristic interference effects for two-fermion states, are described by the classical statistical model. For one-particle states in the non-relativistic approximation we derive the Schrödinger equation for a particle in a potential from the time evolution law for the probability distribution of the Ising-spins. Thus all characteristic quantum features, as interference in a double slit experiment, tunneling or discrete energy levels for stationary states, are derived from a classical statistical ensemble. Concerning the particle-wave-duality of quantum mechanics, the discreteness of particles is traced back to the discreteness of occupation numbers or Ising-spins, while the continuity of the wave function reflects the continuity of the probability distribution for the Ising-spins.
Probabilistic cellular automata with deterministic updating are discrete quantum systems. They are characterized by a probability distribution over initial configurations. We employ the quantum formalism for an investigation of random probabilistic cellular automata for which the properties of the deterministic updating are randomly distributed over space and time. We are interested in a possible continuum limit for a very large number of cells or space points. We consider bits with two colors, moving to the left or right on a linear chain. At randomly distributed scattering points they change direction and color. A numerical simulation reveals the typical features of quantum systems. We find particular initial probability distributions which reemerge periodically after a certain number of time steps, as produced by the periodic evolution of energy eigenstates in quantum mechanics. Using a description in terms of wave functions allows us to introduce statistical observables for momentum and energy. They characterize the probabilistic information without taking definite values for a given bit configuration, with a conceptual status similar to temperature in classical statistical thermal equilibrium. Conservation of energy and momentum are essential ingredients for the understanding of the evolution of the stochastic probabilistic automata. This evolution resembles in some aspects a single Dirac fermion in two dimensions with a generalized random potential.