Classical probabilities for Majorana and Weyl spinors

TL;DR

This paper establishes a correspondence between quantum field theories of Weyl and Majorana fermions and classical statistical ensembles of Ising spins, enabling classical computation of quantum observables.

Contribution

It introduces a Grassmann functional integral framework that maps fermionic quantum theories onto classical Ising models, including complex structures and propagating fermions.

Findings

Classical probability distributions can replicate fermionic quantum behavior.

Expectation values of fermionic observables are computable in classical Ising models.

The approach employs lattice regularization for fermions.

Abstract

We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function $q_\tau(t)$ for the Ising states $\tau$. The time dependent probability distribution of a generalized Ising model obtains as $p_\tau(t)=q^2_\tau(t)$. The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure characteristic for quantum mechanics. Probability distributions of the Ising model which correspond to one or many propagating fermions are discussed explicitly. Expectation values of observables can be computed equivalently in the classical statistical Ising model or in the quantum field theory for fermions.