Classical probabilities for Majorana and Weyl spinors

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 qτ(t)qτ(t) for the Ising states τ  . The time dependent probability distribution of a generalized Ising model obtains as pτ(t)=qτ2(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.

► Map of classical statistical Ising model to fermionic quantum field theory. ► Lattice-regularized real Grassmann functional integral for single Weyl spinor. ► Emerging complex structure characteristic for quantum physics. ► A classical statistical ensemble describes a quantum theory.