Fermions from classical statistics

TL;DR

This paper demonstrates how fermionic quantum systems can be represented and evolved within a classical statistical framework, linking classical probabilities to quantum states and field theories for fermions.

Contribution

It introduces a novel mapping of fermionic quantum states and dynamics to classical statistical ensembles and Grassmann integrals, bridging classical and quantum descriptions.

Findings

Classical probabilities can encode fermionic quantum states.

Unitary quantum evolution corresponds to specific rotations of classical probabilities.

Quantum field theories for fermions can emerge from classical Ising models.

Abstract

We describe fermions in terms of a classical statistical ensemble. The states $\tau$ of this ensemble are characterized by a sequence of values one or zero or a corresponding set of two-level observables. Every classical probability distribution can be associated to a quantum state for fermions. If the time evolution of the classical probabilities $p_\tau$ amounts to a rotation of the wave function $q_\tau(t)=\pm \sqrt{p_\tau(t)}$, we infer the unitary time evolution of a quantum system of fermions according to a Schr\"odinger equation. We establish how such classical statistical ensembles can be mapped to Grassmann functional integrals. Quantum field theories for fermions arise for a suitable time evolution of classical probabilities for generalized Ising models.