Fermions from classical statistics

We describe fermions in terms of a classical statistical ensemble. The states $τ$ 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 $p_τ$ amounts to a rotation of the wave function $q_τ(t)=\pm \sqrt{p_τ(t)}$, we infer the unitary time evolution of a quantum system of fermions according to a Schrödinger 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.

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