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Fermions from classical statistics. (English) Zbl 1206.81117

Summary: 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 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.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
82B05 Classical equilibrium statistical mechanics (general)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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