  The concept of time emerges as an ordering structure in a classical
statistical ensemble. Probability distributions $p_\tau(t)$ at a given time $t$
obtain by integrating out the past and future. We discuss all-time probability
distributions that realize a unitary time evolution as described by rotations
of the real wave function $q_\tau(t)=\pm \sqrt{p_\tau(t)}$. We establish a map
to quantum physics and the Schr\"odinger equation. Suitable classical
observables are mapped to quantum operators. The non-commutativity of the
operator product is traced back to the incomplete statistics of the local-time
subsystem. Our investigation of classical statistics is based on two-level
observables that take the values one or zero. Then the wave functions can be
mapped to elements of a Grassmann algebra. Quantum field theories for fermions
arise naturally from our formulation of probabilistic time.

High Energy Physics - Theory;Statistical Mechanics;General Relativity and Quantum Cosmology;Quantum Physics