Probabilistic time

TL;DR

This paper presents a framework where classical probabilistic ensembles can exhibit quantum-like time evolution, establishing a connection between classical statistics and quantum field theories for fermions.

Contribution

It introduces a method to derive quantum dynamics and operators from classical probability distributions, linking classical statistics with quantum physics.

Findings

All-time probability distributions can produce unitary time evolution.

Classical observables are mapped to quantum operators with non-commutative products.

Quantum field theories for fermions naturally emerge from the classical probabilistic framework.

Abstract

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.