Quantum Mechanics from Ergodic Average of Microstates

TL;DR

This paper proposes a new formulation of quantum mechanics as an ergodic average over microstates, reproducing the Born rule and wave-particle duality, and suggests experimental tests at the Compton time scale.

Contribution

It introduces a microstate-based effective theory of quantum mechanics with ergodic properties, linking time averages to quantum expectations and providing a novel perspective on measurement.

Findings

Reproduces the Born rule through microstate jumps.

Shows ergodicity at quantum scales with time averaging.

Suggests experimental tests at the Compton time scale.

Abstract

We formulate quantum mechanics as an effective theory of an underlying structure characterized by microstates $|{\mathcal M}^j(t)\rangle$, each one defined by the quantum state $|\Psi(t)\rangle$ and a complete set of commutative observables $O^j$. At any time $t$, $|{\mathcal M}^j(t)\rangle$ corresponds to a state $|O^j_k\rangle$, for some $k$ depending on $t$, and jumps after time intervals whose duration, of the order of the Compton time $\tau$, is proportional to the probability $|\langle O_k^j|\Psi(t)\rangle|^2$. This reproduces the Born rule and mimics the wave-particle duality. The theory is based on a partition of time whose flow is characterized by quantum probabilities. Ergodicity arises at ordinary quantum scales with the expectation values corresponding to time averaging over a period $\tau$. The measurement of $O^j$ provides a new partition of time and the outcome is the state $|O_k^j\rangle$ to which $|{\mathcal M}^j(t)\rangle$ corresponds at that time. The formulation, that shares some features with the path integral, can be tested by experiments involving time intervals of order $\tau$.