Quantum Mechanics from Ergodic Average of Microstates

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 $\tau$.