Born's rule from statistical mechanics of classical fields: from hitting times to quantum probabilities

TL;DR

This paper derives quantum probabilities from classical statistical mechanics of fields, showing that detector interactions and hitting times can approximate Born's rule, thus bridging classical and quantum probability frameworks.

Contribution

It introduces a model where quantum probabilities emerge from classical field interactions and stochastic processes, providing an approximate derivation of Born's rule.

Findings

Quantum probabilities can be approximated by classical stochastic processes.

Detector thresholds and random gain are crucial for reproducing quantum statistics.

The model suggests possible experimental tests to distinguish from conventional Born's rule.

Abstract

We show that quantum probabilities can be derived from statistical mechanics of classical fields. We consider Brownian motion in the space of fields and show that such a random field interacting with threshold type detectors produces clicks at random moments of time. And the corresponding probability distribution can be approximately described by the formalism of quantum mechanics. Hence, probabilities in quantum mechanics and classical statistical mechanics differ not so much as it is typically claimed. The temporal structure of the "prequantum random field" (which is the $L_2$-valued Wiener process) plays the crucial role. Moments of detector's clicks are mathematically described as hitting times which are actively used in classical theory of stochastic processes. Born's rule appears as an approximate rule. In principle, the difference between the "precise detection probability rule" derived in this paper and the conventional Born's rule can be tested experimentally. In our model the presence of the random gain in detectors playes a crucial role. We also stress the role of the detection threshold. It is not merely a technicality, but the fundamental element of the model.