Born’s formula from statistical mechanics of classical fields and theory of hitting times

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. The corresponding probability distribution can be approximately described by the same mathematical formalism as is used in quantum mechanics, theory of Hermitian operators in complex Hilbert space. The temporal structure of the “prequantum random field” which is the L2-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 formula appears as an approximate formula. In principle, the difference between the formula derived in this paper and the conventional Born’s formula can be tested experimentally. In our model the presence of the random gain in detectors plays a crucial role. We also stress the role of the detection threshold which is not merely a technicality, but the fundamental element of the model.