An analog of Heisenberg uncertainty relation in prequantum classical field theory

TL;DR

This paper introduces a classical analog of the Heisenberg uncertainty principle within prequantum classical statistical field theory, linking quantum and classical dispersions through a Robertson-like inequality.

Contribution

It derives a classical Robertson-like inequality for dispersions of prequantum fields, offering a new perspective on quantum uncertainties as classical field fluctuations.

Findings

Classical Robertson-like inequality restricts dispersions of prequantum fields.

Quantum dispersion is interpreted as the difference between field and vacuum fluctuations.

The classical inequality is less restrictive than the quantum Heisenberg uncertainty principle.

Abstract

Prequantum classical statistical field theory (PCSFT) is a model which provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system, as averages with respect to fluctuations of classical random fields. PCSFT is a classical model of the wave type. For example, "electron" is described by electronic field. In contrast to QM, this field is a real physical field and not a field of probabilities. An important point is that the prequantum field of e.g. electron contains the irreducible contribution of the background field, vacuum fluctuations. In principle, the traditional QM-formalism can be considered as a special regularization procedure: subtraction of averages with respect to vacuum fluctuations. In this paper we derive a classical analog of the Heisenberg-Robertson inequality for dispersions of functionals of classical (prequantum) fields. PCSFT Robertson-like inequality provides a restriction on the product of classical dispersions. However, this restriction is not so rigid as in QM. The quantum dispersion corresponds to the difference between e.g. the electron field dispersion and the dispersion of vacuum fluctuations. Classical Robertson-like inequality contains these differences. Hence, it does not imply such a rigid estimate from below for dispersions as it was done in QM.