The Quantum-like Face of Classical Mechanics

TL;DR

This paper explores a classical mechanics framework resembling quantum mechanics through a 'classical Schr"odinger equation', revealing new insights into quantum-classical relations, measurement, and decoherence.

Contribution

It introduces an operator method for classical mechanics that predicts classical superpositions without interference, offering a novel perspective on quantum-classical boundary and measurement.

Findings

Classical superpositions are possible without interference patterns.

Measurement with a classical apparatus avoids the quantum measurement problem.

The method offers a new basis for studying quantum-classical correspondence.

Abstract

It is first shown that when the Schr\"{o}dinger equation for a wave function is written in the polar form, complete information about the system's {\em quantum-ness} is separated out in a single term $Q$, the so called `quantum potential'. An operator method for classical mechanics described by a `classical Schr\"{o}dinger equation' is then presented, and its similarities and differences with quantum mechanics are pointed out. It is shown how this operator method goes beyond standard classical mechanics in predicting coherent superpositions of classical states but no interference patterns, challenging deeply held notions of classical-ness, quantum-ness and macro realism. It is also shown that measurement of a quantum system with a classical measuring apparatus described by the operator method does not have the measurement problem that is unavoidable when the measuring apparatus is quantum mechanical. The type of decoherence that occurs in such a measurement is contrasted with the conventional decoherence mechanism. The method also provides a more convenient basis to delve deeper into the area of quantum-classical correspondence and information processing than exists at present.