Phase space trajectories in quantum mechanics

TL;DR

This paper introduces a novel representation of quantum mechanics that decomposes quantum states into classical and quantum parts, revealing how classical dynamics emerge from quantum evolution under specific conditions.

Contribution

It presents a new gauge-based framework that splits quantum states into classical and quantum components, with classical evolution following Hamilton's equations.

Findings

Classical parts of quantum states evolve according to Hamilton's equations.

The representation preserves the splitting under certain Schrödinger evolution conditions.

Provides a gauge transformation approach to connect quantum and classical descriptions.

Abstract

An adapted representation of quantum mechanics sheds new light on the relationship between quantum states and classical states. In this approach the space of quantum states splits into a product of the state space of classical mechanics and a Hilbert space, and expectation values of observables decompose into their classical value plus a quantum correction. The splitting is preserved under time evolution of the Schr\"odinger equation under certain assumptions, and the time evolution of the classical part of a quantum state is governed by Hamilton's equation. The new representation is obtained from the usual Hilbert space representation of quantum mechanics by introducing a gauge degree of freedom in a time-dependent unitary transformation, followed by a non-conventional gauge fixing condition.