Quantum-Classical Dynamical Brackets

TL;DR

This paper develops a consistent framework for hybrid quantum-classical dynamics using a new composition product, ensuring the mathematical properties needed for physical consistency and allowing quantum backreaction in classical equations.

Contribution

It introduces a hybrid composition product that constructs a quantum-classical bracket obeying key algebraic identities, enabling consistent interaction models.

Findings

Hybrid brackets obey Jacobi and Leibniz rules if the composition product is associative.

Pure quantum or classical variables can interact without restrictions from no-go theorems.

Quantum backreaction manifests as quantum-dependent terms in classical equations of motion.

Abstract

We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that product. From this we see that the hybrid bracket will obey the Jacobi identity and the Leibniz rule provided the composition product is associative. This suggests that the set of hybrid variables belonging to an associative subalgebra with the composition product will have consistent quantum-classical dynamics. This restricts the class of allowed quantum-classical interaction Hamiltonians. Furthermore, we show that pure quantum or classical variables can interact in a consistent framework, unaffected by no-go theorems in the literature or the restrictions for hybrid variables. In the proposed scheme, quantum backreaction appears as quantum-dependent terms in the classical equations of motion.