Effective Theories of Coupled Classical and Quantum Variables

TL;DR

This paper explores two methods for modeling interactions between classical and quantum variables, using measurement theory and decoherent histories, providing a framework for understanding their coupling in quantum-classical systems.

Contribution

It introduces two novel approaches for coupling classical and quantum variables, one based on continuous measurement theory and the other on decoherent histories, advancing the theoretical understanding of quantum-classical interactions.

Findings

The measurement-based approach yields sensible results even with superpositions.

The effective theory derived from decoherent histories clarifies classical-quantum coupling.

Both methods provide consistent frameworks for quantum-classical dynamics.

Abstract

We address the issue of coupling variables which are essentially classical to variables that are quantum. Two approaches are discussed. In the first (based on collaborative work with L.Di\'osi), continuous quantum measurement theory is used to construct a phenomenological description of the interaction of a quasiclassical variable $X$ with a quantum variable $x$, where the quasiclassical nature of $X$ is assumed to have come about as a result of decoherence. The state of the quantum subsystem evolves according to the stochastic non-linear Schr\"odinger equation of a continuously measured system, and the classical system couples to a stochastic c-number $\x (t)$ representing the imprecisely measured value of $x$. The theory gives intuitively sensible results even when the quantum system starts out in a superposition of well-separated localized states. The second approach involves a derivation of an effective theory from the underlying quantum theory of the combined quasiclassical--quantum system, and uses the decoherent histories approach to quantum theory.