Classical-quantum systems breaking conservation laws

TL;DR

This work analyzes classical-quantum (CQ) hybrid dynamics with completely positive (CP) evolution as a candidate framework for gravity interacting with quantum matter. It presents a precise CQ formalism in which the joint state $\hat{\rho}(z,t)$ evolves under a GKSL-type master equation with Hamiltonian density $H(z,t)$, Lindblad operators $L_\mu$, and phase-space couplings, emphasizing how backreaction and diffusion are linked by a diffusion–decoherence constraint. A central result is a fully closed, rotationally invariant CQ toy model in which angular momentum is not conserved: even though the underlying equations respect rotational symmetry, the expectation values satisfy $\frac{d}{dt} \langle J_a \rangle = -\kappa \langle J_a \rangle$ with $\langle J_a(\infty) \rangle = 0$, illustrating a fundamental tension between CP dynamics and Noether-type conservation laws. The paper delineates the implications for gravity–matter coupling, showing that CP-based hybrids may violate exact conservation and discussing pathways to reconciliation via limits with vanishing dissipation or embedding into a fully quantum description, with relevance to early-universe physics and quantum gravity phenomenology, all while highlighting core structural limitations of CP CQ hybrids.

Abstract

Whether gravity must be quantized remains one of the biggest open problems in fundamental physics. Classical-quantum hybrid theories have recently attracted attention as a possible framework in which gravity is treated classically yet interacts consistently with quantum matter. Schemes based on completely positive dynamics satisfy most formal consistency requirements and enable a systematic treatment of quantum backreaction, but they also give rise to features that challenge conventional physical intuition, such as the breakdown of conservation laws. To illustrate this issue, we consider a qubit interacting with a classical particle and demonstrate that the corresponding hybrid system violates angular momentum conservation despite rotational symmetry of the underlying equations of motion. This provides an explicit example of a fully closed, rotationally invariant classical-quantum system with completely positive dynamics that violates a conservation law.