Quantum Mechanics Relative to a Quantum Reference System: a Relative State Approach

TL;DR

This paper develops an intrinsic, background-independent quantum framework in which a quantum system evolves relative to a quantum clock, using entangled relative states and a non-trivial fiber-bundle geometry. The dynamics are encoded in a Ricci-flat Kahler-Einstein equation that relates the intrinsic curvature of the system’s subspace to the extrinsic curvature of the clock’s subspace, yielding a covariant, Hamiltonian-free description whose linearized limit recovers the Schrödinger equation. The formalism introduces non-inertial, clock-induced effects through a covariant derivative and Berry-type phases, predicting inertial forces and potential gravity-like phenomena arising from quantum reference frames. Compared to prior relational approaches, this work emphasizes curved-base-space projections and internal geometry, offering a conceptually unified route toward quantum gravity within a relativistic, relational quantum mechanics framework.

Abstract

This paper proposes an intrinsic or background-independent quantum framework based on entangled state rather than absolute quantum state, it describes a quantum relative state between the under-study quantum system and the quantum measuring apparatus as a quantum reference system, without relying on any external absolute parameter. The paper focuses on a simple example, in which a quantum object's one-dimensional position as an under-study quantum system, and a quantum clock as a quantum reference system or quantum measuring apparatus. The evolution equation of the state of the quantum object's position with respect to the state of the quantum clock is given coming from the Ricci-flat Kaehler-Einstein equation. In a linear and non-relativistic approximation, the framework recovers the equation of the standard quantum mechanics, in which an intrinsic potential related to some "inertial force" is automatically incorporated in the covariant derivative. A physical relative probability interpretation and a geometric non-trivial fiber bundle interpretation of the entangled state in this intrinsic quantum framework are given. Furthermore, some non-inertial effects, such as the "inertial force", coming from the general covariance of the intrinsic quantum framework are also discussed. Compared with the functional integral approach which is more easily to generalize the quantum clock to the quantum spacetime reference frame and study quantum gravity, the relative state approach as a canonical description is more suitable for conceptually demonstrating the connections to the standard formalism and interpretation of the quantum mechanics.