[Paper Review] Quantum mechanics for non-inertial observers

Marko Toroš, André Großardt+1 more|arXiv (Cornell University)|Jan 16, 2017

Quantum Mechanics and Applications3 citations

TL;DR

This paper rigorously derives the center-of-mass dynamics of quantum systems in gravitational fields by resolving ambiguities in relativistic center-of-mass coordinates. It demonstrates that the alleged gravitational decoherence effect predicted by Pikovski et al. is not universal but results from relative acceleration between the particle and observer, with external forces fully canceling the coupling in Earth-bound labs.

ABSTRACT

A recent analysis by Pikovski et al. [Nat. Phys. 11, 668 (2015)] has triggered interest in the question of how to include relativistic corrections in the quantum dynamics governing many-particle systems in a gravitational field. Here we show how the center-of-mass motion of a quantum system subject to gravity can be derived more rigorously, addressing the ambiguous definition of relativistic center-of-mass coordinates. We further demonstrate that, contrary to the prediction by Pikovski et al., external forces play a crucial role in the relativistic coupling of internal and external degrees of freedom, resulting in a complete cancellation of the alleged coupling in Earth-bound laboratories for systems supported against gravity by an external force. We conclude that the proposed decoherence effect is an effect of relative acceleration between quantum system and measurement device, rather than a universal effect in gravitational fields.

Motivation & Objective

• To resolve the ambiguity in defining relativistic center-of-mass coordinates that satisfy canonical commutation relations and frame independence.
• To investigate whether internal and center-of-mass degrees of freedom couple in gravitational fields as suggested by Pikovski et al. (2015).
• To determine the role of external forces in relativistic quantum systems constrained in gravitational fields, particularly in Earth-bound laboratories.
• To clarify whether the proposed gravitational decoherence effect is a universal feature of gravity or an artifact of the experimental setup.

Proposed method

• Generalizes the Krajcik-Foldy perturbative method for canonical center-of-mass variables to accelerated observers and homogeneous gravitational fields.
• Applies the equivalence principle to map inertial and accelerated frames, enabling consistent dynamics in non-inertial reference frames.
• Derives the center-of-mass Hamiltonian up to $1/c^2$ order, including relativistic corrections to kinetic and potential energy.
• Introduces a position- and momentum-dependent external potential $\hat{U}_{\text{ext}} = \hat{U}^{(0)}_{\text{ext}} + \frac{1}{c^2}\hat{U}^{(1)}_{\text{ext}}$ to model realistic interactions (e.g., electromagnetic support).
• Imposes the no-acceleration condition on the center-of-mass motion to derive constraints on $\hat{U}^{(1)}_{\text{ext}}$, ensuring vanishing effective acceleration.
• Uses commutator algebra to solve for $\hat{U}^{(1)}_{\text{ext}}$ that cancels the relativistic coupling term, yielding $\hat{U}^{(1)}_{\text{ext}} = -H^{(0)}_{\text{rel}}gX - \frac{g}{4M}\{X,\mathbf{P}^2\}$.

Experimental results

Research questions

• RQ1 Can a consistent relativistic center-of-mass coordinate system be defined for quantum many-body systems in gravitational fields?
• RQ2 Does the internal dynamics of a quantum system couple to its center-of-mass motion in a gravitational field as predicted by Pikovski et al.?
• RQ3 What is the role of external forces (e.g., electromagnetic support) in relativistic quantum systems at rest in Earth’s gravitational field?
• RQ4 Is the proposed gravitational decoherence effect universal, or does it depend on the relative acceleration between the system and observer?
• RQ5 How do relativistic corrections to the Schrödinger equation affect the coupling between internal and center-of-mass degrees of freedom?

Key findings

• The relativistic center-of-mass coordinates can be consistently defined using a perturbative extension of the Krajcik-Foldy method to accelerated frames.
• The coupling between center-of-mass and internal degrees of freedom predicted by Pikovski et al. arises only when there is relative acceleration between the particle and observer.
• In Earth-bound laboratories, where the particle is supported by an external force (e.g., electromagnetic), the relativistic coupling is exactly canceled by the external potential.
• The external potential $\hat{U}^{(1)}_{\text{ext}}$ must include terms linear in $X$ and quadratic in $\mathbf{P}$, specifically $-H^{(0)}_{\text{rel}}gX - \frac{g}{4M}\{X,\mathbf{P}^2\}$, to satisfy the no-acceleration condition.
• The decoherence effect is not a universal feature of gravity but an artifact of the experimental setup involving relative acceleration between system and measurement device.
• The analysis confirms that the alleged universal decoherence in gravitational fields is not physical in stationary laboratory conditions, as the effect vanishes when both system and observer are accelerated.