[Paper Review] Weak Equivalence Principle and Propagation of the Wave Function in Quantum Mechanics

Clovis Jacinto de Matos|arXiv (Cornell University)|Jun 14, 2010

Quantum Mechanics and Applications4 references3 citations

TL;DR

This paper argues that the phase velocity of a quantum particle's wave function is unphysical only if the weak equivalence principle (WEP) holds; however, in quantum condensates like superconductors and superfluids, where phase and group velocities become equal, WEP may be violated, implying a physical phase velocity. The Machian interpretation of rest mass energy links inertial and gravitational mass, showing that WEP violation would allow phase velocity to be physical for non-photon particles.

ABSTRACT

The propagation of the wave function of a particle is characterised by a group and a phase velocity. The group velocity is associated with the particle's classical velocity, which is always smaller than the speed of light, and the phase velocity is associated with the propagation speed of the wave function phase and is treated as being unphysical, since its value is always greater than the speed of light. Here we show, using Sciama's Machian formulation of rest mass energy, that this physical interpretation, for the group and the phase velocity of the wave function, is only valid if the weak equivalence principle strictly holds for the propagating particle, except for the photon. In case this constraint is released the phase velocity of the wave function could acquire a physical meaning in quantum condensates.

Motivation & Objective

• To investigate the physical interpretation of wave function phase velocity in quantum mechanics under the assumption of the weak equivalence principle (WEP).
• To explore the consequences of relaxing WEP for the phase velocity of the wave function in quantum systems.
• To examine whether quantum condensates like superconductors and superfluids could exhibit a physical phase velocity due to potential WEP violation.
• To connect the Machian interpretation of rest mass energy with the propagation characteristics of the wave function.

Proposed method

• Using Sciama’s Machian formulation, the paper derives a relation between the gravitational and inertial mass of a particle, linking rest energy to universal gravitational potential.
• Applying de Broglie’s matter wave relations and Planck’s formula to express the wave function’s wavelength and frequency in terms of particle momentum and energy.
• Deriving the relation $ wv = c^2 $ between group and phase velocity under WEP, showing phase velocity exceeds $ c $, hence unphysical.
• Analyzing the condition $ w = c $ to assess when phase velocity could be physical, leading to $ m_{0g}/m_{0i} = v/c $, which only holds for photons under WEP.
• Applying the canonical momentum relation $ \hbar \nabla \varphi = \vec{p} $ to superconductors and superfluids, where $ \vec{v} = \vec{w} $, implying equal group and phase velocities.
• Deriving the Eötvös factor $ \eta = 1 - (v/c)^2 $ as a measure of WEP violation in quantum condensates, based on the ratio $ m_{0g}/m_{0i} = (v/c)^2 $.

Experimental results

Research questions

• RQ1 Under what conditions is the phase velocity of the wave function physically meaningful rather than unphysical due to superluminal propagation?
• RQ2 How does the validity of the weak equivalence principle affect the interpretation of wave function phase velocity in quantum mechanics?
• RQ3 Can quantum condensates such as superconductors and superfluids exhibit a physical phase velocity due to a breakdown of the weak equivalence principle?
• RQ4 What is the quantitative measure of weak equivalence principle violation in superconducting and superfluid systems based on wave function propagation?

Key findings

• The phase velocity of the wave function is unphysical only if the weak equivalence principle holds, as it then exceeds the speed of light, violating relativistic causality.
• For photons, the phase velocity is physical and equal to $ c $, consistent with $ m_{0g}/m_{0i} = 1 $, but this does not extend to massive particles under WEP.
• In superconductors and superfluids, the phase and group velocities of the condensate wave function are equal, implying $ \vec{v} = \vec{w} $, which allows the phase velocity to be physical.
• The ratio $ m_{0g}/m_{0i} = (v/c)^2 $ in quantum condensates implies a strong violation of the weak equivalence principle, with $ v \ll c $.
• The Eötvös factor for WEP violation in these systems is $ \eta = 1 - (v/c)^2 $, indicating a measurable deviation from unity under the assumption of non-unity $ m_{0g}/m_{0i} $.
• The results are consistent with other studies suggesting gauge invariance breaking in superconductors and superfluids, which may underlie the observed WEP violation.