Local Scale Invariance in Quantum Theory: A Non-Hermitian Pilot-Wave Formulation

TL;DR

This paper demonstrates how local scale invariance can be naturally incorporated into quantum theory through a non-Hermitian pilot-wave formulation, modifying the conserved current and ensuring physical predictions remain scale-invariant.

Contribution

It introduces a novel complexification of the electromagnetic coupling to realize Weyl's local scale invariance within pilot-wave quantum theory, applicable to multiple quantum equations and field interactions.

Findings

Modified conserved current density to include local scale factors

Physical predictions remain invariant under local scale transformations

Trajectories are shown to be unique in the pilot-wave framework

Abstract

We show that Weyl's abandoned idea of local scale invariance has a natural realization at the quantum level in pilot-wave (de Broglie-Bohm) theory. We obtain the Weyl covariant derivative by complexifying the electromagnetic gauge coupling parameter. The resultant non-hermiticity has a natural interpretation in terms of local scale invariance in pilot-wave theory. The conserved current density is modified from $|\psi|^2$ to the local scale invariant, trajectory-dependent ratio $|\psi|^2/ \mathbf 1^2[\mathcal C]$, where $\mathbf 1[\mathcal C]$ is a scale factor that depends on the pilot-wave trajectory $\mathcal C$ in configuration space. All physical predictions are local scale invariant, even in the presence of mass terms. Our approach is general, and we implement it for the Schr\"odinger and Pauli equations, and for the Dirac equation in curved spacetime, each coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique. Our results provide a pivotal understanding of local scale invariance in quantum theory.