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

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 $|ψ|^2$ to the local scale invariant, trajectory-dependent ratio $|ψ|^2/ \mathbf 1^2[\mathcal C]$, 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 Schrödinger 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.

💡 Research Summary

The authors revisit Hermann Weyl’s abandoned idea of local scale (conformal) invariance and show that it can be naturally realized within the de Broglie‑Bohm pilot‑wave formulation of quantum mechanics. The key technical move is to complexify the electromagnetic coupling constant, replacing the real charge e by a complex parameter e_C = e + i e_I. This renders the kinetic term (−i∇ − e_C A)^2/2m non‑Hermitian, thereby breaking the usual Hermiticity postulate of orthodox quantum theory but preserving the canonical commutation relation