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

Abstract

We show that Weyl's abandoned idea of local scale invariance has a natural realization at the quantum level in pilot-wave (deBroglie-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 of the quantum state 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. Our approach is general, and we implement it for the Schrödinger, Pauli, and Dirac equations 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.

Figures (3)

• Figure 1: Computed particle trajectories for the non-hermitian potential $V(x) = i \sin x$. The trajectories crossing $x \in [-5,5]$ in $0.05$ equally-spaced increments at times $t = 1, 2...5$ s are shown. The horizontal (vertical) axis corresponds to $t$ ($x$). The trajectories are required to compute the scale factor $\mathds 1[\mathcal{C}]$ in the conserved current density $|\psi|^2/\mathds 1^2[\mathcal{C}]$. The vertical axis is logarithmically scaled.
• Figure 2: Computed scale factor squared $\mathds 1^2[\mathcal{C}]$ for the non-hermitian potential $V(x) = i \sin x$ at times $t = 1, 2...5$ s. The scale factor $\mathds 1[\mathcal{C}]$ at each point $(x, t)$ is defined as the line integral $\mathds 1[\mathcal{C}] = e^{\int^{(x,t)}_\mathcal{C} \sin x'(t')\textbf{ } dt'}$ over the particle trajectory from $(x_0, 0)$ to $(x, t)$, where $x_0$ is the particle position at $t=0$ s. The particle trajectories are shown in figure \ref{['ftrajectories']} and the density $|\psi|^2/\mathds 1^2[\mathcal{C}]$ is shown in figure \ref{['fdensity']}. The vertical axis is logarithmically scaled.
• Figure 3: Comparison of the computed densities $|\psi|^2/\mathds 1^2[\mathcal{C}]$ (red) and $|\psi|^2$ (blue) for the non-hermitian potential $V(x) = i \sin x$ at times $t = 1, 2...5$ s. The two densities are identical at $t = 0$ but quickly diverge. The density $|\psi|^2/\mathds 1^2[\mathcal{C}]$ is conserved and remains normalized, whereas $|\psi|^2$ is not conserved and grows large over extended regions. Comparison with figure \ref{['fscale']} shows that $\mathds 1^2[\mathcal{C}]$ grows large in the regions where $|\psi|^2$ is large so as to keep the ratio $|\psi|^2/\mathds 1^2[\mathcal{C}]$ normalized. The vertical axis is logarithmically scaled.