Local Scale Invariance in Quantum Theory: Experimental Predictions

TL;DR

The paper develops a non-Hermitian pilot-wave formulation with local scale invariance, encoded by a complex gauge coupling $e_C = e + i e_I$, and derives a scale constant $α_S$ that relates to the gravitational fine-structure constant. It shows $α_S/α \sim 10^{-21}$, explaining why such effects are suppressed in standard experiments, and makes concrete experimental predictions: in an Aharonov-Bohm double-slit setup, the equilibrium density becomes trajectory-dependent via a scale factor $\mathds{1}[\mathcal{C}]$, yielding which-way dependent probabilities for a heavy neutral molecule under large flux. The work reassesses the Weyl-Einstein debate by deriving history-independent spectral frequencies while predicting history-dependent spectral intensities and imaginary-energy contributions to linewidths, providing multiple experimentally testable distinctions from Hermitian quantum mechanics. Overall, the paper argues that trajectory-based quantum predictions are empirically distinguishable and highlights a deep link between local scale invariance, gravity, and gauge structure in quantum foundations.

Abstract

We explore the experimental predictions of the local scale invariant, non-Hermitian pilot-wave (de Broglie-Bohm) formulation of quantum theory introduced in arXiv:2601.03567. We use Weyl's definition of gravitational radius of charge to obtain the fine-structure constant for non-integrable scale effects $α_S$. The minuteness of $α_S$ relative to $α$ ($α_S/α\sim 10^{-21}$) effectively hides the effects in usual quantum experiments. In an Aharonov-Bohm double-slit experiment, the theory predicts that the position probability density depends on which slit the particle trajectory crosses, due to a non-integrable scale induced by the magnetic flux. This experimental prediction can be realistically tested for an electrically neutral, heavy molecule with mass $m \sim 10^{-19} \text{g}$ at a $\sim 10^6 \text{ esu}$ flux regime. We analyse the Weyl-Einstein debate on the second-clock effect using the theory and show that spectral frequencies are history-independent. We thereby resolve Einstein's key objection against local scale invariance, and obtain two further experimental predictions. First, spectral intensities turn out to be history-dependent. Second, energy eigenvalues are modified by tiny imaginary corrections that modify spectral linewidths. We argue that the trajectory dependence of the probabilities renders our theory empirically distinguishable from other quantum formulations that do not use pilot-wave trajectories, or their mathematical equivalents, to derive experimental predictions.

Figures (3)

• Figure 1: Schematic illustration of the which-way-dependent probability density in a double-slit Aharonov-Bohm experiment. Two partial wavepackets (shown by dotted lines) emerging from slits A and B pass on either side of a solenoid enclosing magnetic flux (shown by $\otimes$) and then coherently recombine. A spacetime point $x \equiv (ct, \vec{x})$ in the support of the recombined wavepacket is indicated. Two schematic paths $\mathcal{C}_A$ and $\mathcal{C}_B$ (lying outside the solenoid) connecting the preparation source to $x$ via slit A and B are indicated in blue and red, respectively. If the particle trajectory passing through $x$, as determined by the pilot-wave guidance equation, crosses slit A (B), then non-Hermitian PWT predicts the equilibrium probability density at $x$ to be $|\psi(x)|^2/\mathds 1^2[\mathcal{C}_A]$$(|\psi( x)|^2/\mathds 1^2[\mathcal{C}_B] )$. Note that the guidance equation associates any spacetime point in the support of the recombined wavepacket with a single particle trajectory crossing either slit A or B.
• Figure 2: Schematic comparison of probability density in a) orthodox quantum theory and b) non-Hermitian PWT in Aharonov-Bohm double-slit experiment for a neutral, heavy molecule. Orthodox quantum theory predicts that the probability density $|\psi|^2$ would be identical to the standard double-slit experiment as the particle is electrically neutral. In non-Hermitian PWT, the particle couples geometrically to the gauge field $A^\mu$ via its mass, which amplifies (suppresses) the probability density $|\psi|^2/\mathds 1^2[\mathcal{C}]$ in the region $x \leq -0.7$ ($x > -0.7$) via the scale factor $e^{\frac{m\sqrt G}{\hbar c}\oint_{\mathcal{C}_L} \vec{A}(\vec{x}') \cdot d \vec{x}'}$ ($e^{-\frac{m\sqrt G}{\hbar c}\oint_{\mathcal{C}_L} \vec{A}(\vec{x}') \cdot d \vec{x}'}$) (see equation \ref{['woho']}). The point $x = -0.7$ is chosen for illustration as the separatrix, such that points to its left (right) have particle trajectories incoming from the left (right) slit. The initial wavefunction passing through the slits is taken to be $\psi(x, y, 0) = (8/\pi)(e^{-8(x-1.5)^2} + e^{-8(x+1.5)^2})e^{-8y^2}e^{i5y}$, and the probability densities are shown at $(t = 0.7, y=3.5)$. We have taken $m = G = \hbar = c = 1$ and $\oint_{\mathcal{C}_L} \vec{A}(\vec{x}') \cdot d \vec{x}' = \pi/4$ for illustration.
• Figure 3: History-independence of spectral frequencies. The plot shows that the resonance condition $\omega^R_{\vec{n}\vec{p}} = \omega$ for maximizing $|c_{\vec{n}}^1(t)|^2$ is unaffected by the history of the particle. This implies that Einstein's criticism of local scale invariance that it leads to history-dependent spectral frequencies ("second-clock effect") is incorrect in non-Hermitian PWT. We have taken $\omega^I_{\vec{n} \vec{p}}/\omega^R_{\vec{n} \vec{p}} \approx 10^{-21}$, $\overline{V}_{\vec{n} \vec{p}} = 1$ and $t = 10$ in equation \ref{['tdptm']} for illustration.