Dynamic Quantum Wormholes: Path-Integral State Selection and the Quantum Throat Trajectory in an Exactly Solvable Thin-Shell Model

Abstract

We construct a minisuperspace quantization of a dynamic, spherically symmetric thin-shell wormhole in the spirit of pilot-wave quantum cosmology, treating the throat radius $a(τ)$ as the single degree of freedom of the Israel junction-condition constraint. For a pressureless shell of mass $M$ the Wheeler--DeWitt equation reduces exactly to Whittaker's equation, and to the modified Bessel equation as $M\to0$. When the exterior is asymptotically flat, boundedness of the wave function as $a\to\infty$ selects a unique state that can always be chosen real. Since any real solution has a vanishing Bohmian current, the throat is frozen for any linear equation of state and any operator ordering, independently of the choice between Euclidean and Lorentzian boundary conditions. This state is the exact $ε\to0^+$ limit of the regularized Lorentzian path integral, as shown by resolvent analyticity and by an exact Picard--Lefschetz thimble calculation that agrees with the closed-form solution to $48$ digits. The near-throat self-adjointness ambiguity is resolved in closed form through the Whittaker connection formula. Replacing the exterior by Schwarzschild--de~Sitter removes the obstruction: the potential tends to $-\infty$ at large $a$, and the equation admits a genuinely dynamic, purely outgoing state. Propagated exactly through the classically forbidden barrier, it carries a conserved, nonzero Bohmian current everywhere, including the inner pocket. It describes a long-lived metastable wormhole that tunnels into unbounded expansion at a rate consistent with the WKB estimate. Airy connection formulas leave a residual prefactor constant $κ_\infty=0.814\pm0.004$ for sub-critical coupling, which we do not trace to its analytic origin, and the physical results do not depend on it.