Seam Geometry of Bohmian Mechanics: Geodesics, Foliations, and Nodal Singularities

<title>Abstract</title> This paper reframes Bohmian mechanics in seam variables built from the wavefunction modulus and phase. Writing psi=Rexp\(iS/hbar\), the amplitude seam s=log R encodes the Bohm--Madelung quantum potential via Q=-(R"/R)=-(s"+(s')²) in the stationary one-dimensional (1D) convention used here. An effective seam s:=S-tfrac12 s defines a seam-induced conformal factor that serves as a prototype for higher-dimensional metric constructions.Assuming a complex coordinate z=x+iy on a two-dimensional (2D) configuration plane, the quadratic differential (QD) w=\(partial_z^2logpsi\) dz² is introduced and its induced horizontal direction field is compared to probability-current streamlines. For the Gaussian vortex state psi=ze^-|z|2/2, exact alignment is proved: away from the node, the QD-horizontal direction field coincides with the probability-current direction field. More generally, a local "vortex dominance" proposition gives asymptotic direction-field alignment near any isolated simple node under a mild factorisation hypothesis, and explicit limitations are documented via a two-node non-example.Reproducible numerical tests for a centered vortex, a shifted vortex, and a two-node state quantify when alignment persists and when it fails.