Determinantal polynomial wave functions induced by random matrices

TL;DR

This paper introduces a novel model of vortex dynamics using determinantal polynomial wave functions derived from random matrices, incorporating quaternionic structures to describe complex topological defect interactions.

Contribution

It develops a new framework linking random matrix theory with vortex dynamics, including anti-vortices and topological defects, and explores their interactions and topological reactions.

Findings

Model captures vortex-antivortex interactions and defect reactions.

Quaternionic structures enable interpretation of vortex processes as quaternionic states.

Identifies a time-energy uncertainty principle in vortex dynamics.

Abstract

Random-matrix eigenvalues have a well-known interpretation as a gas of like-charge particles. We make use of this to introduce a model of vortex dynamics by defining a time-dependent wave function as the characteristic polynomial of a random matrix with a parameterized deformation, the zeros of which form a gas of interacting vortices in the phase. By the introduction of a quaternionic structure, these systems are generalized to include anti-vortices and non-vortical topological defects: phase maxima, phase minima and phase saddles. The commutative group structure for complexes of such defects generates a hierarchy, which undergo topologically-allowed reactions. Several special cases, including defect-line bubbles and knots, are discussed from both an analytical and computational perspective. Finally, we return to the quaternion structures to provide an interpretation of two-vortex fundamental processes as states in a quaternionic space, where annihilation corresponds to scattering out of real space, and identify a time--energy uncertainty principle.