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arXiv · 1807.01173

Determinantal polynomial wave functions induced by random matrices

Anthony Mays·Anita K. Ponsaing·David M. Paganin
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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.

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Anthony Mays, Anita K. Ponsaing, David M. Paganin. 2018-10-29. Determinantal polynomial wave functions induced by random matrices. https://doi.org/10.1103/physreva.98.063813

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