Geometrical maps in the hydrodynamic/quantum description of the evolution equations

Abstract

In this work, the topological and geometrical structure of the evolution equations of physical systems is treated and analyzed from the point of view of the set of solutions, in particular the singular ones. To this end, a generalization of the Wigner function is introduced for the case of multidimensional quadratic Hamiltonians of the type of our previous work [A. I. Aptekarev, Yu. G. Rykov and D. J. Cirilo-Lombardo, Diffeomorphic structure of evolution equations, J. Geom. Phys. 202 (2024) 105234]. The phase space solutions from the generalized Wigner equation are compared with those from the Schrodinger-type equation where the Madelung transformation was introduced. We show that singular delta-type solutions coming from the Wigner–Liouville system certainly exist, intuiting accordingly that they could be mapped to a hydrodynamic representation, in sharp contrast with the Schrodinger (wave mechanical) case, where delta-like solutions become inconsistent. This is achieved by considering a topological limit to the coherent subspace of solutions of the Wigner–Liouville equation, where the kernel (acting as quasidistribution) of these new coherent functions (coherent states) takes the main role. Our results are compared in the context of the wave mechanical approach to cosmic structure formation considering the Zeldovich approach of [P. Coles and K. Spencer, A wave-mechanical approach to cosmic structure formation, Mon. Not. R. Astron. Soc. 342 (2003) 176]. The possibility of obtaining a one-to-one functional relationship between the singular solution space of the hydrodynamic system and the Wigner–Liouville one is briefly discussed.