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International Journal of Geometric Methods in Modern Physics

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

Alexander Ivanovich Aptekarev, Diego Julio Cirilo-Lombardo, Yu. G. Rykov

Published May 16, 2025• 0 citations
Publisher

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.

DOI
10.1142/s0219887825502123

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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.
Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.
In this paper we discuss various geometric aspects related to the Schrödinger and the Pauli equations. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schrödinger equation. The probability current provides an equivariant moment map for the group [Formula: see text] of volume-preserving diffeomorphisms of [Formula: see text] (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalized Schrödinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural “coherent state” representations of [Formula: see text]. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schrödinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
Abstract The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.
The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.