The Madelung Picture as a Foundation of Geometric Quantum Theory

David Shale

Ryszard Paweł Kostecki

D. H. Delphenich

Ordinarily, the stress tensor that one derives for a Madelung fluid is not regarded as being coupled to a strain tensor, which is consistent with the fluid hypothesis. However, based upon earlier work regarding the geometric nature of the quantum potential, one can, in fact, define a strain tensor, which is not, however, due to a deformation of a spatial region, but to a deformation of a frame field on that region. When one expresses the Madelung stress tensor as a function of the strain tensor and its derivatives, one then defines a constitutive law for the Madelung medium that might lead to a more detailed picture of its elementary structure. It is pointed out that the resulting constitutive law is strongly analogous to laws that were presented by Kelvin and Tait for the bending and torsion of elastic wires and plates, as well as the Einstein equations for gravitation if one takes the viewpoint of metric elasticity.

James Thomas Wheeler

Andika Kusuma Wijaya, Arief Hermanto, Muhammad Farchani Rosyid

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.

J. Anandan