The symplectic geometrical formulation of quantum hydrodynamics
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 fro...
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International Journal of Geometric Methods in Modern Physics
2022
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| _version_ | 1797037848023531520 |
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| author | Wijaya, Andika Kusuma Hermanto, Arief Rosyid, Muhammad Farchani |
| author_facet | Wijaya, Andika Kusuma Hermanto, Arief Rosyid, Muhammad Farchani |
| author_sort | Wijaya, Andika Kusuma |
| collection | UGM |
| description | 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 ρ and phase function S 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. © 2022 World Scientific Publishing Company. |
| first_indexed | 2024-03-14T00:10:14Z |
| format | Other |
| id | oai:generic.eprints.org:284425 |
| institution | Universiti Gadjah Mada |
| last_indexed | 2024-03-14T00:10:14Z |
| publishDate | 2022 |
| publisher | International Journal of Geometric Methods in Modern Physics |
| record_format | dspace |
| spelling | oai:generic.eprints.org:2844252023-12-19T06:35:48Z https://repository.ugm.ac.id/284425/ The symplectic geometrical formulation of quantum hydrodynamics Wijaya, Andika Kusuma Hermanto, Arief Rosyid, Muhammad Farchani Quantum Physics 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 ρ and phase function S 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. © 2022 World Scientific Publishing Company. International Journal of Geometric Methods in Modern Physics 2022 Other NonPeerReviewed Wijaya, Andika Kusuma and Hermanto, Arief and Rosyid, Muhammad Farchani (2022) The symplectic geometrical formulation of quantum hydrodynamics. International Journal of Geometric Methods in Modern Physics. https://www.worldscientific.com/doi/10.1142/S0219887822501948 10.1142/S0219887822501948 |
| spellingShingle | Quantum Physics Wijaya, Andika Kusuma Hermanto, Arief Rosyid, Muhammad Farchani The symplectic geometrical formulation of quantum hydrodynamics |
| title | The symplectic geometrical formulation of quantum hydrodynamics |
| title_full | The symplectic geometrical formulation of quantum hydrodynamics |
| title_fullStr | The symplectic geometrical formulation of quantum hydrodynamics |
| title_full_unstemmed | The symplectic geometrical formulation of quantum hydrodynamics |
| title_short | The symplectic geometrical formulation of quantum hydrodynamics |
| title_sort | symplectic geometrical formulation of quantum hydrodynamics |
| topic | Quantum Physics |
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