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 $\rho$ 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.

由马德隆方程描述的量子流体动力学是在辛几何的框架中分析的，即在几何场论的协变相空间方法中。提供描述量子流体哈密顿动力学的相空间的前辛流形是由马德隆方程的所有解及其相应的拉格朗日密度的集合构成的。正在考虑的马德隆方程是与薛定谔方程相关的马德隆方程（在非相对论的情况下）和与克莱因-戈登方程相关的马德隆方程（在相对论的情况下）。这里还考虑了存在与电磁场耦合的情况。我们的辛公式不同于 [M. Spera，薛定谔和泡利方程的矩图和规范几何方面，诠释。J.几何。方法国防部。物理。 13 (2016) 1-36] 在基础领域或变量的选择中。这里我们考虑密度函数$\rho$和相位函数$\mathrm{小号}$不是作为规范对，而是作为理论的基本领域。对应于一个可观测量的哈密顿向量场是从由可观测量产生的哈密顿方程获得的。然后，两个可观察量的泊松括号由与每个可观察量相关的哈密顿向量场确定。一般来说，两个可观察量的泊松括号不是唯一的，因为每个可观察量都有多个对应的哈密顿向量场。需要指出的是，泊松括号在所考虑的马德隆方程的前辛流形上定义的所有可观测量集合的某个子集上具有唯一值。