Poisson Geometric Formulation of Quantum Mechanics

We study the Poisson geometrical formulation of quantum mechanics for finite dimensional mixed and pure states. Equivalently, we show that quantum mechanics can be understood in the language of classical mechanics. We review the symplectic structure of the Hilbert space and identify its canonical coordinates. We extend the geometric picture to the space of density matrices $D_N^+$. We find it is not symplectic but admits a linear $\mathfrak{su}(N)$ Poisson structure. We identify Casimir surfaces of $D_N^+$ and show that the space of pure states $P_N \equiv \mathbb{C}P^{N-1}$ is one of its symplectic submanifolds which is an intersection of primitive Casimirs. We identify generic symplectic submanifolds of $D_N^+$ and calculate their dimensions. We find that $D_N^+$ is singularly foliated by the symplectic leaves of varying dimensions, also known as coadjoint orbits. We also find an ascending chain of Poisson submanifolds $D_N^M \subset D_N^{M+1}$ for $ 1 \leq M \leq N-1$. Each such Poisson submanifold $D_N^M$ is obtained by tracing out the $\mathbb{C}^M$ states from the bipartite system $\mathbb{C}^N \times \mathbb{C}^M$ and is an intersection of $N-M$ primitive Casimirs of $D_N^+$. Their Poisson structure is induced from the symplectic structure of the bipartite system. We also show their foliations. Finally, we study the positive semi-definite geometry of the symplectic submanifold $E_N^M$ consisting of the mixed states with maximum entropy in $D_N^M$.