I'm reading "Symplectic geometry and geometric quantization" by Matthias Blau and he introduces a complete set of observables for the classical case:
The functions $q^k$ and $p_l$ form a complete set of observables in the sense that any function which Poisson commutes (has vanishing Poisson brackets) with all of them is a constant.
I wonder why is it so? That is why do we call it a complete set of observables? As I understand it means the functions satisfying the condition above form coordinates on a symplectic manifold, but I don't see how.
Best Answer
Blau called them a 'complete set' in analogy to the quantum mechanical picture, where a observable commuting (read Poission-commuting in the classical case) with a complete set of commuting observables is proportional to the unit, i.e. a 'constant'. This is called (first) Schur's lemma.