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I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

Qmechanic says:

Note that the coordinate components are the same $$X^Q_H[z^I]~\stackrel{(4)}{=}~X_H[z^I],\tag{6}$$ which is part of the problem. Also in eq. (1) we have for convenience defined a divergence $${\rm div}_{\rho} X~:=~ \rho^{-1}\frac{\partial(\rho X[z^I])}{\partial z^I}\tag{7}$$ of a possibly higher-order differential operator $X$. Eq. (7) is not a geometric object, which foretells the doom of eq. (1).

Question: what are the mathematical objects at play here?

  1. Is $z^I$ a point on the manifold?
  2. What do they mean by coordinate components?
  3. Do $X_H$ and $X_H^Q$ live in the tangent space?
  4. Why do we have this form for ${\rm div}_{\rho} X$?
  5. Why is Eq. (7) not a geometric object?

(I think my questions stem from not knowing enough about differential geometry and symplectic manifolds which I'm in the process of learning about. Open to any reference suggestions! Just started in on An Introduction to Manifolds by Loring Tu.)

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