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S1k. Quantization in non-Cartesian coordinates
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Textbook quantization rules assume (often silently, without warning)
Cartesian coordinates. The rules derived there are based on
canonical commutation rules and are invalid for systems
described in other coordinate systems.
In particular, a Hamiltonian alone does not have a physical meaning
since it can be quite arbitrarily transformed by coordinate
transformations. The Hamiltonian needs to be combined with the
correct Poisson bracket to yield the correct dynamical equations.
Only if the classical Poisson bracket satisfies the canonical
commutation rules, the quantum mechanics is obtained by imposing
canonical commutation rules on the commutators.
The standard quantization procedure assumes that the symplectic form
underlying the Hamiltonian description has the standard form
p dq - q dp. Under a coordinate transformation, the symplectic form
changes into something nonstandard, and naive quantization gives
wrong results.
To get correct results, one has to take account of the correct
symplectic structure, more precisely of the Poisson bracket defined
by it. This is most naturally done in a differential geometric
setting, in terms of symplectic manifolds and Poisson manifolds.
To proceed, one must quantize a symplectic (or a Poisson) manifold
together with a Hamiltonian defined on it.
This combination is invariant under coordinate transformations
and hence has a coordinate-independent geometric meaning.
How to quantize Hamiltonians on a symplectic (or a Poisson) manifold
is the subject of geometric quantization, about which there is a
significant literature.