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Quantization in practice
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In classical mechanics, the observables belong to a commutative 
Poisson algebra of functions, equipped with the Poisson bracket as 
Lie bracket. The corresponding Lie group is the group of canonical 
transformations.

In quantum mechanics, the observables belong to a non-commutative 
Poisson algebra of linear operators on a dense subspace of a Hilbert 
space, equipped with i/hbar times the commutator as Lie bracket. 
The corresponding Lie group is the group of unitary operators. 

Quantization of a classical theory corrresponds to assigning to 
selected classical observables (so-called canonical variables) 
corresponding quantum operators, and promoting the important canonical 
transformations to appropriate unitary operators. This cannot be done 
in a fully isomorphic way, but key relations must be preserved.


In the cases of practical interest, the canonical transformations are 
given in terms of their infinitesimal generators (or rewritten as 
exponentials, producing corresponding infinitesimal generators).
The infinitesimal generators are typically low degree polynomials in 
the canonical variables. Then one expresses things in terms of ladder
operators (which exist also classically), normally orders the result 
to get a unique representative expression, discards the constant 
(which only provides a phase and usually has no effect, except when 
anomalies are present), replaces the classical ladder operators by 
their quantum version, and exponentiates the result - hoping that it 
defines a self-adjoint operator (needed to make the exponential 
well-defined). (Note: Doing the same for a canonical transformation 
directly rather than the infinitesimal generator will usually _not_ 
produce a unitary operator, even when the transformation is polynomial!)
 
This recipe works whenever the the classical variables generate a 
finite-dimensional Lie algebra with a triangular decomposition (so that 
ladder operators are well-defined) and the quantum space carries a 
corresponding unitary representation. In the standard mechanical case,
the Lie algebra is a direct sum of Heisenberg algebras; for the 
spinning top, it is SO(3). 
 
In field theories, the recipe works only on the formal level since the 
Lie algebra is infinite-dimensional, and the topological side 
conditions are much stronger. Unfortunately, self-adjointness fails 
miserably for quantum fields in dimensions 1+d, d>1 and makes 
renormalization necessary.