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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.