- The space of all functions onMis far too big, allowing states localized in
both position and coordinate variables in the caseM=R^2 d. One needs
some way to cut down this space to something like a space of functions
depending on only half the variables (e.g., just the positions, or just the
momenta). This requires finding an appropriate choice of a so-called “po-
larization” that will accomplish this. - To get an inner product on the space of states, one needs to introduce
a twist by a “square root” of a certain line bundle, something called the
“metaplectic correction”.
For more details, see for instance [41] or [103].
Geometric quantization focuses on finding an appropriate state space. An-
other general method, the method of “deformation quantization” focuses instead
on the algebra of operators, with a quantization given by finding an appropriate
non-commutative algebra that is in some sense a deformation of a commuta-
tive algebra of functions. To first order the deformation in the product law is
determined by the Poisson bracket.
Starting with any Lie algebrag, in principle 15.14 can be used to get a Pois-
son bracket on functions on the dual spaceg∗, and then one can take the quan-
tization of this to be the algebra of operators known as the universal enveloping
algebraU(g). This will in general have many different irreducible representa-
tions and corresponding possible quantum state spaces. The co-adjoint orbit
philosophy posits an approximate matching between orbits ing∗under the dual
of the adjoint representation (which are symplectic manifolds) and irreducible
representations. Geometric quantization provides one possible method for trying
to associate representations to orbits. For more details, see [51].
None of the general methods of quantization is fully satisfactory, with each
running into problems in certain cases, or not providing a construction with all
the properties that one would want.
17.6 For further reading
Just about all quantum mechanics textbooks contain some version of the discus-
sion here of canonical quantization starting with classical mechanical systems
in Hamiltonian form. For discussions of quantization from the point of view of
representation theory, see [8] and chapters 14-16 of [37]. For a detailed discus-
sion of the Heisenberg group and Lie algebra, together with their representation
theory, also see chapter 2 of [51].