the Heisenberg group and the Weil representation of the symplectic group
(more precisely, its metaplectic covering). The theorem of Groenewold and
van Hove then says that this quantization is maximal; that is, it cannot be
extended to polynomials of higher degree.
The remainder of the fifth chapter consists in laying the groundwork
for the general situation, which essentially follows KlRiLLOV [Ki]. Here a
subalgebra p, the primary quantities, comes into play, which for the case
of M = T*Q turns out to be the arbitrary functions in q and the linear
functions in p. Here yet more functional analysis and topology are required
in order to demonstrate the result of Kirillov that for a symplectic manifold,
with an algebra p in ^(M) of primary quantities relative to the Poisson
bracket, a quantization is possible. That is, there is a map which assigns
to each / E p a self-adjoint operator / on Hilbert space TL satisfying the
(1) the function 1 corresponds to the identity id^,
(2) the Poisson bracket of the two functions corresponds to the Lie
bracket of operators, and
(3) the algebra of operators operates irreducibly.
There is a one-to-one correspondence between the set of equivalence classes
of such representations of p and the cohomology group
In the first two appendices, manifolds, vector bundles, Lie groups and
algebras, vector fields, tensors, differential forms and their basic handling
are covered. In particular, the various derivation processes are covered so
that one may follow the proofs in the cited literature. A quick reading of
this synopsis is perhaps recommended as an entrance to the second chapter.
In Chapter 2 some material about cohomology groups will also be required.
The third appendix presents some of the rudiments of cohomology theory.
In the final appendix, the central concept of coadjoint orbits is prepared by
a consideration of the fundamental concepts and constructions of represen-
As already mentioned, somewhat more from the theory of differential
equations than is usually presented in a beginner's course on the topic, in
particular Frobenius' theorem, is required to fully follow the treatment of
symplectic geometry given here. Since in these cases the difficulty is not
in grasping the statements, this material is left out of the appendices and
simply used in the text as needed, though again without proof.
It is not the intention of this text to compete with the treatment of the
classical and current literature over the research in the various subtopics
of symplectic geometry as can be found, for example, in the books by