These notes are based on a series of ten lectures given by Professor
David Simms in London, in the autumn of 197^5 in which he outlined the
"geometric quantization" programme^ of 3- Kostant and J-M. Souriau.
The aim of the programme is to find a way of formulating the relationship
between classical and quantum mechanics in a geometric language, as a
relationship between symplecticmanifolds (classical phase spaces) and
Hilbert spaces (quantum phase spaces). Though it is unlikely that a
completely general and intrinsic quantization construction will ever
be found, work on this programme has lead to new insights into the
connection between the concepts of symmetry in classical and in quantum
mechanics and into the ambiguities involved in the physicist's concept
of quantization.