I have some questions about the relationship between Heisenberg groups/algebras and symplectic vector spaces. This is my first time properly dealing with many of these topics, so please be patient if the questions are poorly posed.
- Is there some way to visualize or intuit the fact every symplectic vector space has an associated Heisenberg group?
- What property do symplectic vector spaces have over other vector spaces that allows them to have an associated Heisenberg group?
- How does this relate (if it does) to the Poisson algebra of smooth functions on the symplectic space having a Heisenberg lie subalgebra?
To give some background, I've recently been studying the geometric quantization of affine symplectic spaces. In a paper, Witten, Axelrod and Della Pietra state that it is sufficient to construct a Hilbert space representation of only the Heisenberg Lie algebra associated with the affine symplectic space. From there, the construction can be extended to geometrically quantize the entire space. Trying to understand how the Heisenberg group/algebra has arisen in this context at led me to the simpler questions above.