A Kähler structure on a symplectic manifold induces a polarization, and geometric quantization with respect to such Kähler polarizations works particularly well and has numerous examples (e.g. the orbit method and quantization of Chern-Simons theory).
An almost Kähler structure does not induce a polarization in the usual sense, unless it is actually a Kähler structure. Nevertheless, via geometric quantization by push-forward there is still a concept of geometric quantization of almost Kähler structures (in the case of finite-dimensional manifolds), which is “as good” as Kähler quantization (Borthwick-Uribe 96).
Examples of almost-Kähler structures on phase spaces include the phase space of the scalar field with reasonable interaction terms, such as the free field or the phi^4 interaction (Collini 16, section 3.2.2).
Fedosov deformation quantization naturally applies to almost-Kähler structures, too
A discussion of a kind of geometric quantization via push-forward on finite-dimensional almost Kähler manifolds is in
Discussion of the almost-Kähler structure on the phase space of the scalar field is in
On symplectic reduction in the almost-Kähler case:
Last revised on August 17, 2017 at 13:42:30. See the history of this page for a list of all contributions to it.