In 1999, Karaali wrote a review of formal deformation quantization for a class she took with Weinstein. She ends the paper with the following remark:
Another question that remains involves the infinite dimensional case: Kontsevich‘s results settle the problem in the finite dimensional case, but there are places in physics where we have to deal with infinite dimensional Poisson manifolds, i.e infinite dimensional manifolds with a Poisson structure on them. This case involves new problems and perhaps may shed light on a better mathematical understanding of quantum field theory.
Since then, it seems that techniques constructed by Schlichenmaier settled this problem for infinite-dimensional symplectic manifolds, as claimed in nLab.
More recently, in a paper by Hawkins and Rejzner, a deformation quantization for infinite-dimensional affine manifolds has been constructed. They also review the problem stated above, and mention that it is still open.
- What is the current status of formal deformation quantization of infinite-dimensional Poisson manifolds?