Types of quantum field thories
-algebraic deformation quantization or strict deformation quantization is a refinement of deformation quantization which produces quantum algebras of observables not just in the space of formal power series, but produces actual C-star algebras that have a chance to genuinely constitute an algebraic quantum field theory.
Typically the -algebraic deformation takes the quantum algebra to be a suitable convolution algebra of suitably polarized sections over a Lie groupoid that Lie integrates a Poisson Lie algebroid which encodes the original Poisson bracket to be quantized.
Examples of sequences of infinitesimal and local structures
|first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|derivative||Taylor series||germ||smooth function|
|tangent vector||jet||germ of curve||curve|
|Lie algebra||formal group||local Lie group||Lie group|
|Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|
The general idea is discussed in
Marc Rieffel, Deformation quantization and operator algebras, in: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), 411–423, Proc. Sympos. Pure Math. 51, Part 1, Amer. Math. Soc. 1990, MR91h:46120; (pdf)