nLab
C-star algebraic deformation quantization

Idea

$C^{*}$-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 $C^{*}$-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.

Related concepts

• geometric quantization of symplectic groupoids

Examples of sequences of infinitesimal and local structures

first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
	$\leftarrow$ differentiation		integration $\to$
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

References

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)

• Marc Rieffel, Deformation quantization for actions of ${ℝ}^d$, Mem. Amer. Math. Soc. 106 (1993), no. 506, x+93 pp. MR94d:46072

• Marc Rieffel, Quantizaton and $C^{*}$-algebras, Contemporary mathematics vol. 167 (1994) (pdf)

For the special case of Poisson manifolds that are total spaces of Lie algebroids, discussion is in

• Klaas Landsman, B. Ramazan, Quantization of Poisson algebras associated to Lie algebroids (arXiv:math-ph/0001005)

• Klaas Landsman, Strict deformation quantization of a particle in external gravitational and Yang-Mills fields, Journal of Geometry and Physics 12:2, p. 93-132 (web)