Contents

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.

Examples of sequences of infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$←$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict 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)

Revised on February 6, 2013 18:44:28 by Urs Schreiber (82.113.106.234)