C* algebraic deformation quantization



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What is called strict or C*C*-algebraic deformation quantization is an attempt to formalize quantization of phase spaces or more generally of Poisson manifolds by continuously deforming, in a precise sense, their commutative algebras of functions (algebras of observables) to non-commutative C*-algebras whose commutators are, to “first order” in a suitable sense, determined by the given Poisson bracket.

This is in contrast to formal deformation quantization, where one asks not for C*-algebras but just for formal power series algebras. Where formal deformation quantization is perturbative quantization (perturbation in Planck's constant, see Collini 16), strict deformation quantization is meant to reflect non-perturbative quantization.

While there are good examples of strict C *C^\ast-algebraic deformation quantization for toy examples such as low spacetime dimension (notably quantum mechanics) to date no examples of interacting field theories in spacetime dimension 4\geq 4 have a known non-perturbative quantization. (For the case of Yang-Mills theory/QCD the construction of its non-perturbative quantization is one of the open “Millenium Problems” listed by the Clay Mathematics Institute, see at quantization of Yang-Mills theory.)

Typically the C *C^\ast-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 (Hawkins 06), see at geometric quantization of symplectic groupoids.


Relation to formal deformation quantization

Under favorable circumstances, one can form from a strict C *C^\ast-algebraic deformation quantization given by a continuous field of C*-algebras over a subset of the interval the “differentiation” as [0,1]\hbar \in [0,1] tends to 0, such that this reproduces a formal deformation quantization.

Conversely, a natural intuition might be that given a formal deformation quantization then the subalgebra of converging power series inside all formal power series has a completion to a C*-algebra which constitutes a strict deformation quantization.

While this seems natural, the only actual example where this is understood to date seems to be the simple case of the standard Poisson structure on 2n\mathbb{R}^{2n} with its Weyl algebra star product. (See this MO discussion).


Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization


Convergence of formal power series in formal deformation quantization is discussed for instance in

  • M. Bordemann, M. Brischle, C. Emmrich, Stefan Waldmann, Subalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for P n\mathbb{C}P^n (arXiv:q-alg/9512019)

  • H. Omori, Y. Maeda, N. Miyazaki, A. Yoshida, Deformation quantization of Fréchet-Poisson algebras of Heisenberg type 2001 (pdf)

and in the textbook

The notion of strict C *C^\ast-algebraic deformation quantization was introduced in

  • Marc Rieffel, Quantization and C *C^\ast-algebras, Contemporary mathematics vol. 167 (1994) (pdf)

A brief review with a list of open questions is in

More details are 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 d\mathbb{R}^d, Mem. Amer. Math. Soc. 106 (1993), no. 506, x+93 pp. MR94d:46072

Discussion of strict deformation quantization in terms of geometric quantization of symplectic groupoids via polarized twisted groupoid convolution algebras is in

For the special case of Moyal deformation quantization (Hawkins 06, section 6.2) this construction had been suggested without proof in

  • Alan Weinstein, in P. Donato et al. (eds.) Symplectic Geometry and Mathematical Physics, (Birkhäuser, Basel, 1991); p. 446.

and a detailed proof was given in

A review of variants of the definition of strict deformation quantization is in section 2 of

see also

  • Martin Bordemann, Eckhard Meinrenken, Martin Schlichenmaier, Toeplitz Quantization of Kähler Manifolds and gl(N)gl(N) NN\to\infty, Commun.Math.Phys. 165 (1994) 281-296 (arXiv:hep-th/9309134)

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 September 20, 2017 18:24:17 by Urs Schreiber (