nLab deformation quantization of the 2-sphere

Idea

The 2-dimensional sphere naturally carries the structure of a Poisson manifold, in fact of a symplectic manifold, with its standard volume form serving as the symplectic form. As such one may consider the deformation quantization of its Poisson algebra of functions.

Properties

Strict deformation quantization

A strict deformation quantization of the 2-sphere is obained as follows.

Take the volume of the 2-sphere to be a natural number. Then there is a prequantum line bundle $(L,\nabla)$ on $S^2$ whose curvature 2-form is the symplectic form, hence the volume form, and which is a holomorphic line bundle with respect to the standard complex manifold structure of the 2-sphere (the Riemann sphere).

For $N \in \mathbb{N}_+$ a positive natural number, the geometric quantization of the 2-sphere for Planck's constant $\hbar = 1/N$ produced the space of quantum states

$$\mathcal{H}_N = \Gamma_{hol}(S^2, L^{\otimes N})$$

which is the space of holomorphic sections of the $N$th tensor power of the prequantum line bundle. See at geometric quantization of the 2-sphere.

This is a finite-dimensional complex Hilbert space, hence the matrix algebra $Mat_N(\mathbb{C})$ canonically acts on it.

One finds that the assignment

$$\left\{ 0 , \cdots, \frac{1}{N}, \cdots, \frac{1}{3}, \frac{1}{2}, 1 \right\} \longrightarrow C* Alg$$

which sends $\frac{1}{N}$ to $Mat_N(\mathbb{C})$ is a strict deformation quantization of the 2-sphere (Hawkins 07, section 4).

Related concepts

• quantization of the 2-sphere

  • geometric quantization of the 2-sphere

  • deformation quantization of the 2-sphere

References

Section 4 of

• Eli Hawkins, An Obstruction to Quantization of the Sphere (arXiv:0706.2946)

following

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