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 (strict) deformation quantization of its Poisson algebra of functions.

Details

There are various notions of deformation quantization which one may consider. By default one often means formal deformation quantization with values in formal power series in Planck's constant $\hbar$. One definition of strict deformation quantization takes it to be such a formal deformation quantization which happens to converge for finite real number values of $\hbar$. This is not known to exist for the 2-sphere.

However, there are other notions of strict quantization via deformation of the algebra of functions (review in Hawkins (2007), Section 2).

Formal deformation quantization

(…)

Strict deformation quantization

One kind of strict deformation quantization of the 2-sphere is obtained as follows [Hawkins (2007), section 4].

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 becomes a module over the matrix algebra $Mat_N(\mathbb{C})$.

One finds that the assignment

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

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

Related concepts

• quantization of the 2-sphere

  • geometric quantization of the 2-sphere

  • deformation quantization of the 2-sphere

References

• Eli Hawkins, Section 4 of: An Obstruction to Quantization of the Sphere, Communications in Mathematical Physics 283 (2008) 675–699 [arXiv:0706.2946, doi:10.1007/s00220-008-0517-2]

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, doi:10.1007/BF02099772]