# nLab deformation quantization of the 2-sphere

Contents

## Surveys, textbooks and lecture notes

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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).

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### 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.

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)