deformation quantization of the 2-sphere




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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.


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,)(L,\nabla) on S 2S^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 +N \in \mathbb{N}_+ a positive natural number, the geometric quantization of the 2-sphere for Planck's constant =1/N\hbar = 1/N produced the space of quantum states

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

which is the space of holomorphic sections of the NNth 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()Mat_N(\mathbb{C}) canonically acts on it.

One finds that the assignment

{0,,1N,,13,12,1}C*Alg \left\{ 0 , \cdots, \frac{1}{N}, \cdots, \frac{1}{3}, \frac{1}{2}, 1 \right\} \longrightarrow C* Alg

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


Section 4 of


  • 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)

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