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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.
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
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
which sends $\frac{1}{N}$ to $Mat_N(\mathbb{C})$ is a strict deformation quantization of the 2-sphere (Hawkins 07, section 4).
deformation quantization of the 2-sphere
Section 4 of
following
Last revised on June 27, 2019 at 06:45:20. See the history of this page for a list of all contributions to it.