physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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 on 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 a positive natural number, the geometric quantization of the 2-sphere for Planck's constant produced the space of quantum states
which is the space of holomorphic sections of the 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 canonically acts on it.
One finds that the assignment
which sends to 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.