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deformation quantization of the 2-sphere

Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

AQFT

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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 deformation quantization of its Poisson algebra of functions.

Properties

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).

References

Section 4 of

following

  • 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)

Created on September 4, 2013 at 05:25:39. See the history of this page for a list of all contributions to it.