Geometric and deformation quantization through examples

Abstract

We study the problem of quantization of classical mechanical systems from both geometric and algebraic perspectives, with the aim of understanding how the passage from commutative to non-commutative observable algebras is controlled by the geometric structures of the underlying phase space. We begin with the 𝐶∗-algebraic formulation of classical and quantum mechanics, showing how non-commutativity of the observable algebra leads to intrinsic uncertainty and the failure of simultaneous determinate values for incompatible observables. We then present geometric quantization via prequantization and polarization, followed by Berezin quantization on complex projective spaces using coherent states and reproducing kernel Hilbert spaces, including the induced quantization scheme of Dey-Ghosh for compact manifolds embedded in ℂℙ𝑛. Turning to the algebraic approach, we develop deformation quantization as the problem of constructing associative star products on Poisson manifolds. We present Fedosov’s construction for symplectic manifolds using flat connections on the Weyl bundle, and Kontsevich’s formality theorem for general Poisson manifolds via 𝐿∞ morphisms between Polyvector fields and polydifferential operators. We unify both perspectives through the Maurer-Cartan equation, identifying the flatness of connections, the Jacobi identity for Poisson structures, and the associativity of star products as instances of the same algebraic condition. In the main original contribution, we construct 𝐺-invariant star products with quantum moment maps on Poisson manifolds by embedding them into a formal symplectic groupoid and performing Fedosov quantization in the ambient symplectic manifold. The star product on the Poisson manifold is obtained by descending through an algebra isomorphism between the Weyl bundle on the ambient space and a Weyl bundle on the original manifold. When the manifold carries a Hamiltonian 𝐺-action with a 𝐺-invariant connection, we show that the descended star product is 𝐺-invariant and admits quantum moment maps. We conclude with a discussion of how the algebraic structures appearing in deformation quantization : Maurer-Cartan equations and differential graded Lie algebras, have a higher categorical interpretation.