Wick Rotations in Deformation Quantization

TL;DR

This paper explores deformation quantizations of manifolds derived from phase space reduction, demonstrating how Wick rotation links different signatures and providing explicit constructions and isomorphisms of star product algebras.

Contribution

It generalizes deformation quantization results to manifolds obtained via Wick rotation, establishing explicit star product descriptions and isomorphisms between different signatures.

Findings

Explicit construction of formal and non-formal star products.

Existence of convergent subalgebras of polynomial functions.

Isomorphism between quantizations of different signatures via Wick rotation.

Abstract

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from $\mathbb{C}^{1+n}$ with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space $\mathbb{CP}^n$ and the complex hyperbolic disc $\mathbb{D}^n$. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of "polynomial" functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking e.g. the star products on $\mathbb{CP}^n$ and $\mathbb{D}^n$. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the *-involution of pointwise complex conjugation.