TY - JOUR

T1 - Wick rotations in deformation quantization

AU - Schmitt, Philipp

AU - Schötz, Matthias

N1 - Publisher Copyright: © 2021 World Scientific Publishing Company.

PY - 2022

Y1 - 2022

N2 - We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from C1+n with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space CPn and the complex hyperbolic disc 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 CPn and 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.

AB - We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from C1+n with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space CPn and the complex hyperbolic disc 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 CPn and 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.

KW - Deformation quantization

KW - Fréchet algebra

KW - strict quantization

KW - symmetry reduction

KW - Wick rotation

UR - http://www.scopus.com/inward/record.url?scp=85111425420&partnerID=8YFLogxK

U2 - 10.1142/S0129055X21500355

DO - 10.1142/S0129055X21500355

M3 - Journal article

AN - SCOPUS:85111425420

VL - 34

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 1

M1 - 2150035

ER -