Quantum spaces, central extensions of Lie groups and related quantum field theories

Abstract

Quantum spaces with $\mathfrak{su}(2)$ noncommutativity can be modelled by using a family of $SO(3)$-equivariant differential ${}^{\ast }$-representations. The quantization maps are determined from the combination of the Wigner theorem for $SU(2)$ with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to $\mathfrak{su}(2)$ is obtained from a subfamily of differential ${}^{\ast }$-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ${\varphi }^4$ theory on ${\mathbb{R}}^3$ are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed.

12 pages

References in corpus (14)

• Group field theory with non-commutative metric variables
• Kappa-Minkowski space-time and the star product realizations
• Noncommutative Induced Gauge Theory
• kappa-Minkowski spacetime as the result of Jordanian twist deformation
• Twisting all the way: from Classical Mechanics to Quantum Fields
• Vacuum configurations for renormalizable non-commutative scalar models
• Twisted Noncommutative Field Theory with the Wick-Voros and Moyal Products
• Noncommutative field theory on ${\mathbb{R}}_{\lambda }^3$
• On the vacuum states for noncommutative gauge theory
• Noncommutative field theories on $R_{\lambda }^3$: Towards UV/IR mixing freedom
• Star products made (somewhat) easier
• Noncommutative $R^d$ via closed star product
• Exact Partition Functions for Gauge Theories on ${\mathbb{R}}_{\lambda }^3$
• Involutive representations of coordinate algebras and quantum spaces

Cited by in corpus (1)

• Gauge theories on quantum spaces