Deformation Quantization and Quantum Field Theory on Curved Spaces: the Case of Two-Sphere

TL;DR

This paper explores scalar quantum field theory on noncommutative two-spheres using deformation quantization, revealing conditions for unitarity and the impact of topological classes on the physical viability of the models.

Contribution

It provides a detailed analysis of deformation quantization on curved spaces, especially the two-sphere, and identifies topological obstructions to unitarity in the resulting quantum field theories.

Findings

Fuzzy sphere is a special case with finite degrees of freedom.

Unitarity fails when the cohomology class is non-integer valued.

Topological formula characterizes unitarity obstructions.

Abstract

We study the scalar quantum field theory on a generic noncommutative two-sphere as a special case of noncommutative curved space, which is described by the deformation quantization algebra obtained from symplectic reduction and parametrized by $H^2(S^2, \QR)$. The fuzzy sphere is included as a special case parametrized by the integer two-cohomology class $H^2(S^2, \QZ)$, which has finite number of degrees of freedom and the field theory has a well defined Hilbert space. When the two-cohomology class is not integer valued, the scalar quantum field theory based on the deformation algebra is not unitary: the signature of the inner product on the space of functions is indefinite. Hence the existence of deformation quantization does not guarantee a physically acceptable deformed geometric background. For the deformation quantization on a general curved space, this obstruction of unitarity can be given by an explicit topological formula.