Paper

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

Zhou, Chengang

2001

Overview

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, )\\). The fuzzy sphere is included as a special case parametrized by the integer two-cohomology class \\(H^2(S^2, )\\), 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.

• Publisher: Cornell University Library, arXiv.org
• Subject: Deformation / Field theory / Hilbert space / Homology / Magnetic fields / Mathematical analysis / Measurement / Parameterization / Quantum field theory / Quantum theory