One-loop Effects on the Fuzzy Sphere

The two-point function for scalar field theory on the fuzzy sphere is calculated at one loop, including planar and nonplanar contributions. It turns out that there is no UV/IR mixing on the fuzzy sphere. The limit of the commutative sphere is regular without UV/IR mixing; however quantization does not commute with the commutative limit, and a finite ``noncommutative anomaly'' survives in the commutative limit. In a different limit, the noncommutative plane R^2_theta is obtained, and the UV/IR mixing appears. This provides an explanation of the UV/IR mixing as an infinite variant of the ``noncommutative anomaly''.