On the origin of the UV-IR mixing in non-commutative matrix geometry

Abstract

Scalar field theories with quartic interaction are quantized on fuzzy S 2 and fuzzy S 2× S 2 to obtain the 2- and 4-point correlation functions at one-loop. Different continuum limits of these non-commutative matrix spheres are then taken to recover the quantum non-commutative field theories on the non-commutative planes R 2 and R 4 , respectively. The canonical limit of large stereographic projection leads to the usual theory on the non-commutative plane with the well-known singular UV-IR mixing. A new planar limit of the fuzzy sphere is defined in which the non-commutativity parameter θ, beside acting as a short distance cut-off, acts also as a conventional cut-off Λ= 2 θ in the momentum space. This non-commutative theory is characterized by absence of UV-IR mixing. The new scaling is implemented through the use of an intermediate scale that demarcates the boundary between commutative and non-commutative regimes of the scalar theory. We also comment on the continuum limit of the 4-point function.