Finite quantum field theory in noncommutative geometry

Section: Discussionmentioning

confidence: 99%

“…This paper focuses on a particular kind of noncommutative geometry and associated spaces, known as fuzzy spaces, which serve as regulators in field theories [4]. The advantage of this approach over the traditional lattice discretisation lies in the fact that fuzzy spaces retain the isometries of the underlying space.…”

Section: Introductionmentioning

confidence: 99%

A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces

Dolan

Huet

Murray

et al. 2008

“…The integral of a function F ∈ S 2 N over the fuzzy sphere is given by [7][8][9][10] 12) and the inner product can be defined by…”

Section: Quantized Scalar Field On Fuzzy Sphere: Feynman Rulementioning

confidence: 99%

“…The problems of the one loop renormalization of the scalar field on the fuzzy sphere have been studied in some recent papers [7][8][9][10]. The phenomena of the UL/IR mixing [11,12] on the fuzzy sphere are different from those in the noncommutative plane.…”

Section: Introductionmentioning

confidence: 99%

Effective Potential on Fuzzy Sphere

Huang

2002

The effective potential of quantized scalar field on fuzzy sphere is evaluated to the twoloop level. We see that one-loop potential behaves like that in the commutative sphere and the Coleman-Weinberg mechanism of the radiatively symmetry breaking could be also shown in the fuzzy sphere system. In the two-loop level, we use the heavy-mass approximation and the high-temperature approximation to perform the evaluations. The results show that both of the planar and nonplanar Feynman diagrams have inclinations to restore the symmetry breaking in the tree level. However, the contributions from planar diagrams will dominate over those from nonplanar diagrams by a factor N 2 . Thus, at heavy-mass limit or hightemperature system the quantum field on the fuzzy sphere will behave like those on the commutative sphere. We also see that there is a drastic reduction of the degrees of freedom in the nonplanar diagrams when the particle wavelength is smaller than the noncommutativity scale.

“…There is now a substantial body of work on S 2 F , starting from the works of [6,11]. Solitons and monopoles in non-linear σ-models on S 2 F were studied by [8] (see also [7]), while topological issues such as instantons, θ-term and derivation of the chiral anomaly were discussed in [9].…”

Section: Introductionmentioning

confidence: 99%

Scalar Multi-Solitons on the Fuzzy Sphere

Vaidya

2002

We study solitons in scalar theories with polynomial interactions on the fuzzy sphere. Such solitons are described by projection operators of rank k, and hence the moduli space for the solitons is the Grassmannian Gr(k, 2j + 1). The gradient term of the action provides a non-trivial potential on Gr(k, 2j+1), thus reducing the moduli space. We construct configurations corresponding to wellseparated solitons, and show that although the solitons attract each other, the attraction vanishes in the limit of large j. In this limit, it is argued that the moduli space is (CP 1 ) ⊗k /S k ≃ CP k .For the k-soliton bound state, the moduli space is simply CP 1 , all other moduli being lifted. We find that the moduli space of multi-solitons is smooth and that there are no singularities as several solitons coalesce. When the fuzzy S 2 is flattened to a noncommutative plane, we find agreement with the known results, modulo some operator-ordering ambiguities. This suggests that the fuzzy sphere is a natural way to regulate the noncommutative plane both in the ultraviolet and infrared.