Field Theory on the q-deformed Fuzzy Sphere II: Quantization

TL;DR

This paper develops a second quantization framework for field theory on the q-deformed fuzzy sphere, maintaining symmetry and positivity, and provides methods for calculating correlators with applications to scalar and gauge fields.

Contribution

It introduces a systematic path-integral quantization approach on the q-deformed fuzzy sphere, including techniques for correlator computation and extensions to gauge fields and higher dimensions.

Findings

Successfully computes 4-point correlator for free scalar field

Analyzes planar tadpole contribution in  theory

Provides a framework applicable beyond 2 dimensions

Abstract

We study the second quantization of field theory on the q-deformed fuzzy sphere for real q. This is performed using a path-integral over the modes, which generate a quasiassociative algebra. The resulting models have a manifest U_q(su(2)) symmetry with a smooth limit q -> 1, and satisfy positivity and twisted bosonic symmetry properties. A systematic way to calculate n-point correlators in perturbation theory is given. As examples, the 4-point correlator for a free scalar field theory and the planar contribution to the tadpole diagram in \phi^4 theory are computed. The case of gauge fields is also discussed, as well as an operator formulation of scalar field theory in 2_q + 1 dimensions. An alternative, essentially equivalent approach using associative techniques only is also presented. The proposed framework is not restricted to 2 dimensions.