Quantum Field Theory in a Non-Commutative Space: Theoretical Predictions and Numerical Results on the Fuzzy Sphere

Abstract

Abstract. We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to the scalar model. Then we comment on the results recently obtained from Monte Carlo simulations, and show a preview of new numerical data, which are consistent with the expected transition between two phases characterised by the topology of the support of a matrix eigenvalue distribution.

Spontaneous Symmetry Breakdown in Fuzzy Spheres

“…Earlier studies on simulations have focused on scalar fields on fuzzy spheres (S 2 F ) and led to the demonstration of new phases characterised as nonuniform ordered phase [7,9,10,11]. In the continuum infinite volume limit this nonuniform phase appears as stripe phase in the GroenewoldMoyal space [12,13].…”

“…This suggests the possibility that fuzzy spaces may serve as novel regulators of field theories and bring out new features which were absent in the conventional lattice regularisations. This program falls well within the front line of research activity of study of noncommutative geometry [1,2,3,4,6,7,8,9,10,11].…”

The numerical approach to quantum field theory in a noncommutative space

“…Several works [13,14,15,16,25,49] studied the phase structure of NC φ 4 theory in two dimensions, finding numerical evidence for a striped phase [50] characterized by non-uniform order. As an example, fig.…”

“…This type of calculation has been carried out by Monte Carlo methods in a number of works [13,14,15,16,17] (including a very recent study of the entanglement entropy [18]). A different way to map NC scalar field theory in two dimensions to a finite-dimensional matrix model was introduced in ref.…”

SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry

“…Field theory models on supersphere have already been proposed; a non-linear sigma model and a scalar field model were explored in [19] and [20], respectively. Numerical calculations on fuzzy spaces have also been carried out (see [21] as a review). The SUSY Landau problems on higher dimensional coset supermanifolds were developed in [22,23].…”