Fuzzy scalar field theory as a multitrace matrix model

Abstract

We develop an analytical approach to scalar field theory on the fuzzy sphere based on considering a perturbative expansion of the kinetic term. This expansion allows us to integrate out the angular degrees of freedom in the hermitian matrices encoding the scalar field. The remaining model depends only on the eigenvalues of the matrices and corresponds to a multitrace hermitian matrix model. Such a model can be solved by standard techniques as e.g. the saddlepoint approximation. We evaluate the perturbative expansion up to second order and present the one-cut solution of the saddle-point approximation in the large N limit. We apply our approach to a model which has been proposed as an appropriate regularization of scalar field theory on the plane within the framework of fuzzy geometry.

Fuzzy spaces and new random matrix ensembles

“…Similarly, the distribution of eigenvalues of B is also a Wigner semicircle with radius R = 2 √ g. In this sense, the Wigner semicircle is very robust, arising basically from the planarity property of expectation values of matrix observables in the large-N limit. This result is at odds with the result of [8] which gives a polynomially deformed Wigner distribution in the presence of the kinetic term in the action. The source of the discrepancy could be the fact that that the kinetic term becomes dominant in the large-N limit, while it was treated perturbatively in [8].…”

“…This result is at odds with the result of [8] which gives a polynomially deformed Wigner distribution in the presence of the kinetic term in the action. The source of the discrepancy could be the fact that that the kinetic term becomes dominant in the large-N limit, while it was treated perturbatively in [8].…”

“…Not surprisingly, there have been attempts to understand the role of and generalize the Wigner distribution for fuzzy spaces [8,9]. The authors of [8], in particular, start with the mass term as the leading term of the action and integrate out the angular modes which appear in the kinetic term, treating this term as a perturbation.…”

“…The authors of [8], in particular, start with the mass term as the leading term of the action and integrate out the angular modes which appear in the kinetic term, treating this term as a perturbation. There is, however, no guarantee of continuously connecting this to the zero mass limit because of possible phase transitions as a function of N and the lack of summability of the perturbation series.…”

The numerical approach to quantum field theory in a noncommutative space

“…[52]), in a tridimensional version of the model [53,54], at finite temperature [55], in a multi-trace formulation [56,57], et c. 1 The results of these numerical simulations can be compared with several analytical studies, including refs. [5,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76].…”