A numerical approach to harmonic non-commutative spectral field theory

International Journal of Modern Physics A (Impact Factor: 1.7). 11/2011; 27(14). DOI: 10.1142/S0217751X12500753
Source: arXiv


We present a first numerical investigation of a non-commutative gauge theory
defined via the spectral action for Moyal space with harmonic propagation. This
action is approximated by finite matrices. Using Monte Carlo simulation we
study various quantities such as the energy density, the specific heat density
and some order parameters, varying the matrix size and the independent
parameters of the model. We find a peak structure in the specific heat which
might indicate possible phase transitions. However, there are mathematical
arguments which show that the limit of infinite matrices is very different from
the original spectral model.

Full-text preview

Available from: ArXiv
  • Source
    • "However, we can expect some advantages of the fuzzy approach with the simulations of other field theories, in particular the fermionic theories or super symmetric models [14]. Beside another recent field of application the matrix models are the Yang-Mills actions [15] [16] [17] in which appear the concept of the emerging geometry. "
    [Show abstract] [Hide abstract]
    ABSTRACT: The fuzzy disc is a discretization of the algebra of functions on the two dimensional disc using finite matrices which preserves the action of the rotation group. We define a $\varphi^4$ scalar field theory on it and analyze numerically for three different limits for the rank of the matrix going to infinity. The numerical simulations reveal three different phases: uniform and disordered phases already the present in the commutative scalar field theory and a nonuniform ordered phase as a noncommutative effects. We have computed the transition curves between phases and their scaling. This is in agreement with studies on the fuzzy sphere, although the speed of convergence for the disc seems to be better. We have performed also three the limits for the theory in the cases of the theory going to the commutative plane or commutative disc. In this case the theory behaves differently, showing the intimate relationship between the nonuniform phase and noncommutative geometry.
    Preview · Article · Jul 2012 · International Journal of Modern Physics A
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We review the matrix bases for a family of noncommutative $\star$ products based on a Weyl map. These products include the Moyal product, as well as the Wick-Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix bases. We discuss the uses of these matrix bases for field theory, fuzzy spaces and emergent gravity.
    Full-text · Article · Mar 2014 · Symmetry Integrability and Geometry Methods and Applications
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is presented, and their implications are reviewed. In addition, we also discuss how related numerical techniques have been recently applied in computer simulations of dimensionally reduced supersymmetric theories.
    Preview · Article · Jan 2016