Abstract
We describe a new way of rewriting the partition function of scalar field theory on fuzzy complex projective spaces as a solvable multitrace matrix model. This model is given as a perturbative high-temperature expansion. At each order, we present an explicit analytic expression for most of the arising terms; the remaining terms are computed explicitly up to fourth order. The method presented here can be applied to any model of hermitian matrices. Our results confirm constraints previously derived for the multitrace matrix model by Polychronakos. A further implicit expectation about the shape of the multitrace terms is however shown not to be true.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [INSPIRE].
H. Grosse, C. Klimčík and P. Prešnajder, Towards finite quantum field theory in noncommutative geometry, Int. J. Theor. Phys. 35 (1996) 231 [hep-th/9505175] [INSPIRE].
X. Martin, A matrix phase for the ϕ 4 scalar field on the fuzzy sphere, JHEP 04 (2004) 077 [hep-th/0402230] [INSPIRE].
F. Garcia Flores, D. O’Connor and X. Martin, Simulating the scalar field on the fuzzy sphere, PoS(LAT2005)262 [hep-lat/0601012] [INSPIRE].
M. Panero, Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere, JHEP 05 (2007) 082 [hep-th/0608202] [INSPIRE].
F. Garcia Flores, X. Martin and D. O’Connor, Simulation of a scalar field on a fuzzy sphere, Int. J. Mod. Phys. A 24 (2009) 3917 [arXiv:0903.1986] [INSPIRE].
B. Ydri, New algorithm and phase diagram of noncommutative ϕ 4 on the fuzzy sphere, JHEP 03 (2014) 065 [arXiv:1401.1529] [INSPIRE].
H. Steinacker, A non-perturbative approach to non-commutative scalar field theory, JHEP 03 (2005) 075 [hep-th/0501174] [INSPIRE].
D. O’Connor and C. Sämann, Fuzzy scalar field theory as a multitrace matrix model, JHEP 08 (2007) 066 [arXiv:0706.2493] [INSPIRE].
C. Sämann, The multitrace matrix model of scalar field theory on fuzzy CP n, SIGMA 6 (2010) 050 [arXiv:1003.4683] [INSPIRE].
M. Ihl, C. Sachse and C. Sämann, Fuzzy scalar field theory as matrix quantum mechanics, JHEP 03 (2011) 091 [arXiv:1012.3568] [INSPIRE].
A.P. Polychronakos, Effective action and phase transitions of scalar field on the fuzzy sphere, Phys. Rev. D 88 (2013) 065010 [arXiv:1306.6645] [INSPIRE].
J. Tekel, Uniform order phase and phase diagram of scalar field theory on fuzzy CP n, JHEP 10 (2014) 144 [arXiv:1407.4061] [INSPIRE].
B. Ydri, A multitrace approach to noncommutative Φ 42 , arXiv:1410.4881 [INSPIRE].
V.P. Nair, A.P. Polychronakos and J. Tekel, Fuzzy spaces and new random matrix ensembles, Phys. Rev. D 85 (2012) 045021 [arXiv:1109.3349] [INSPIRE].
F.A. Berezin, General concept of quantization, Commun. Math. Phys. 40 (1975) 153 [INSPIRE].
J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. thesis, http://hdl.handle.net/1721.1/15717, MIT, U.S.A. (1982).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1412.6255
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Sämann, C. Bootstrapping fuzzy scalar field theory. J. High Energ. Phys. 2015, 44 (2015). https://doi.org/10.1007/JHEP04(2015)044
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2015)044