Fuzzy scalar field theory as matrix quantum mechanics

  • Matthias Ihl
  • Christoph Sachse
  • Christian Sämann
Article

Abstract

We study the phase diagram of scalar field theory on a three dimensional Euclidean spacetime whose spatial component is a fuzzy sphere. The corresponding model is an ordinary one-dimensional matrix model deformed by terms involving fixed external matrices. These terms can be approximated by multitrace expressions using a group theoretical method developed recently. The resulting matrix model is accessible to the standard techniques of matrix quantum mechanics.

Keywords

Matrix Models Non-Commutative Geometry Field Theories in Lower Dimensions 

References

  1. [1]
    F.A. Berezin, General concept of quantization, Commun. Math. Phys. 40 (1975) 153 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  3. [3]
    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] [SPIRES].MATHCrossRefGoogle Scholar
  4. [4]
    H. Steinacker, A non-perturbative approach to non-commutative scalar field theory, JHEP 03 (2005) 075 [hep-th/0501174] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    H. Steinacker, Quantization and eigenvalue distribution of noncommutative scalar field theory, hep-th/0511076 [SPIRES].
  6. [6]
    D. O’Connor and C. Sämann, Fuzzy scalar field theory as a multitrace matrix model, JHEP 08 (2007) 066 [arXiv:0706.2493] [SPIRES].CrossRefGoogle Scholar
  7. [7]
    D. O’Connor and C. Sämann, A multitrace matrix model from fuzzy scalar field theory, arXiv:0709.0387[SPIRES].
  8. [8]
    C. Sämann, The multitrace matrix model of scalar field theory on fuzzy CP n, SIGMA 6 (2010) 050 [arXiv:1003.4683] [SPIRES].Google Scholar
  9. [9]
    X. Martin, A matrix phase for the ϕ4 scalar field on the fuzzy sphere, JHEP 04 (2004) 077 [hep-th/0402230] [SPIRES].ADSCrossRefGoogle Scholar
  10. [10]
    F. Garcia Flores, D. O’Connor and X. Martin, Simulating the scalar field on the fuzzy sphere, PoS(LAT2005)262 [hep-lat/0601012] [SPIRES].
  11. [11]
    M. Panero, Numerical simulations of a non-commutative theory: The scalar model on the fuzzy sphere, JHEP 05 (2007) 082 [hep-th/0608202] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    M. Panero, Quantum field theory in a non-commutative space: theoretical predictions and numerical results on the fuzzy sphere, SIGMA 2 (2006) 081 [hep-th/0609205] [SPIRES].MathSciNetGoogle Scholar
  13. [13]
    C.R. Das, S. Digal and T.R. Govindarajan, Finite temperature phase transition of a single scalar field on a fuzzy sphere, Mod. Phys. Lett. A 23 (2008) 1781 [arXiv:0706.0695] [SPIRES].ADSGoogle Scholar
  14. [14]
    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] [SPIRES].ADSGoogle Scholar
  15. [15]
    E. Brezin, C. Itzykson, G. Parisi and J.B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  16. [16]
    A. Jevicki and B. Sakita, The quantum collective field method and its application to the planar limit, Nucl. Phys. B 165 (1980) 511 [SPIRES].ADSCrossRefGoogle Scholar
  17. [17]
    F. Sugino and O. Tsuchiya, Critical behavior in c =1 matrix model with branching interactions, Mod. Phys. Lett. A9 (1994) 3149 [hep-th/9403089] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    S.S. Gubser and I.R. Klebanov, A modified c =1 matrix model with new critical behavior, Phys. Lett. B 340 (1994) 35 [hep-th/9407014] [SPIRES].MathSciNetADSGoogle Scholar
  19. [19]
    I.R. Klebanov, Touching random surfaces and Liouville gravity, Phys. Rev. D 51 (1995) 1836 [hep-th/9407167] [SPIRES].MathSciNetADSGoogle Scholar
  20. [20]
    W. Bietenholz, F. Hofheinz and J. Nishimura, Phase diagram and dispersion relation of the non-commutative lambda ϕ4 model in D = 3, JHEP 06 (2004) 042 [hep-th/0404020] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    J. Medina, W. Bietenholz, F. Hofheinz and D. O’Connor, Field theory simulations on a fuzzy sphere: An alternative to the lattice, PoS(LAT2005)263 [hep-lat/0509162] [SPIRES].
  22. [22]
    J. Medina, W. Bietenholz and D. O’Connor, Probing the fuzzy sphere regularisation in simulations of the 3D λϕ4 model, JHEP 04 (2008) 041 [arXiv:0712.3366] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    C. Iuliu-Lazaroiu, D. McNamee and C. Sämann, Generalized Berezin quantization, Bergman metrics and fuzzy laplacians, JHEP 09 (2008) 059 [arXiv:0804.4555] [SPIRES].ADSCrossRefGoogle Scholar
  24. [24]
    P.H. Ginsparg and G.W. Moore, Lectures on 2D gravity and 2D string theory, hep-th/9304011 [SPIRES].
  25. [25]
    J.A. Shapiro, A test of the collective field method for the N →∞ limit, Nucl. Phys. B 184 (1981) 218 [SPIRES].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Matthias Ihl
    • 1
  • Christoph Sachse
    • 2
  • Christian Sämann
    • 3
    • 4
  1. 1.Instituto de FísicaUniversidade Federal do Rio de JaneiroRio de JaneiroBrasil
  2. 2.Fachbereich Mathematik, Bereich Algebra und ZahlentheorieUniversität HamburgHamburgDeutschland
  3. 3.Department of MathematicsHeriot-Watt UniversityEdinburghU.K.
  4. 4.Maxwell Institute for Mathematical SciencesEdinburghU.K.

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