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Lectures on Matrix Field Theory pp 207–275Cite as

The Multitrace Approach

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Part of the book series: Lecture Notes in Physics ((LNP,volume 929))

Abstract

In this chapter we apply the powerful multitrace approach to noncommutative \(\Phi ^{4}\) theory on the Moyal-Weyl plane \(\mathbf{R}_{\theta,\Omega }^{2}\) and on the fuzzy sphere S N 2 and employ random matrix theory to solve for the phase structure of the theory. Then a discussion of the planar theory is given in some detail.

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Notes

  1. 1.

    In this article we make the identification T r N  ≡ T r.

  2. 2.

    In this case, the kinetic term is independent of the identity mode in the scalar field \(\hat{\Phi }\).

  3. 3.

    This can be expanded to any order in an obvious way which will be discussed in the next section.

  4. 4.

    The possibility of g negative is more relevant to quantum gravity in two dimensions. Indeed, a second order phase transition occurs at the value g = −μ 2∕12. This is the pure gravity critical point. The corresponding density of eigenvalues ρ(λ) is given by

    $$\displaystyle{ \rho (\lambda ) = \frac{2g} {\pi } (\lambda ^{2} -\delta ^{2})^{\frac{2} {3} }. }$$
    (5.193)
  5. 5.

    This is equivalent, from Eq. (5.179), to the requirement that the eigenvalues density ρ(λ) must be normalized to one.

  6. 6.

    This can be seen by computing the specific heat and observing that its derivative is discontinuous at this point.

  7. 7.

    Again, this is because the potential is even under M ⟶ − M.

  8. 8.

    The case η = 0 can occur, with a non-zero kinetic term, for particular values of \(\sqrt{\omega }\) which can be determined in an obvious way.

  9. 9.

    The range of \(\sqrt{\omega }\) for which η is positive can be determined quite easily.

  10. 10.

    The case of the Moyal-Weyl is similar.

References

  1. Ambjorn, J., Catterall, S.: Stripes from (noncommutative) stars. Phys. Lett. B 549, 253 (2002) [hep-lat/0209106]

    Google Scholar 

  2. Balachandran, A.P., Kurkcuoglu, S., Vaidya, S.: Lectures on fuzzy and fuzzy SUSY physics (2005). arXiv:hep-th/0511114

    Google Scholar 

  3. Bars, I.: U(n) integral for generating functional in lattice gauge theory. J. Math. Phys. 21, 2678 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  4. Bietenholz, W., Hofheinz, F., Nishimura, J.: Phase diagram and dispersion relation of the noncommutative lambda phi**4 model in d = 3. J. High Energy Phys. 0406, 042 (2004) [hep-th/0404020]

    Article  ADS  MathSciNet  Google Scholar 

  5. Brezin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Connes, A.: Noncommutative Geometry. Academic Press, London (1994)

    MATH  Google Scholar 

  7. Creutz, M.: On invariant integration over Su(n). J. Math. Phys. 19, 2043 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Creutz, M.: Quarks, Gluons and Lattices. Cambridge Monographs on Mathematical Physics, 169p. Cambridge University Press, Cambridge (1983)

    Google Scholar 

  9. Das, S.R., Dhar, A., Sengupta, A.M., Wadia, S.R.: New critical behavior in d = 0 large N matrix models. Mod. Phys. Lett. A 5, 1041 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Das, C.R., Digal, S., Govindarajan, T.R.: Finite temperature phase transition of a single scalar field on a fuzzy sphere. Mod. Phys. Lett. A 23, 1781 (2008) [arXiv:0706.0695 [hep-th]]

    Google Scholar 

  11. Eynard, B.: Random matrices. Cours de Physique Theorique de Saclay. Unpublished Lectures

    Google Scholar 

  12. Filev, V.G., O’Connor, D.: On the phase structure of commuting matrix models. J. High Energy Phys. 1408, 003 (2014). doi:10.1007/JHEP08(2014)003. arXiv:1402.2476 [hep-th]

    Google Scholar 

  13. Frohlich, J., Gawedzki, K.: Conformal field theory and geometry of strings. In: Gawedzki, K. (ed.) Proceedings, Mathematical quantum theory, vol. 1, pp. 57–97, 44 p. Vancouver (1993). Preprint

    Google Scholar 

  14. Fulton, W., Harris, J.: Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)

    Google Scholar 

  15. Garcia Flores, F., O’Connor, D., Martin, X.: Simulating the scalar field on the fuzzy sphere. PoS LAT 2005, 262 (2006) [hep-lat/0601012]

    Google Scholar 

  16. Garcia Flores, F., Martin, X., O’Connor, D.: Simulation of a scalar field on a fuzzy sphere. Int. J. Mod. Phys. A 24, 3917 (2009) [arXiv:0903.1986 [hep-lat]]

    Google Scholar 

  17. Golner, G.R.: Calculation of the critical exponent eta via renormalization-group recursion formulas. Phys. Rev. B 8, 339 (1973)

    Article  ADS  Google Scholar 

  18. Gracia-Bondia, J.M., Varilly, J.C.: Algebras of distributions suitable for phase space quantum mechanics. 1. J. Math. Phys. 29, 869 (1988)

    Google Scholar 

  19. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 5th edn. Academic, New York (1980)

    MATH  Google Scholar 

  20. Grosse, H., Wulkenhaar, R.: Renormalization of phi**4 theory on noncommutative R**2 in the matrix base. J. High Energy Phys. 0312, 019 (2003) [hep-th/0307017]

    Article  ADS  MathSciNet  Google Scholar 

  21. Grosse, H., Wulkenhaar, R.: Power counting theorem for nonlocal matrix models and renormalization. Commun. Math. Phys. 254, 91 (2005) [hep-th/0305066]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Grosse, H., Wulkenhaar, R.: Renormalization of phi**4 theory on noncommutative R**4 in the matrix base. Commun. Math. Phys. 256, 305 (2005) [hep-th/0401128]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Gubser, S.S., Sondhi, S.L.: Phase structure of noncommutative scalar field theories. Nucl. Phys. B 605, 395 (2001) [hep-th/0006119]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Hoppe, J.: Quantum theory of a massless relativistic surface and a two-dimensional bound state problem. Ph.D. thesis, MIT (1982)

    Google Scholar 

  25. Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168 (2002) [hep-th/0202039]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of noncommutative field theory in background magnetic fields. Phys. Lett. B 569, 95 (2003) [hep-th/0303082]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of quantum field theory on noncommutative phase spaces. J. High Energy Phys. 0401, 017 (2004) [hep-th/0308043]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Lizzi, F., Spisso, B.: Noncommutative field theory: numerical analysis with the fuzzy disc. Int. J. Mod. Phys. A 27, 1250137 (2012) [arXiv:1207.4998 [hep-th]]

    Google Scholar 

  29. Madore, J.: The fuzzy sphere. Classical Quantum Gravity 9, 69 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Martin, X.: A Matrix phase for the phi**4 scalar field on the fuzzy sphere. J. High Energy Phys. 0404, 077 (2004) [hep-th/0402230]

    Article  ADS  MathSciNet  Google Scholar 

  31. Medina, J., Bietenholz, W., O’Connor, D.: Probing the fuzzy sphere regularisation in simulations of the 3d lambda phi**4 model. J. High Energy Phys. 0804, 041 (2008) [arXiv:0712.3366 [hep-th]]

    Google Scholar 

  32. Mejía-Díaz, H., Bietenholz, W., Panero, M.: The continuum phase diagram of the 2d non-commutative lambda phi**4 model. J. High Energy Phys. 1410, 56 (2014). doi:10.1007/JHEP10(2014)056. arXiv:1403.3318 [hep-lat]

    Google Scholar 

  33. Montvay, I., Munster, G.: Quantum Fields on a Lattice. Cambridge Monographs on Mathematical Physics, 491p. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  34. Nair, V.P., Polychronakos, A.P., Tekel, J.: Fuzzy spaces and new random matrix ensembles. Phys. Rev. D 85, 045021 (2012) [arXiv:1109.3349 [hep-th]]

    Google Scholar 

  35. O’Connor, D., Saemann, C.: Fuzzy scalar field theory as a multitrace matrix model. J. High Energy Phys. 0708, 066 (2007) [arXiv:0706.2493 [hep-th]]

    Google Scholar 

  36. Panero, M.: Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere. J. High Energy Phys. 0705, 082 (2007) [hep-th/0608202]

    Article  ADS  MathSciNet  Google Scholar 

  37. Polychronakos, A.P.: Effective action and phase transitions of scalar field on the fuzzy sphere. Phys. Rev. D 88, 065010 (2013). doi:10.1103/PhysRevD.88.065010. arXiv:1306.6645 [hep-th]

    Google Scholar 

  38. Saemann, C.: The multitrace matrix model of scalar field theory on fuzzy CP**n. SIGMA 6, 050 (2010) [arXiv:1003.4683 [hep-th]]

    Google Scholar 

  39. Shimamune, Y.: On the phase structure of large n matrix models and gauge models. Phys. Lett. B 108, 407 (1982)

    Article  ADS  Google Scholar 

  40. Smit, J.: Introduction to quantum fields on a lattice: a robust mate. Camb. Lect. Notes Phys. 15, 1 (2002)

    MathSciNet  MATH  Google Scholar 

  41. Steinacker, H.: A Non-perturbative approach to non-commutative scalar field theory. J. High Energy Phys. 0503, 075 (2005) [hep-th/0501174]

    Article  ADS  MathSciNet  Google Scholar 

  42. Tekel, J.: Random matrix approach to scalar fields on fuzzy spaces. Phys. Rev. D 87 (8), 085015 (2013) [arXiv:1301.2154 [hep-th]]

    Google Scholar 

  43. Tekel, J.: Uniform order phase and phase diagram of scalar field theory on fuzzy CP**n. J. High Energy Phys. 1410, 144 (2014). doi:10.1007/JHEP10(2014)144. arXiv:1407.4061 [hep-th]

    Google Scholar 

  44. Wilson, K.G., Kogut, J.B.: The Renormalization group and the epsilon expansion. Phys. Rep. 12, 75 (1974)

    Article  ADS  Google Scholar 

  45. Ydri, B.: New algorithm and phase diagram of noncommutative ϕ 4 on the fuzzy sphere. J. High Energy Phys. 1403, 065 (2014) [arXiv:1401.1529 [hep-th]]

    Google Scholar 

  46. Ydri, B.: A multitrace approach to noncommutative \(\Phi _{2}^{4}\). Phys. Rev. D 93 (6), 065041 (2016). doi:10.1103/PhysRevD.93.065041. arXiv:1410.4881 [hep-th]

    Google Scholar 

  47. Ydri, B., Bouchareb, A.: The fate of the Wilson-Fisher fixed point in non-commutative ϕ 4. J. Math. Phys. 53, 102301 (2012) [arXiv:1206.5653 [hep-th]]

    Google Scholar 

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  1. Institute of Physics, BM Annaba University, Annaba, Algeria

    Badis Ydri

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Ydri, B. (2017). The Multitrace Approach. In: Lectures on Matrix Field Theory. Lecture Notes in Physics, vol 929. Springer, Cham. https://doi.org/10.1007/978-3-319-46003-1_5

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