Abstract

Phase transition problems in the one-matrix models in QCD are discussed in this book by using string equations. The phase transition models are formulated by using eigenvalue density which represents the momentum operator described in quantum mechanics. By using orthogonal polynomials, the eigenvalue densities in various matrix models can be unified. The string equation establishes a connection between the position and momentum aspects described in the Heisenberg uncertainty principle, which is important and usually hard to find in other methods. The recursion formula of the orthogonal polynomials can be applied to introduce an index folding technique to reorganize the wave functions in order to achieve a renormalization in the momentum aspect. It will be discussed that the critical phenomenon associated with the Gross-Witten model can be found by using Toda lattice and double scaling. The hypergeometric-type differential equations improve on some shortages of integrable systems to work on physical problems, such as the fact that a soliton system does not have a differential equation along the spectrum direction, and illustrate a new background to study the singularities of physical quantities, such as mass.

Keywords

Critical point Gross-Witten model Heisenberg uncertainty principle Matrix model String equation Unified model 

References

  1. 1.
    Akhiezer, N.I.: Elements of the Theory of Elliptic Functions. Translations of Math. Monographs, vol. 79. AMS, Providence (1990) MATHGoogle Scholar
  2. 2.
    Bertola, M.: Free energy of the two-matrix model/dToda tau-function. Nucl. Phys. B 669, 435–461 (2003) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bertola, M., Marchal, O.: The partition of the two-matrix models as an isomonodromic τ function. J. Math. Phys. 50, 013529 (2009) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bleher, P., Eynard, B.: Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations. J. Phys. A 36, 3085–3106 (2003) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brézin, E., Hikami, S.: Intersection theory from duality and replica. Commun. Math. Phys. 283, 507–521 (2008) MATHCrossRefGoogle Scholar
  6. 6.
    Brézin, E., Kazakov, V.A.: Exactly solvable field theories of closed strings. Phys. Lett. B 236, 144–150 (1990) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35–51 (1978) MATHCrossRefGoogle Scholar
  8. 8.
    Chekhov, L., Mironov, A.: Matrix models vs. Seiberg–Witten/Whitham theories. Phys. Lett. B 552, 293–302 (2003) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dijkgraaf, R., Moore, G.W., Plesser, R.: The partition function of 2-D string theory. Nucl. Phys. B 394, 356–382 (1993) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Douglas, M.R.: Strings in less than one dimension and the generalized KdV hierarchies. Phys. Lett. B 238, 176–180 (1990) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Douglas, M.R., Kazakov, V.A.: Large N phase transition in continuum QCD in two-dimensions. Phys. Lett. B 319, 219–230 (1993) CrossRefGoogle Scholar
  12. 12.
    Douglas, M.R., Shenker, S.H.: Strings in less than one dimension. Nucl. Phys. B 335, 635–654 (1990) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dutkay, D.E., Jorgensen, P.E.T.: Hilbert spaces built on a similarity and on dynamical renormalization. J. Math. Phys. 47(5), 053504 (2006) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Eynard, B.: Large-N expansion of the 2 matrix model. J. High Energy Phys. 1, 051 (2003) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Eynard, B., Orantin, N.: Mixed correlation functions in the 2-matrix model, and the Bethe ansatz. J. High Energy Phys. 08, 028 (2005) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fokas, A.S., Its, A.R., Kitaev, A.V.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142, 313–344 (1991) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Fuji, H., Mizoguchi, S.: Remarks on phase transitions in matrix models and N=1 supersymmetric gauge theory. Phys. Lett. B 578, 432–442 (2004) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten exact solution. Phys. Lett. B 355, 466–477 (1995) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gross, D.J.: Some remarks about induced QCD. Phys. Lett. B 293, 181–186 (1992) CrossRefGoogle Scholar
  20. 20.
    Gross, D.J., Migdal, A.A.: Nonperturbative two-dimensional quantum gravity. Phys. Rev. Lett. 64, 127–130 (1990) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Gross, D.J., Newman, M.J.: Unitary and Hermitian matrices in an external field II: the Kontsevich model and continuum Virasoro constraints. Nucl. Phys. B 380, 168–180 (1992) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gross, D.J., Witten, E.: Possible third-order phase transition in the large-N lattice gauge theory. Phys. Rev. D 21, 446–453 (1980) CrossRefGoogle Scholar
  23. 23.
    Harish-Chandra: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87–120 (1957) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Hille, E.: Analytic Function Theory, vols. 1, 2. Chelsea, New York (1974) Google Scholar
  25. 25.
    Hisakado, M.: Unitary matrix model and the Painlevé III. Mod. Phys. Lett. A 11, 3001–3010 (1996) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Hisakado, M., Wadati, M.: Matrix models of two-dimensional gravity and discrete Toda theory. Mod. Phys. Lett. A 11, 1797–1806 (1996) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Itzykson, C., Zuber, J.B.: The planar approximation II. J. Math. Phys. 21, 411–421 (1980) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. Physica D 2, 306–352 (1981) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Jorgensen, P.E.T.: Analysis and Probability: Wavelets, Signals, Fractals. Graduate Texts in Mathematics, vol. 234. Springer, New York (2006) Google Scholar
  30. 30.
    Jorgensen, P.E.T., Song, M.-S.: Comparison of Discrete and Continuous Wavelet Transforms. Springer Encyclopedia of Complexity and Systems Science. Springer, Berlin (2008) Google Scholar
  31. 31.
    Jurkiewicz, J.: Regularization of one-matrix models. Phys. Lett. B 235, 178–184 (1990) MathSciNetGoogle Scholar
  32. 32.
    Kazakov, V.A., Migdal, A.A.: Induced QCD at large N. Nucl. Phys. B 397, 214–238 (1993) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Landau, L.D., Lifshitz, E.M.: Statistical Physics, Part 1. Course of Theoretical Physics, vol. 5, 3rd edn. Butterworth–Heinemann, Oxford (1980) Google Scholar
  34. 34.
    Lee, M.H.: Solutions of the generalized Langevin equation by a method of recurrence relations. Phys. Rev. B 26, 2547–2551 (1982) MathSciNetCrossRefGoogle Scholar
  35. 35.
    Loutsenko, I.M., Spiridonov, V.P.: A critical phenomenon in solitonic Ising chains. SIGMA 3, 059 (2007) MathSciNetGoogle Scholar
  36. 36.
    Marcenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Mat. Sb. 72(114)(4), 507–536 (1967) MathSciNetGoogle Scholar
  37. 37.
    McKay, B.D.: The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40, 203–216 (1981) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    McLeod, J.B., Wang, C.B.: Discrete integrable systems associated with the unitary matrix model. Anal. Appl. 2, 101–127 (2004) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    McLeod, J.B., Wang, C.B.: Eigenvalue density in Hermitian matrix models by the Lax pair method. J. Phys. A, Math. Theor. 42, 205205 (2009) MathSciNetCrossRefGoogle Scholar
  40. 40.
    McMullen, C.T.: Complex Dynamics and Renormalization. Annals of Mathematics Studies, vol. 135. Princeton University Press, Princeton (1994) Google Scholar
  41. 41.
    Mehta, M.L.: A method of integration over matrix variables. Commun. Math. Phys. 79, 327–340 (1981) MATHCrossRefGoogle Scholar
  42. 42.
    Mehta, M.L.: Random Matrices, 3rd edn. Academic Press, New York (2004) MATHGoogle Scholar
  43. 43.
    Migdal, A.A.: Phase transitions in induced QCD. Mod. Phys. Lett. A 8, 153–166 (1993) MathSciNetCrossRefGoogle Scholar
  44. 44.
    Mironov, A., Morozov, A., Semeno, G.W.: Unitary matrix integrals in the framework of generalized Kontsevich model. 1. Brézin-Gross-Witten model. Int. J. Mod. Phys. A 11, 5031–5080 (1996) MATHCrossRefGoogle Scholar
  45. 45.
    Newell, A.C.: Solitons in Mathematics and Physics. SIAM, Philadelphia (1985) CrossRefGoogle Scholar
  46. 46.
    Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett. 64, 1326–1329 (1990) CrossRefGoogle Scholar
  47. 47.
    Periwal, V., Shevitz, D.: Exactly solvable unitary matrix models: multicritical potentials and correlations. Nucl. Phys. B 344, 731–746 (1990) MathSciNetCrossRefGoogle Scholar
  48. 48.
    Rossi, P., Campostrini, M., Vicari, E.: The large-N expansion of unitary-matrix models. Phys. Rep. 302, 143–209 (1998) MathSciNetCrossRefGoogle Scholar
  49. 49.
    Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994). Erratum, ibid. B 430, 485–486 (1994) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Sen, S.: Exact solution of the Heisenberg equation of motion for the surface spin in a semi-infinite \(S={1 \over2} X Y\) chain at infinite temperatures. Phys. Rev. B 44, 7444–7450 (1991) CrossRefGoogle Scholar
  51. 51.
    Sengupta, A.M., Mitra, P.P.: Distributions of singular values for some random matrices. Phys. Rev. E 60, 3389–3392 (1991) CrossRefGoogle Scholar
  52. 52.
    Shimamune, Y.: On the phase structure of large N matrix models and gauge models. Phys. Lett. B 108, 407–410 (1982) CrossRefGoogle Scholar
  53. 53.
    Simon, B.: Orthogonal Polynomials on the Unit Circle, vol. 1: Classical Theory. AMS Colloquium Series. AMS, Providence (2005) Google Scholar
  54. 54.
    Simon, B.: Orthogonal Polynomials on the Unit Circle, vol. 2: Spectral Theory. AMS Colloquium Series. AMS, Providence (2005) Google Scholar
  55. 55.
    Stanley, H.E.: Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, Oxford (1971) Google Scholar
  56. 56.
    Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society Colloquium Publications, vol. 23. AMS, Providence (1975) MATHGoogle Scholar
  57. 57.
    Vafa, C.: Geometry of grand unification arXiv:0911.3008 (2009)
  58. 58.
    Wang, C.B.: Orthonormal polynomials on the unit circle and spatially discrete Painlevé II equation. J. Phys. A 32, 7207–7217 (1999) MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955) MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–328 (1958) MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Yeomans, J.M.: Statistical Mechanics of Phase Transitions. Oxford University Press, London (1994) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • C. B. Wang
    • 1
  1. 1.Institute of AnalysisTroyUSA

Personalised recommendations