Advertisement

Journal of Statistical Physics

, Volume 10, Issue 4, pp 259–297 | Cite as

Statistical theory of energy levels and random matrices in physics

  • M. Carmeli
Review Article

Abstract

In this paper the physical aspects of the statistical theory of the energy levels of complex physical systems and their relation to the mathematical theory of random matrices are discussed. After a preliminary introduction we summarize the symmetry properties of physical systems. Different kinds of ensembles are then discussed. This includes the Gaussian, orthogonal, and unitary ensembles. The problem of eigenvalue-eigenvector distributions of the Gaussian ensemble is then discussed, followed by a discussion on the distribution of the widths. In the appendices we discuss the symplectic group and quaternions, and the Gaussian ensemble in detail.

Key words

Random matrices eigenvalues multivariate analysis distributions energy levels nuclear physics statistical theory ensembles 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. I. Schiff,Quantum Mechanics, 2nd ed. McGraw-Hill, New York (1955).Google Scholar
  2. 2.
    L. S. Kisslinger and R. A. Sorensen,Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd 32, No. 9 (1960).Google Scholar
  3. 3.
    M. Baranger, Extension of the Shell Model for Heavy Spherical Nuclei,Phys. Rev. 120:957 (1960).Google Scholar
  4. 4.
    F. J. Dyson, Statistical Theory of the Energy Levels of Complex Systems. I,J. Math. Phys. 3:140 (1962).Google Scholar
  5. 5.
    J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Slow Neutron Resonance Spectroscopy. I. U238,Phys. Rev. 118:687 (1960); Slow Neutron Resonance Spectroscopy. II. Ag, Au, Ta,Phys. Rev. 120:2214 (1960).Google Scholar
  6. 6.
    E. P. Wigner, Random Matrices in Physics,SIAM Review 9:1 (1957).Google Scholar
  7. 7.
    H. Goldstein,Classical Mechanics, Addison-Wesley, Cambridge, Massachusetts (1950).Google Scholar
  8. 8.
    B. O. Koopman, Hamiltonian Systems and Transformations in Hilbert Space,Proc. Nat. Acad. Sci. U.S.A. 17:315 (1931).Google Scholar
  9. 9.
    G. D. Birkhoff, Proof of a Recurrence Theorem for Strongly Transitive Systems,Proc. Nat. Acad. Sci. U.S.A. 17:650 (1931).Google Scholar
  10. 10.
    G. D. Birkhoff, Proof of the Ergodic Theorem,Proc. Nat. Acad. Sci. U.S.A. 17:656 (1931).Google Scholar
  11. 11.
    J. von Neumann, Proof of the Quasi-Ergodic Hypothesis,Proc. Nat. Acad. Sci. U.S.A. 18:70 (1932).Google Scholar
  12. 12.
    G. D. Birkhoff and B. O. Koopman, Recent Contributions to the Ergodic Theory,Proc. Nat. Acad. Sci. U.S.A. 18:279 (1932).Google Scholar
  13. 13.
    F. Ajzenberg and T. Lauritsen, Energy Levels of Light Nuclei, V,Rev. Mod. Phys. 27:77 (1955).Google Scholar
  14. 14.
    E. P. Wigner, Statistical Properties of Real Symmetric Matrices with Many Dimensions, inCan. Math. Congr. Proc., Univ. of Toronto Press, Toronto, Canada (1957), p. 174.Google Scholar
  15. 15.
    J. W. Mihelich, G. Scharff-Goldhaber, and M. McKeown, Decay Scheme of the 5.5-Hr Isomer of Hf180,Phys. Rev. 94:794 (1954).Google Scholar
  16. 16.
    D. J. Hughes and J. A. Harvey, Neutron Cross Sections, Brookhaven National Laboratory Technical Report No. 325 (1955).Google Scholar
  17. 17.
    J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Evidence for the 2π decay of theK 2 0 meson, Phys. Rev. Letters 13:138 (1964).Google Scholar
  18. 18.
    E. W. Beier, D. A. Buchholz, A. K. Mann, W. K. McFarlane, S. H. Parker, and J. B. Roberts, Experimental Tests of Discrete Symmetries in the DecaysK ± →π + π e +- v e,Phys. Rev. Letters 29:511 (1972).Google Scholar
  19. 19.
    C. E. Porter, Fluctuations of Quantal Spectra, inStatistical Theories of Spectra: Fluctuations, C. E. Porter, ed., Academic Press, New York (1965).Google Scholar
  20. 20.
    B. van der Pol,Phil. Mag. 26 (7): 921 (1938).Google Scholar
  21. 21.
    B. van der Pol and H. Bremmer,Operational Calculus, Cambridge University Press, London and New York (1959).Google Scholar
  22. 22.
    D. N. Lehmer,List of Prime Numbers from 1 to 10,006,721, Hafner, New York (1956).Google Scholar
  23. 23.
    L. D. Landau and E. M. Lifshitz,Mechanics, Pergamon Press, New York (1960).Google Scholar
  24. 24.
    E. P. Wigner,Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York (1959).Google Scholar
  25. 25.
    N. Rosenzweig, inStatistical Physics, K. W. Ford, ed., W. A. Benjamin, New York (1963).Google Scholar
  26. 26.
    A. Messiah,Quantum Mechanics, Vols. I and II, Wiley, New York (1961).Google Scholar
  27. 27.
    G. C. Wick, Invariance Principles of Nuclear Physics,Ann. Rev. Nucl. Soc. 8:1 (1958).Google Scholar
  28. 28.
    H. A. Kramers,Proc. Acad. Sci. Amsterdam 33:959 (1930).Google Scholar
  29. 29.
    M. Tinkham,Group Theory and Quantum Mechanics, McGraw-Hill, New York (1964).Google Scholar
  30. 30.
    K. Wilson, Proof of a Conjecture by Dyson,J. Math. Phys. 3:1040 (1962).Google Scholar
  31. 31.
    L. K. Hua,Am. J. Math. 66:470 (1944).Google Scholar
  32. 32.
    C. Chevalley,Theory of Lie Groups, Princeton University Press, Princeton, New Jersey (1946).Google Scholar
  33. 33.
    H. Weyl,The Classical Groups, Princeton University Press, Princeton, New Jersey (1946).Google Scholar
  34. 34.
    F. J. Dyson, The Threefold Way: Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics,J. Math. Phys. 3:1199 (1962).Google Scholar
  35. 35.
    E. P. Wigner, On a Class of Analytic Functions from the Quantum Theory of Collisions,Ann. Math. 53:36 (1951).Google Scholar
  36. 36.
    E. P. Wigner, Characteristic Vectors of Bordered Matrices with Infinite Dimensions,Ann. Math. 62:548 (1955).Google Scholar
  37. 37.
    E. P. Wigner, Characteristic Vectors of Bordered Matrices with Infinite Dimensions II,Ann. Math. 65:203 (1957).Google Scholar
  38. 38.
    E. P. Wigner, On the Distribution of the Roots of Certain Symmetric Matrices,Ann. Math. 67:325 (1958).Google Scholar
  39. 39.
    C. E. Porter and N. Rosenzweig, Statistical Properties of Atomic and Nuclear Spectra,Suomalaisen Tiedeakatemian Toimituksia (Ann. Acad. Sci. Fennicae) AVI, No. 44 (1960).Google Scholar
  40. 40.
    C. E. Porter and N. Rosenzweig, Repulsion of Energy Levels in Complex Atomic Spectra,Phys. Rev. 120:1698 (1960).Google Scholar
  41. 41.
    B. V. Baronk, Accuracy of the Semicircle Approximation for the Density of Eigenvalues of Random Matrices,J. Math. Phys. 5:215 (1964).Google Scholar
  42. 42.
    F. Coeseter, The Symmetry of theS Matrix,Phys. Rev. 89:619 (1953).Google Scholar
  43. 43.
    J. B. Grag, J. Rainwater, J. S. Petersen, and W. W. Havens, Jr., Neutron Resonance Spectroscopy, III. Th233 and U238,Phys. Rev. 134:B985 (1964).Google Scholar
  44. 44.
    J. von Neumann and E. P. Wigner,Phys. Z. 30:467 (1929).Google Scholar
  45. 45.
    L. Landan and Ya. Smorodinsky,Lectures on the Theory of the Atomic Nucleus. State Tech.-Theoret. Lit. Press, Moscow (1955).Google Scholar
  46. 46.
    F. J. Dyson, Statistical Theory of the Energy Levels of Complex Systems, II,J. Math. Phys. 3:157 (1962).Google Scholar
  47. 47.
    M. Gaudin, Sur la loi limite de l'espacement des valeurs propres d'une matrice aleatoire,Nucl. Phys. 25:447 (1961).Google Scholar
  48. 48.
    E. P. Wigner,Conference on Neutron Physics by Time of Flight (Gatlinburg, Tennessee, November 1956), Oak Ridge National Lab. Report ORNL-2309 (1957).Google Scholar
  49. 49.
    M. L. Mehta, On the Statistical Properties of the Level-spacings in Nuclear Spectra,Nucl. Phys. 18:395 (1960).Google Scholar
  50. 50.
    M. L. Mehta and M. Gaudin, On the Density of Eigenvalues of a Random Matrix,Nucl. Phys. 18:420 (1960).Google Scholar
  51. 51.
    F. J. Dyson, Statistical Theory of Energy Levels of Complex Systems, III,J. Math. Phys. 3:166 (1962).Google Scholar
  52. 52.
    P. S. Kahn, Energy Level Spacing Distributions,Nucl. Phys. 41:459 (1963).Google Scholar
  53. 53.
    C. E. Porter, Random Matrix Diagonalization-Some Numerical Computations,J. Math. Phys. 4:1039 (1963).Google Scholar
  54. 54.
    F. J. Dyson and M. L. Mehta, Statistical Theory of the Energy Levels of Complex Systems, IV,J. Math. Phys. 4:701 (1963).Google Scholar
  55. 55.
    J. Gunson, Proof of a Conjecture by Dyson in the Statistical Theory of Energy Levels,J. Math. Phys. 3:752 (1962).Google Scholar
  56. 56.
    M. L. Mehta and F. J. Dyson, Statistical Theory of the Energy Levels of Complex Systems, V,J. Math. Phys. 4:713 (1963).Google Scholar
  57. 57.
    C. E. Porter, Further Remarks on Energy Level Spacings,Nucl. Phys. 40:167 (1963).Google Scholar
  58. 58.
    P. B. Khan and C. E. Porter, Statistical Fluctuations of Energy Levels: The Unitary Ensemble,Nucl. Phys. 48:385 (1963).Google Scholar
  59. 59.
    H. S. Leff, Systematic Characterization ofmth-Order Energy Level Spacing distributions,J. Math. Phys. 5:756 (1964).Google Scholar
  60. 60.
    H. S. Leff, Class of Ensembles in the Statistical Theory of Energy Level Spectra,J. Math. Phys. 5:763 (1964).Google Scholar
  61. 61.
    D. Fox and P. B. Khan, Identity of thenth-Order Spacing Distributions for a Class of Hamiltonian,Phys. Rev. 134:B1151 (1964).Google Scholar
  62. 62.
    I. Gurevich and M. I. Pevsher, Repulsion of Nuclear Levels,Nucl. Phys. 2:575 (1956).Google Scholar
  63. 63.
    A. M. Lane, inConference on Neutron Physics by Time of Flight (Gatlinburg, Tennessee, November 1956), Oak Ridge National Lab. Report ORNL-2309 (1957).Google Scholar
  64. 64.
    R. E. Trees, Repulsion of Energy Levels in Complex Atomic Spectra,Phys. Rev. 123:1293 (1961).Google Scholar
  65. 65.
    L. Dresner, Spacings of Nuclear Energy Levels,Phys. Rev. 113:232 (1959).Google Scholar
  66. 66.
    F. J. Dyson, A Brownian, Motion Model for the Eigenvalues of a Random Matrix,J. Math. Phys. 3:1191 (1962).Google Scholar
  67. 67.
    L. D. Favro, P. B. Khan, and M. L. Mehta, Concerning Polynomials Encountered in the Study of the Distribution Function of Spacings, Brookhaven National Lab. Report No. BNL-757 (T-280), September 1932.Google Scholar
  68. 68.
    J. M. Scott,Phil. Mag. 45:1322 (1952).Google Scholar
  69. 69.
    C. E. Porter and R. G. Thomas, Fluctuations of Nuclear Reaction Widths,Phys. Rev. 104:483 (1956).Google Scholar
  70. 70.
    J. A. Harvey, D. J. Hughes, R. S. Carter, and V. E. Pilcher, Spacings and Neutron Widths of Nuclear Energy Levels,Phys. Rev. 99:10 (1955).Google Scholar
  71. 71.
    D. J. Hughes and J. A. Harvey, Size Distribution of Neutron Widths,Phys. Rev. 99:1032 (1955).Google Scholar
  72. 72.
    C. E. Porter, Statistics of Atomic Radiative Transition Probabilities,Phys. Letters 2:292 (1962).Google Scholar
  73. 73.
    N. Ullah and C. E. Porter, Expectation Value Fluctuations in the Unitary Ensemble,Phys. Rev. 132:948 (1963).Google Scholar
  74. 74.
    N. Ullah and C. E. Porter, Invariance Hypothesis and Hamiltonian Matrix Elements Correlations,Phys. Letters 6:301 (1963).Google Scholar
  75. 75.
    N. Rosenzweig, Anomalous Statistics of Partial Radiation Widths,Phys. Letters 6:123 (1963).Google Scholar
  76. 76.
    J. B. Grag,Statistical Properties of Nuclei, Plenum Press, New York (1972).Google Scholar
  77. 77.
    E. P. Wigner, Introductory Talk, in Ref. 78, p. 7.Google Scholar
  78. 78.
    G. Joos,Introduction to Theoretical Physics, Hafner Publication Co., New York (1958).Google Scholar
  79. 79.
    J. B. French and S. S. Wong, Validity of Random Matrix Theories for ManyParticle Systems,Phys. Letters 33B:449 (1970).Google Scholar
  80. 80.
    O. Bohigrs and J. Flores, Two-Body Random Hamiltonian and Level Density,Phys. Letters 34B:261 (1971).Google Scholar
  81. 81.
    J. B. French and F. S. Chang, Distribution Theory of Nuclear Level Densities and Related Quantities, Ref. 78, p. 405.Google Scholar
  82. 82.
    O. Bohigrs and J. Flores, Some Properties of Level Spacing Distributions, Ref. 78, p. 195.Google Scholar
  83. 83.
    R. Balian, Random Matrices and Information Theory,Nuovo Cimento B57:183 (1968).Google Scholar
  84. 84.
    S. N. Roy,Some Aspects of Multivariate Analysis, Wiley (1957).Google Scholar
  85. 85.
    A. T. James, Distributions of Matrix Variates and Latent Roots Derived from Normal Samples,Ann. Math. Stat. 35:475 (1964).Google Scholar
  86. 86.
    P. R. Krishnaiah and T. C. Chang, On the Exact Distributions of the Extreme Roots of the Wishart and Manova Matrices,J. Multivariate Analysis 1:108 (1971).Google Scholar
  87. 87.
    P. R. Krishnaiah and V. B. Waikar, Exact Distributions of Any Few Ordered Roots of a Class of Random Matrices,J. Multivariate Analysis 1:308 (1971).Google Scholar
  88. 88.
    P. R. Krishnaiah and A. K. Chattopadhyay, review article in preparation.Google Scholar
  89. 89.
    L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, inTranslations of Mathematical Monographs, Vol. 6, Am. Math. Soc., Providence, Rhode Island (1963); L. K. Hua,American Mathematical Society Translations, Vols. 32 and 33, Am. Math. Soc., Providence, Rhode Island (1963).Google Scholar
  90. 90.
    M. A. Naimark,Linear Representations of the Lorentz Group, Pergamon Press, London (1964).Google Scholar

Copyright information

© Plenum Publishing Corporation 1974

Authors and Affiliations

  • M. Carmeli
    • 1
  1. 1.Department of PhysicsUniversity of NegevBeer ShevaIsrael

Personalised recommendations