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Quantum Signatures of Chaos pp 111–203Cite as

Random-Matrix Theory

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Abstract

A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces .

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Notes

  1. 1.

    The reader’s attention is drawn to the overall factor 2 in the exponent of (5.2.19); it stems from Kramers’ degeneracy. When dealing with the GSE, some authors choose to redefine the trace operation as one half the usual trace, so as to account for only a single eigenvalue in each Kramers’ doublet; correspondingly, these authors take the determinant as the square root of the usual one. Throughout this book we will keep the standard definitions.

  2. 2.

    In Sect. 10.8, an alternative method for dealing with problems of this kind will be presented.

  3. 3.

    Actually, (5.12.13) requires l 1≠l 2≠ … ≠l n. When some of the l i coincide such that k different values are assumed, 1 ≤ k ≤ n, with multiplicities n 1, n 2, … , n k and \(\sum n_{i} = n ,\) the factor 1∕n! must be replaced by 1∕(n 1!n 2! … n k!). For the Taylor coefficients E l, given in (5.12.18) or (5.12.21), the distinction in question is irrelevant, due to the appearance of another factor n 1! … n k!∕n! accompanying the summation over the configurations {l i}.

  4. 4.

    For the sake of typographical convenience, δ(m, n) here denotes the Kronecker symbol usually written as δ mn.

  5. 5.

    Analytic continuation to imaginary a ij yields the Fresnel integrals to be met in Chap. 10.

  6. 6.

    The two-point function of the GUE will be treated in Chap. 6.

  7. 7.

    The (N →∞) limit K(τ) of the form factor is related to the Fourier transform \(\tilde {Y}(\tau )\) of the cluster function Y (e) by \(K(\tau )= \delta (\tau )+1-\tilde {Y}(\tau )\).

  8. 8.

    By dynamic rather than random-matrix arguments, a case could be construed for a possibly competing scale, the “Ehrenfest” time \(\tau _E=N^{-1}\ln N\) alias \(n_E=\ln N\), i.e., the time scale on which a perturbation of the Floquet operator F becomes noticeable in expectation values of observables, given global classical chaos.

  9. 9.

    Newton was concerned with relationships between different symmetric functions of N variables [40].

  10. 10.

    Note that in (5.15.3), (5.15.4), (5.15.5), in contrast to the proceeding subsection, we depart from Mehta’s sign convention; readers not willing to put up with such wicked confusion may return to the random-matrix path of virtue by C n → (−1)n−1C n.

  11. 11.

    The symbol \([\frac {N}{2}]\) denotes N∕2 for even N and the next smaller integer for odd N.

  12. 12.

    Like most authors in random-matrix theory, Mehta [7] included, we keep here to a sign tradition differing from the otherwise more widespread one by the factor(−1)n−1 for the cumulant C n.

  13. 13.

    To remove the following quotation marks, think of the rows of the determinant swapped pairwise, the first with the last, the second with the last but one, etc.

  14. 14.

    Indeed, generalizing the joint distribution of eigenvalues (5.4.1) to arbitrary real β = n − 1, one is led to a Wigner distribution \(P_{\beta }(S)=AS^{\beta }\mathrm {e}^{-BS^2}\) with A and B fixed by normalization and \(\overline {S}=1\) which for β →∞ approaches the delta function δ(S − 1).

  15. 15.

    The reader might (and should!) wonder whether the “orthogonal” version of (5.20.1) goes “symplectic” for half-integer j. It does not. The reason is T 2 = +1 for all j in that case since two components of an angular momentum can simultaneously be given real representations.

  16. 16.

    The reader is invited to check the equivalence of (5.17.21) and (5.20.4).

References

  1. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984); J. de Phys. Lett. 45, 1015 (1984)

    Google Scholar 

  2. M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A 356, 375 (1977)

    Article  ADS  Google Scholar 

  3. S.W. McDonald, A.N. Kaufman, Phys. Rev. Lett. 42, 1189 (1979)

    Article  ADS  Google Scholar 

  4. G. Casati, F. Valz-Gris, I. Guarneri, Lett. Nouvo Cimento 28, 279 (1980)

    Article  Google Scholar 

  5. M.V. Berry, Ann. Phys. (USA) 131, 163 (1981)

    Article  ADS  Google Scholar 

  6. A. Altland, M.R. Zirnbauer, Phys. Rev. B 55, 1142 (1997)

    Article  ADS  Google Scholar 

  7. M.L. Mehta, Random Matrices (Academic, New York, 1967); 2nd edition 1991; 3rd edition (Elsevier, 2004)

    Google Scholar 

  8. C.E. Porter (ed.), Statistical Theories of Spectra (Academic, New York, 1965)

    Google Scholar 

  9. T. Guhr, A. Müller-Groeling, H.A. Weidenmüller, Phys. Rep. 299, 189 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  10. M.A. Stephanov, J.J.M. Verbaarschot, T. Wettig, arXiv:hep-ph/0509286v1

    Google Scholar 

  11. E.P. Wigner, Proceedings of the 4th Canadian Mathematical Congress, Toronto, p. 174 (1959)

    Google Scholar 

  12. F.J. Dyson, J. Math. Phys. 3(1), 140, 157, 166 (1962)

    Google Scholar 

  13. M. Kuś, J. Mostowski, F. Haake, J. Phys. A: Math. Gen. 21, L1073–1077 (1988)

    Article  ADS  Google Scholar 

  14. F. Haake, K. Zyczkowski, Phys. Rev. A 42, 1013 (1990)

    Article  ADS  Google Scholar 

  15. M. Feingold, A. Peres, Phys. Rev. A 34, 591 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  16. B. Mehlig, K. Müller, B. Eckhardt: Phys. Rev. E 59, 5272 (1999)

    Article  ADS  Google Scholar 

  17. C.E. Porter, R.G. Thomas, Phys. Rev. 104, 483 (1956)

    Article  ADS  Google Scholar 

  18. T.A. Brody, J. Floris, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981)

    Article  ADS  Google Scholar 

  19. M.V. Berry, J. Phys. A 10, 2083 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  20. V.N. Prigodin, Phys. Rev. Lett. 75, 2392 (1995)

    Article  ADS  Google Scholar 

  21. H.-J. Stöckmann, Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999)

    Book  Google Scholar 

  22. A. Pandey, Ann. Phys. 119, 170–191 (1979)

    Article  ADS  Google Scholar 

  23. M.L. Mehta, Commun. Math. Phys. 20, 245 (1971)

    Article  ADS  Google Scholar 

  24. F.J. Dyson, J. Math. Phys. (N.Y.) 3, 166 (1962)

    Google Scholar 

  25. F.J. Dyson, Commun. Math. Phys. 19, 235 (1970)

    Article  ADS  Google Scholar 

  26. B. Dietz, PhD thesis, Essen (1991)

    Google Scholar 

  27. B. Dietz, F. Haake, Z. Phys. B 80, 153 (1990)

    Google Scholar 

  28. F.J. Dyson, Commun. Math. Phys. 47, 171–183 (1976)

    Article  ADS  Google Scholar 

  29. B. Riemann, Monatsberichte d. Preuss. Akad. d. Wissensch., Berlin 671 (1859)

    Google Scholar 

  30. H.M. Edwards, Riemann’s Zeta Function (Academic, New York, 1974)

    MATH  Google Scholar 

  31. H.L. Montgomery, Proc. Sympos. Pure Math. 24, 181 (1973)

    Article  Google Scholar 

  32. A.M. Odlyzko, Math. Comput. 48, 273 (1987)

    Article  Google Scholar 

  33. M.V. Berry, J.P. Keating, Supersymmetry and Trace Formulae: Chaos and Disorder, ed. by J.P. Keating, I.V. Lerner (Plenum, New York, 1998)

    Google Scholar 

  34. P. Bourgade, J.P. Keating, Chaos, ed. by B. Duplantier, S. Nonnenmacher, V. Rivasseau. Progress in Mathematical Physics, vol. 66 (Birkhäuser, Basel, 2013)

    Google Scholar 

  35. J. Keating, Random Matrices and Number Theory: Some Recent Themes. Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School, vol.104, July 2015 (2017)

    Google Scholar 

  36. C. Itzykson, J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1890)

    MATH  Google Scholar 

  37. F.A. Berezin, Method of Second Quantization (Academic, New York, 1966)

    MATH  Google Scholar 

  38. F.A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987)

    Book  Google Scholar 

  39. K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997)

    MATH  Google Scholar 

  40. I. Newton: Universal Arithmetic (1707); see e.g., W.W. Rouse Ball: A Short Account of the History of Mathematics, 4th edition (1908) or www.maths.tcd.ie/pub/HistMath/People/Newton/RouseBall/RB_Newton.html

  41. A. Mostowski, M. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964)

    MATH  Google Scholar 

  42. M. Marden, Geometry of Polynomials (American Mathematical Society, Providence, 1966)

    MATH  Google Scholar 

  43. E. Bogomolny, O. Bohigas, P. Leboeuf, Phys. Rev. Lett. 68, 2726 (1992); J. Stat. Phys. 85, 639 (1996)

    Google Scholar 

  44. M. Kuś, F. Haake, B. Eckhardt, Z. Physik B 92, 221 (1993)

    Article  ADS  Google Scholar 

  45. F. Haake, M. Kuś, H.-J. Sommers, H. Schomerus, K. Życkowski, J. Phys. A: Math. Gen. 29, 3641 (1996)

    Article  ADS  Google Scholar 

  46. M.V. Berry, J.P. Keating, J. Phys. A 23, 4839 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  47. E. Bogomolny, Comments At. Mo. Phys. 25, 67 (1990); Nonlinearity 5, 805 (1992)

    Google Scholar 

  48. F. Haake, H.-J. Sommers, J. Weber, J. Phys. A: Math. Gen. 32, 6903 (1999)

    Article  ADS  Google Scholar 

  49. H.-J. Sommers, F. Haake, J. Weber, J. Phys. A: Math. Gen. 31, 4395 (1998)

    Article  ADS  Google Scholar 

  50. P. Braun, F. Haake, J. Phys. A: Math. Gen. 48, 135101 (2015)

    Article  ADS  Google Scholar 

  51. J.P. Conrey, D.W. Farmer, M.R. Zirnbauer, arXiv:math-ph/0511024v2 (2007)

    Google Scholar 

  52. R.E. Hartwig, M.E. Fischer, Arch. Ration. Mech. Anal. 32, 190 (1969)

    Article  Google Scholar 

  53. U. Smilansky, Physica D 109, 153 (1997)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Universität Duisburg-Essen, Essen, Germany

    Fritz Haake

  2. School of Mathematical Science, University of Nottingham, Nottingham, UK

    Sven Gnutzmann

  3. Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland

    Marek Kuś

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Haake, F., Gnutzmann, S., Kuś, M. (2018). Random-Matrix Theory. In: Quantum Signatures of Chaos. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-97580-1_5

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