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Decoherence, einselection, and the quantum origins of the classical
 REVIEWS OF MODERN PHYSICS 75, 715. AVAILABLE ONLINE AT HTTP://ARXIV.ORG/ABS/QUANTPH/0105127
, 2003
"... The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) ..."
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Cited by 46 (1 self)
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The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environmentinduced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal "Schrödingercat states." The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information
Quantum chaotic dynamics and random polynomials
 J. Stat. Phys
, 1996
"... We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of ”quantum chaotic dynamics”. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plan ..."
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Cited by 32 (0 self)
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We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of ”quantum chaotic dynamics”. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of selfinversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity. Special attention is devoted all over the paper to the role of symmetries in the distribution of roots of random polynomials.
Universal Results for Correlations of Characteristic Polynomials
 RiemannHilbert Approach. Commun. Math. Phys
, 2003
"... Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same ..."
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Cited by 26 (6 self)
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Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the RiemannHilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via DeiftZhou steepestdescent/stationary phase method for RiemannHilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of β = 2 symmetry class. 1.
Limiting eigenvalue distribution of random matrices with correlated entries
 Markov Proc. Rel. Fields 2
, 1996
"... Some recent results. ..."
Semiclassical Form Factor for Chaotic Systems With Spin 1/2
, 1999
"... . We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled ..."
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Cited by 20 (16 self)
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. We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory. PACS numbers: 03.65.Sq, 05.45.Mt k Email address: bol@physik.uniulm.de  Email address: kep@physik.uniulm.de + Address after 1 October 1999: Abteilung Theoretische Physik, Universitat Ulm, AlbertEinsteinAllee 11, D89069 Ulm, G...
Random matrices, magic squares and matching polynomials
 Research Paper
"... Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zetafunction, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the c ..."
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Cited by 19 (3 self)
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Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zetafunction, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed. 1
On asymptotics of large Haar distributed unitary matrices
"... this paper random unitary matrices are studied whose entries must be correlated. A unitary matrix U = (U ij ) is a matrix with complex entries and UU # = U # U = I ..."
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Cited by 17 (1 self)
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this paper random unitary matrices are studied whose entries must be correlated. A unitary matrix U = (U ij ) is a matrix with complex entries and UU # = U # U = I
2003 Random matrix theory and the zeros of ζ
 Preprint mathph/0207044
"... Abstract. We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to ..."
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Cited by 10 (0 self)
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Abstract. We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accurate description for the horizontal distribution of the zeros of ζ ′ (s) to the right of the critical line. We show that as N → ∞ the fraction of roots of Z ′ (U, z) that lie in the region 1−x/(N −1) ≤ z  < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x → ∞ and x → 0 and compare them with numerical experiments. Mathematics Subject Classification: 15A52, 11M99Random matrix theory and the zeros of ζ ′ (s) 2 1.
Quantum and classical ergodicity of spinning particles
, 2001
"... We give a formulation of quantum ergodicity for Pauli Hamiltonians with arbitrary spin in terms of a WignerWeyl calculus. The corresponding classical phase space is the direct product of the phase space of the translational degrees of freedom and the twosphere. On this product space we introduce a ..."
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Cited by 9 (7 self)
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We give a formulation of quantum ergodicity for Pauli Hamiltonians with arbitrary spin in terms of a WignerWeyl calculus. The corresponding classical phase space is the direct product of the phase space of the translational degrees of freedom and the twosphere. On this product space we introduce a combination of the translational motion and classical spin precession. We prove quantum ergodicity under the condition that this product flow is ergodic.