Statistical theory of nuclear reactions and the Gaussian orthogonal ensemble

Statistical theory of nuclear reactions and the Gaussian orthogonal ensemble

,ANNALS OF PIIYSICS 157. 537-538 Abstracts (1984) of Papers Ecaluating Fermion Determinants Physics. Brandeis University. to Appear in Future ...

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,ANNALS

OF PIIYSICS

157. 537-538

Abstracts

(1984)

of Papers

Ecaluating Fermion Determinants Physics. Brandeis University.

to Appear

in Future

through the Chiral Anomaly. RAFAEL Waltham. Massachusetts 02254.

Issues

1. NEPOMECHIE.

Department

of

A class of fermion operators whose determinants can be calculated exactly has recently been noted. It is observed that typically such operators can be chirally rotated into the free Dirac operator; hence. their determinants are given by the chiral anomaly. Four dimensional fermion determinants of this type are computed; the appearance of the Wess-Zumino anomaly term is noted.

Replica Variables, VERRAARSCHO~ Germany.

Loop Expansion and Spectral Rigidity of Random-Matri.r Ensembles. J. J. M. AND M. R. ZIRNBAUER. Max-Planck-Institut fur Kernphysik, Heidelberg, West

The repltca trick of statistical mechanics is used to derive integral representations of n-point Green’s functions both for the GOE and the EGOE. These integral representations are particularly suited for perturbative evaluation (loop expansion). Using the floop correction to the GOE fpoint function. we find that the density of states at the edge of the semicircle scales as -N “‘p(N”‘d) where N is the dimension of the matrix ensemble. For the n-point functions with n 2 2, the existence of the microscopic limit to all orders in N-’ is proved by decomposing the integration variables into massive (i.e., macroscopm) and massless (microscopic) components. Evaluation of the EGOE Z-point function to leading order in the inverse local distance variable yields the first analytic evidence that the long-range correlations of EGOE spectra are similar to the GOE but nonstationary.

Statistical Theory of Nuclear MOLLER. Max-Planck-lnstitut

Reactions and the Gaussian fur Kernphysik. Heidelberg,

Orthogonal Ensemble. West Germany.

HANS

A.

WEIDEN-

Using methods developed in field theory and statistical mechanics, especially in the context of the Anderson model as generalized by Wegner. a novel approach to the statistical theory of nuclear reactions is developed. A finite set of N bound states. coupled to each other by an ensemble of Gaussian orthogonal matrices. and coupled to a set of channels via fixed coupling matrix elements is considered. The ensemble average and the variance of the elements of the nuclear scattering matrix. using the method of a generating function combined with the replica trick. followed by the Hubbard-Stratonovitch transformation and a modified loop expansion are evaluated. In the limit N + co, it is shown quite generally that. aside from a trivial dependence on average S-matrix elements, the variance depends only on the transmission coefficients. and that the correlation width of a pair of S-matrix elements is given by a universal function of the transmission coefficients. A modified loop expansion yields an asymptotic series valid for strong absorption. The terms in this series are partly novel, and partly coincide with results obtained earlier in the framework of a model which did not take account of the GOE eigenvalue fluctuations. This suggests that average cross sections are mainly sensitive to the stiffness of the GOE spectrum. Fluctuation properties are also derived. and the link to Ericson fluctuation theory is established. 537 0003.4916/84

$7.50

CopyrIght c 1984 by Academic Press, Inc. All rights of reproduction m any form reserved.