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GRASSMANN INTEGRATION IN STOCHASTIC QUANTUM PHYSICS - THE CASE OF COMPOUND NUCLEUS SCATTERING
VERBAARSCHOT JJM, WEIDENMULLER HA, ZIRNBAUER MR
PHYSICS REPORTS-REVIEW SECTION OF PHYSICS LETTERS
129: (6) 367-438 DEC 1985
 Document type: Article Language: English Cited References:  22 Times Cited:  412 (October 28, 2004)

Abstract:
Using a stochastic model for N compound-nucleus resonances coupled to the channels, we calculate the limit N \rightarrow \infty of the ensemble average of the S-matrix (the "one-point function") and of the product of an S-matrix element with the complex conjugate of another, both taken at different energies (the "two-point function"). Using a generating function involving both commuting and anticommuting integration variables, we evaluate the ensemble averages trivially. The problem of carrying out the remaining integrations is solved with the help of the Hubbard-Stratonovitch transformation. We put special emphasis on the convergence properties of this transformation, and on the unerlying symmetries of the stochastic model for the compound nucleus. These two features together completely define the parametrization of the composite variables in terms of a group of transformations. This group is compact in the "Fermion-Fermion block" and non-compact in the "Boson-Boson block". The limit $N\rightarrow/infty$ is taken with the help of the saddle-point approximation. After integration over the massive "modes", we show that the two-point function can be expressed in terms of the transmission coefficients. In this way we prove that the fluctuation properties of the nuclear S-matrix are the same over the entire spectrum of the random Hamiltonian describing the compound nucleus. The integration over the saddle-point manifold is carried out using symmetry properties of the random Hamiltonian. We finally obtain a closed form expression for the two-point function in terms of a threefold integral over real variables. This expression can easily be evaluated numerically.

KeyWords Plus:
COMPOUND NUCLEUS, SUPER-SYMMETRY, RANDOM MATRIX THEORY