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
412 (October 28, 2004)
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