Det-Det Correlations for Quantum Maps: Dual Pair and Saddle-Point Analyses

Details Det-Det Correlations for Quantum Maps: Dual Pair and Saddle-Point Analyses Det-Det Correlations for Quantum Maps: Dual Pair and Saddle-Point Analyses

2001-09-24

arxiv.org/abs/math-ph/0109025v2

J. Math. Phys. 43, 2214-2240 (2002)

Physics

Mathematical Physics

Scientific paper

An attempt is made to clarify the ballistic non-linear sigma model formalism recently proposed for quantum chaotic systems, by the spectral determinant Z(s)=Det(1-sU) of a quantized map U element of U(N). More precisely, we study the correlator omega_U(s)=<|Z(st)|^2> (averaging t over the unit circle). Identifying the group U(N) as one member of a dual pair acting in the spinor representation of Spin(4N), omega_U(s) is expanded in terms of irreducible characters of U(N). In close analogy with the ballistic non-linear sigma model, a coherent-state integral representation of omega_U(s) is developed. We show that the leading-order saddle-point approximation reproduces omega_U(s) exactly, up to a constant factor; this miracle can be explained by interpreting omega_U(s) as a character of U(2N), for which the saddle-point expansion yields the Weyl character formula. Unfortunately, this decomposition behaves non-smoothly in the semiclassical limit, and to make further progress some averaging over U needs to be introduced. Several averaging schemes are investigated. In general, a direct application of the saddle-point approximation to these schemes is demonstrated to give incorrect results; this is not the case for a `semiclassical averaging scheme', for which all loop corrections vanish identically. As a side product of the dual pair decomposition, we compute a crossover between the Poisson and CUE ensembles for omega_U(s).

Affiliated with

Also associated with

No associations

Rating

Det-Det Correlations for Quantum Maps: Dual Pair and Saddle-Point Analyses does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Det-Det Correlations for Quantum Maps: Dual Pair and Saddle-Point Analyses, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Det-Det Correlations for Quantum Maps: Dual Pair and Saddle-Point Analyses will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-308186