74,828 research outputs found
Counting loop diagrams: computational complexity of higher-order amplitude evaluation
We discuss the computational complexity of the perturbative evaluation of
scattering amplitudes, both by the Caravaglios-Moretti algorithm and by direct
evaluation of the individual diagrams. For a self-interacting scalar theory, we
determine the complexity as a function of the number of external legs. We
describe a method for obtaining the number of topologically inequivalent
Feynman graphs containing closed loops, and apply this to one- and two-loop
amplitudes. We also compute the number of graphs weighted by their symmetry
factors, thus arriving at exact and asymptotic estimates for the average
symmetry factor of diagrams. We present results for the asymptotic number of
diagrams up to 10 loops, and prove that the average symmetry factor approaches
unity as the number of external legs becomes large.Comment: 27 pages, 17 table
Quantum Graphs: A model for Quantum Chaos
We study the statistical properties of the scattering matrix associated with
generic quantum graphs. The scattering matrix is the quantum analogue of the
classical evolution operator on the graph. For the energy-averaged spectral
form factor of the scattering matrix we have recently derived an exact
combinatorial expression. It is based on a sum over families of periodic orbits
which so far could only be performed in special graphs. Here we present a
simple algorithm implementing this summation for any graph. Our results are in
excellent agreement with direct numerical simulations for various graphs.
Moreover we extend our previous notion of an ensemble of graphs by considering
ensemble averages over random boundary conditions imposed at the vertices. We
show numerically that the corresponding form factor follows the predictions of
random-matrix theory when the number of vertices is large---even when all bond
lengths are degenerate. The corresponding combinatorial sum has a structure
similar to the one obtained previously by performing an energy average under
the assumption of incommensurate bond lengths.Comment: 8 pages, 3 figures. Contribution to the conference on Dynamics of
Complex Systems, Dresden (1999
Chaotic Scattering on Individual Quantum Graphs
For chaotic scattering on quantum graphs, the semiclassical approximation is
exact. We use this fact and employ supersymmetry, the colour-flavour
transformation, and the saddle-point approximation to calculate the exact
expression for the lowest and asymptotic expressions in the Ericson regime for
all higher correlation functions of the scattering matrix. Our results agree
with those available from the random-matrix approach to chaotic scattering. We
conjecture that our results hold universally for quantum-chaotic scattering
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs I: construction and numerical results
In a series of two papers we investigate the universal spectral statistics of
chaotic quantum systems in the ten known symmetry classes of quantum mechanics.
In this first paper we focus on the construction of appropriate ensembles of
star graphs in the ten symmetry classes. A generalization of the
Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry
classes. The conjecture is supported by numerical results that demonstrate the
fidelity of the spectral statistics of star graphs to the corresponding
Gaussian random-matrix theories.Comment: 15 page
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs II: semiclassical approach
A semiclassical approach to the universal ergodic spectral statistics in
quantum star graphs is presented for all known ten symmetry classes of quantum
systems. The approach is based on periodic orbit theory, the exact
semiclassical trace formula for star graphs and on diagrammatic techniques. The
appropriate spectral form factors are calculated upto one order beyond the
diagonal and self-dual approximations. The results are in accordance with the
corresponding random-matrix theories which supports a properly generalized
Bohigas-Giannoni-Schmit conjecture.Comment: 15 Page
Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs
We present exact, explicit, convergent periodic-orbit expansions for
individual energy levels of regular quantum graphs. One simple application is
the energy levels of a particle in a piecewise constant potential. Since the
classical ray trajectories (including ray splitting) in such systems are
strongly chaotic, this result provides the first explicit quantization of a
classically chaotic system.Comment: 25 pages, 5 figure
Quantum Graphs: A simple model for Chaotic Scattering
We connect quantum graphs with infinite leads, and turn them to scattering
systems. We show that they display all the features which characterize quantum
scattering systems with an underlying classical chaotic dynamics: typical
poles, delay time and conductance distributions, Ericson fluctuations, and when
considered statistically, the ensemble of scattering matrices reproduce quite
well the predictions of appropriately defined Random Matrix ensembles. The
underlying classical dynamics can be defined, and it provides important
parameters which are needed for the quantum theory. In particular, we derive
exact expressions for the scattering matrix, and an exact trace formula for the
density of resonances, in terms of classical orbits, analogous to the
semiclassical theory of chaotic scattering. We use this in order to investigate
the origin of the connection between Random Matrix Theory and the underlying
classical chaotic dynamics. Being an exact theory, and due to its relative
simplicity, it offers new insights into this problem which is at the fore-front
of the research in chaotic scattering and related fields.Comment: 28 pages, 13 figures, submitted to J. Phys. A Special Issue -- Random
Matrix Theor
Universal Chaotic Scattering on Quantum Graphs
We calculate the S-matrix correlation function for chaotic scattering on
quantum graphs and show that it agrees with that of random--matrix theory
(RMT). We also calculate all higher S-matrix correlation functions in the
Ericson regime. These, too, agree with RMT results as far as the latter are
known. We concjecture that our results give a universal description of chaotic
scattering.Comment: 4 page
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