33,708 research outputs found
Renormalization in the Henon family, I: universality but non-rigidity
In this paper geometric properties of infinitely renormalizable real
H\'enon-like maps in are studied. It is shown that the appropriately
defined renormalizations converge exponentially to the one-dimensional
renormalization fixed point. The convergence to one-dimensional systems is at a
super-exponential rate controlled by the average Jacobian and a universal
function . It is also shown that the attracting Cantor set of such a map
has Hausdorff dimension less than 1, but contrary to the one-dimensional
intuition, it is not rigid, does not lie on a smooth curve, and generically has
unbounded geometry.Comment: 42 pages, 5 picture
Semiclassical Series from Path Integrals
We derive the semiclassical series for the partition function in Quantum
Statistical Mechanics (QSM) from its path integral representation. Each term of
the series is obtained explicitly from the (real) minima of the classical
action. The method yields a simple derivation of the exact result for the
harmonic oscillator, and an accurate estimate of ground-state energy and
specific heat for a single-well quartic anharmonic oscillator. As QSM can be
regarded as finite temperature field theory at a point, we make use of Feynman
diagrams to illustrate the non-perturbative character of the series: it
contains all powers of and graphs with any number of loops; the usual
perturbative series corresponds to a subset of the diagrams of the
semiclassical series. We comment on the application of our results to other
potentials, to correlation functions and to field theories in higher
dimensions.Comment: 18 pages, 4 figures. References update
Semiclassical Statistical Mechanics
We use a semiclassical approximation to derive the partition function for an
arbitrary potential in one-dimensional Quantum Statistical Mechanics, which we
view as an example of finite temperature scalar Field Theory at a point. We
rely on Catastrophe Theory to analyze the pattern of extrema of the
corresponding path-integral. We exhibit the propagator in the background of the
different extrema and use it to compute the fluctuation determinant and to
develop a (nonperturbative) semiclassical expansion which allows for the
calculation of correlation functions. We discuss the examples of the single and
double-well quartic anharmonic oscillators, and the implications of our results
for higher dimensions.Comment: Invited talk at the La Plata meeting on `Trends in Theoretical
Physics', La Plata, April, 1997; 14 pages + 5 ps figures. Some cosmetical
modifications, and addition of some references which were missing in the
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