150 research outputs found
The Concept of Particle Weights in Local Quantum Field Theory
The concept of particle weights has been introduced by Buchholz and the
author in order to obtain a unified treatment of particles as well as (charged)
infraparticles which do not permit a definition of mass and spin according to
Wigner's theory. Particle weights arise as temporal limits of physical states
in the vacuum sector and describe the asymptotic particle content. Following a
thorough analysis of the underlying notion of localizing operators, we give a
precise definition of this concept and investigate the characteristic
properties. The decomposition of particle weights into pure components which
are linked to irreducible representations of the quasi-local algebra has been a
long-standing desideratum that only recently found its solution. We set out two
approaches to this problem by way of disintegration theory, making use of a
physically motivated assumption concerning the structure of phase space in
quantum field theory. The significance of the pure particle weights ensuing
from this disintegration is founded on the fact that they exhibit features of
improper energy-momentum eigenstates, analogous to Dirac's conception, and
permit a consistent definition of mass and spin even in an infraparticle
situation.Comment: PhD thesis, 124 pages, amslatex, mathpt
Entropies of non positively curved metric spaces
We show the equivalences of several notions of entropy, such as a version of
the topological entropy of the geodesic flow and the Minkowski dimension of the
boundary, in metric spaces with convex geodesic bicombings satisfying a uniform
packing condition. Similar estimates will be given in case of closed subsets of
the boundary of Gromov-hyperbolic metric spaces with convex geodesic
bicombings. A uniform Ahlfors regularity of the limit set of
quasiconvex-cocompact actions on Gromov-hyperbolic packed metric spaces with
convex geodesic bicombing will be shown, implying a uniform rate of convergence
to the entropy. As a consequence we prove the continuity of the critical
exponent for quasiconvex-cocompact groups with bounded codiameter
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
Review on computational methods for Lyapunov functions
Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function
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