111 research outputs found
On the problem of mass-dependence of the two-point function of the real scalar free massive field on the light cone
We investigate the generally assumed inconsistency in light cone quantum
field theory that the restriction of a massive, real, scalar, free field to the
nullplane is independent of mass \cite{LKS}, but the
restriction of the two-point function depends on it (see, e.g., \cite{NakYam77,
Yam97}). We resolve this inconsistency by showing that the two-point function
has no canonical restriction to in the sense of distribution theory.
Only the so-called tame restriction of the two-point function exists which we
have introduced in \cite{Ull04sub}. Furthermore, we show that this tame
restriction is indeed independent of mass. Hence the inconsistency appears only
by the erroneous assumption that the two-point function would have a
(canonical) restriction to .Comment: 10 pages, 2 figure
Off-Critical SLE(2) and SLE(4): a Field Theory Approach
Using their relationship with the free boson and the free symplectic fermion,
we study the off-critical perturbation of SLE(4) and SLE(2) obtained by adding
a mass term to the action. We compute the off-critical statistics of the source
in the Loewner equation describing the two dimensional interfaces. In these two
cases we show that ratios of massive by massless partition functions,
expressible as ratios of regularised determinants of massive and massless
Laplacians, are (local) martingales for the massless interfaces. The
off-critical drifts in the stochastic source of the Loewner equation are
proportional to the logarithmic derivative of these ratios. We also show that
massive correlation functions are (local) martingales for the massive
interfaces. In the case of massive SLE(4), we use this property to prove a
factorisation of the free boson measure.Comment: 30 pages, 1 figures, Published versio
An exactly solved model for mutation, recombination and selection
It is well known that rather general mutation-recombination models can be
solved algorithmically (though not in closed form) by means of Haldane
linearization. The price to be paid is that one has to work with a multiple
tensor product of the state space one started from.
Here, we present a relevant subclass of such models, in continuous time, with
independent mutation events at the sites, and crossover events between them. It
admits a closed solution of the corresponding differential equation on the
basis of the original state space, and also closed expressions for the linkage
disequilibria, derived by means of M\"obius inversion. As an extra benefit, the
approach can be extended to a model with selection of additive type across
sites. We also derive a necessary and sufficient criterion for the mean fitness
to be a Lyapunov function and determine the asymptotic behaviour of the
solutions.Comment: 48 page
Coherent States of the q--Canonical Commutation Relations
For the -deformed canonical commutation relations for in some Hilbert
space we consider representations generated from a vector
satisfying , where .
We show that such a representation exists if and only if .
Moreover, for these representations are unitarily equivalent
to the Fock representation (obtained for ). On the other hand
representations obtained for different unit vectors are disjoint. We
show that the universal C*-algebra for the relations has a largest proper,
closed, two-sided ideal. The quotient by this ideal is a natural -analogue
of the Cuntz algebra (obtained for ). We discuss the Conjecture that, for
, this analogue should, in fact, be equal to the Cuntz algebra
itself. In the limiting cases we determine all irreducible
representations of the relations, and characterize those which can be obtained
via coherent states.Comment: 19 pages, Plain Te
LERW as an example of off-critical SLEs
Two dimensional loop erased random walk (LERW) is a random curve, whose
continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter
kappa=2. In this article we study ``off-critical loop erased random walks'',
loop erasures of random walks penalized by their number of steps. On one hand
we are able to identify counterparts for some LERW observables in terms of
symplectic fermions (c=-2), thus making further steps towards a field theoretic
description of LERWs. On the other hand, we show that it is possible to
understand the Loewner driving function of the continuum limit of off-critical
LERWs, thus providing an example of application of SLE-like techniques to
models near their critical point. Such a description is bound to be quite
complicated because outside the critical point one has a finite correlation
length and therefore no conformal invariance. However, the example here shows
the question need not be intractable. We will present the results with emphasis
on general features that can be expected to be true in other off-critical
models.Comment: 45 pages, 2 figure
Reflections upon separability and distillability
We present an abstract formulation of the so-called Innsbruck-Hannover
programme that investigates quantum correlations and entanglement in terms of
convex sets. We present a unified description of optimal decompositions of
quantum states and the optimization of witness operators that detect whether a
given state belongs to a given convex set. We illustrate the abstract
formulation with several examples, and discuss relations between optimal
entanglement witnesses and n-copy non-distillable states with non-positive
partial transpose.Comment: 12 pages, 7 figures, proceedings of the ESF QIT Conference Gdansk,
July 2001, submitted to special issue of J. Mod. Op
Optimization of entanglement witnesses
An entanglement witness (EW) is an operator that allows to detect entangled
states. We give necessary and sufficient conditions for such operators to be
optimal, i.e. to detect entangled states in an optimal way. We show how to
optimize general EW, and then we particularize our results to the
non-decomposable ones; the latter are those that can detect positive partial
transpose entangled states (PPTES). We also present a method to systematically
construct and optimize this last class of operators based on the existence of
``edge'' PPTES, i.e. states that violate the range separability criterion
[Phys. Lett. A{\bf 232}, 333 (1997)] in an extreme manner. This method also
permits the systematic construction of non-decomposable positive maps (PM). Our
results lead to a novel sufficient condition for entanglement in terms of
non-decomposable EW and PM. Finally, we illustrate our results by constructing
optimal EW acting on H=\C^2\otimes \C^4. The corresponding PM constitute the
first examples of PM with minimal ``qubit'' domain, or - equivalently - minimal
hermitian conjugate codomain.Comment: 18 pages, two figures, minor change
Calculus on manifolds of conformal maps and CFT
In conformal field theory (CFT) on simply connected domains of the Riemann
sphere, the natural conformal symmetries under self-maps are extended, in a
certain way, to local symmetries under general conformal maps, and this is at
the basis of the powerful techniques of CFT. Conformal maps of simply connected
domains naturally have the structure of an infinite-dimensional groupoid, which
generalizes the finite-dimensional group of self-maps. We put a topological
structure on the space of conformal maps on simply connected domains, which
makes it into a topological groupoid. Further, we (almost) extend this to a
local manifold structure based on the infinite-dimensional Frechet topological
vector space of holomorphic functions on a given domain A. From this, we
develop the notion of conformal A-differentiability at the identity. Our main
conclusion is that quadratic differentials characterizing cotangent elements on
the local manifold enjoy properties similar to those of the holomorphic
stress-energy tensor of CFT; these properties underpin the local symmetries of
CFT. Applying the general formalism to CFT correlation functions, we show that
the stress-energy tensor is exactly such a quadratic differential. This is at
the basis of constructing the stress-energy tensor in conformal loop ensembles.
It also clarifies the relation between Cardy's boundary conditions for CFT on
simply connected domains, and the expression of the stress-energy tensor in
terms of metric variations.Comment: v1: 51 pages, 5 figures. v2: 56 pages, corrections and
clarifications. v3: 53 pages, one substantial addition (groupoid structure),
discussion further clarified and simplified. v4: 59 pages, introduction
improved, with a discussion on the relations with previous works. Published
versio
A proof of the Mazur-Orlicz theorem via the Markov-Kakutani common fixed point theorem, and vice versa
In this paper, we present a new proof of the Mazur-Orlicz theorem, which uses the Markov-Kakutani common fixed point theorem, and a new proof of the Markov-Kakutani common fixed point theorem, which uses the Mazur-Orlicz theorem
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