131 research outputs found
Quantum astrometric observables II: time delay in linearized quantum gravity
A clock synchronization thought experiment is modeled by a diffeomorphism
invariant "time delay" observable. In a sense, this observable probes the
causal structure of the ambient Lorentzian spacetime. Thus, upon quantization,
it is sensitive to the long expected smearing of the light cone by vacuum
fluctuations in quantum gravity. After perturbative linearization, its mean and
variance are computed in the Minkowski Fock vacuum of linearized gravity. The
na\"ive divergence of the variance is meaningfully regularized by a length
scale , the physical detector resolution. This is the first time vacuum
fluctuations have been fully taken into account in a similar calculation.
Despite some drawbacks this calculation provides a useful template for the
study of a large class of similar observables in quantum gravity. Due to their
large volume, intermediate calculations were performed using computer algebra
software. The resulting variance scales like , where
is the Planck length and is the distance scale separating the ("lab" and
"probe") clocks. Additionally, the variance depends on the relative velocity of
the lab and the probe, diverging for low velocities. This puzzling behavior may
be due to an oversimplified detector resolution model or a neglected second
order term in the time delay.Comment: 30 pages, 8 figures, revtex4-1; v3: minor updates and corrections,
close to published versio
On the numerical evaluation of finite-part integrals involving an algebraic singularity
Thesis (PhD)--Stellenbosch University, 1975.ENGLISH ABSTRACT: Some problems of applied mathematics, for instance in the fields of
aerodynamics or electron optics, involve certain singular integrals
which do not exist classically. The problems can, however, be solved
pLovided that such integrals are interpreted as finite-part integrals.
Although the concept of a finite-part integral has existed for
about fifty years, it was possible to define it rigorously only by means
of distribution theory, developed about twenty-five years ago. But, to
the best of our knowledge, no quadrature formula for the numerical eva=
luation of finite-part integrals ha~ been given in the literature.
The main concern of this thesis is the study and discussion of.two
kinds of quadrature formulae for evaluating finite-part integrals in=
volving an algebraic singularity.
Apart from a historical introduction, the first chapter contains
some physical examples of finite-part integrals and their definition
based on distribution theory. The second chapter treats the most im=
portant properties of finite-part integrals; in particular we study
their behaviour under the most common rules for ordinary integrals.
In chapters three and four we derive a quadrature formula for equispaced
stations and one which is optimal in the sense of the Gauss-type quadra=
ture. In connection with the latter formula, we also study a new class
of orthogonal polynomials. In the fifth and.last chapter we give a
derivative-free error bound for the equispaced quadrature formula. The
error quantities which are independent of the integrand were computed
for the equispaced quadrature formula and are also given. In the case
of some examples, we compare the computed error bounds with the actual
errors.
~esides this theoretical investigation df finite-part integrals,
we also computed - for several orders of the algebraic singularity
the coefficients for both of the aforesaid quadrature formulae, in
which the number of stations ranges from three up to twenty. In the
case of the equispaced quadrature fortnu1a,we give the weights and -
for int~ger order of the singularity - the coefficients for a numerical
derivative of the integrand function. For the Gauss-type quadrature,
we give the stations, the corresponding weights and the coefficients of
the orthogonal polynomials.
These data are being published in a separate report [18] which
also contains detailed instructions on the use of the tables
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page
Physics of the Analytic S-Matrix
You might've heard about various mathematical properties of scattering
amplitudes such as analyticity, sheets, branch cuts, discontinuities, etc. What
does it all mean? In these lectures, we'll take a guided tour through simple
scattering problems that will allow us to directly trace such properties back
to physics. We'll learn how different analytic features of the S-matrix are
really consequences of causality, locality of interactions, unitary
propagation, and so on.
These notes are based on a series of lectures given in Spring 2023 at the
Institute for Advanced Study in Princeton and the Higgs Centre School of
Theoretical Physics in Edinburgh.Comment: 162 page
Quasianalytic Monogenic Solutions of a Cohomological Equation
We prove that the solutions of a cohomological equation of complex dimension one and in the
analytic category have a monogenic dependence on the parameter, and we investigate the question
of their quasianalyticity. This cohomological equation is the standard linearized conjugacy equation
for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue
of the linear part, denoted by q.
Borel\u2019s theory of non-analytic monogenic functions has been first investigated by Arnol\u2019d and
Herman in the related context of the problem of linearization of analytic diffeomorphisms of the
circle close to a rotation. Herman raised the question whether the solutions of the cohomological
equation had a quasianalytic dependence on the parameter q. Indeed they are analytic for q 08 CS
1
,
the unit circle S
1 appears as a natural boundary (because of resonances, i.e. roots of unity), but
the solutions are still defined at points of S
1 which lie \u201cfar enough from resonances\u201d. We adapt to
our case Herman\u2019s construction of an increasing sequence of compacts which avoid resonances and
prove that the solutions of our equation belong to the associated space of monogenic functions ;
some general properties of these monogenic functions and particular properties of the solutions are
then studied.
For instance the solutions are defined and admit asymptotic expansions at the points of S
1
which satisfy some arithmetical condition, and the classical Carleman Theorem allows us to answer
negatively to the question of quasianalyticity at these points. But resonances (roots of unity) also
lead to asymptotic expansions, for which quasianalyticity is obtained as a particular case of Ecalle\u2019s \ub4
theory of resurgent functions. And at constant-type points, where no quasianalytic Carleman class
contains the solutions, one can still recover the solutions from their asymptotic expansions and
obtain a special kind of quasianalyticity.
Our results are obtained by reducing the problem, by means of Hadamard\u2019s product, to the
study of a fundamental solution (which turns out to be the so-called q-logarithm or \u201cquantum
logarithm\u201d). We deduce as a corollary of our work the proof of a conjecture of Gammel on the
monogenic and quasianalytic properties of a certain number-theoretical Borel-Wolff-Denjoy series
Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes
This monograph provides a largely self--contained and broadly accessible
exposition of two cosmological applications of algebraic quantum field theory
(QFT) in curved spacetime: a fundamental analysis of the cosmological evolution
according to the Standard Model of Cosmology and a fundamental study of the
perturbations in Inflation. The two central sections of the book dealing with
these applications are preceded by sections containing a pedagogical
introduction to the subject as well as introductory material on the
construction of linear QFTs on general curved spacetimes with and without gauge
symmetry in the algebraic approach, physically meaningful quantum states on
general curved spacetimes, and the backreaction of quantum fields in curved
spacetimes via the semiclassical Einstein equation. The target reader should
have a basic understanding of General Relativity and QFT on Minkowski
spacetime, but does not need to have a background in QFT on curved spacetimes
or the algebraic approach to QFT. In particular, I took a great deal of care to
provide a thorough motivation for all concepts of algebraic QFT touched upon in
this monograph, as they partly may seem rather abstract at first glance. Thus,
it is my hope that this work can help non--experts to make `first contact' with
the algebraic approach to QFT.Comment: 123 pages, 4 figures, to appear as SpringerBriefs in Mathematical
Physic
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