2,816 research outputs found
On the polarizability and capacitance of the cube
An efficient integral equation based solver is constructed for the
electrostatic problem on domains with cuboidal inclusions. It can be used to
compute the polarizability of a dielectric cube in a dielectric background
medium at virtually every permittivity ratio for which it exists. For example,
polarizabilities accurate to between five and ten digits are obtained (as
complex limits) for negative permittivity ratios in minutes on a standard
workstation. In passing, the capacitance of the unit cube is determined with
unprecedented accuracy. With full rigor, we develop a natural mathematical
framework suited for the study of the polarizability of Lipschitz domains.
Several aspects of polarizabilities and their representing measures are
clarified, including limiting behavior both when approaching the support of the
measure and when deforming smooth domains into a non-smooth domain. The success
of the mathematical theory is achieved through symmetrization arguments for
layer potentials.Comment: 33 pages, 7 figure
Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I (The two antagonistic localizations and their asymptotic compatibility)
It is shown that there are significant conceptual differences between QM and
QFT which make it difficult to view the latter as just a relativistic extension
of the principles of QM. At the root of this is a fundamental distiction
between Born-localization in QM (which in the relativistic context changes its
name to Newton-Wigner localization) and modular localization which is the
localization underlying QFT, after one separates it from its standard
presentation in terms of field coordinates. The first comes with a probability
notion and projection operators, whereas the latter describes causal
propagation in QFT and leads to thermal aspects of locally reduced finite
energy states. The Born-Newton-Wigner localization in QFT is only applicable
asymptotically and the covariant correlation between asymptotic in and out
localization projectors is the basis of the existence of an invariant
scattering matrix. In this first part of a two part essay the modular
localization (the intrinsic content of field localization) and its
philosophical consequences take the center stage. Important physical
consequences of vacuum polarization will be the main topic of part II. Both
parts together form a rather comprehensive presentation of known consequences
of the two antagonistic localization concepts, including the those of its
misunderstandings in string theory.Comment: 63 pages corrections, reformulations, references adde
Classical and vector sturm—liouville problems: recent advances in singular-point analysis and shooting-type algorithms
AbstractSignificant advances have been made in the last year or two in algorithms and theory for Sturm—Liouville problems (SLPs). For the classical regular or singular SLP −(p(x)u′)′ + q(x)u = λw(x)u, a < x < b, we outline the algorithmic approaches of the recent library codes and what they can now routinely achieve.For a library code, automatic treatment of singular problems is a must. New results are presented which clarify the effect of various numerical methods of handling a singular endpoint.For the vector generalization −(P(x)u′)′+Q(x)u = λW(x)u where now u is a vector function of x, and P, Q, W are matrices, and for the corresponding higher-order vector self-adjoint problem, we outline the equally impressive advances in algorithms and theory
2D growth processes: SLE and Loewner chains
This review provides an introduction to two dimensional growth processes.
Although it covers a variety processes such as diffusion limited aggregation,
it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner
evolutions (SLE) which are Markov processes describing interfaces in 2D
critical systems. It starts with an informal discussion, using numerical
simulations, of various examples of 2D growth processes and their connections
with statistical mechanics. SLE is then introduced and Schramm's argument
mapping conformally invariant interfaces to SLE is explained. A substantial
part of the review is devoted to reveal the deep connections between
statistical mechanics and processes, and more specifically to the present
context, between 2D critical systems and SLE. Some of the SLE remarkable
properties are explained, as well as the tools for computing with SLE. This
review has been written with the aim of filling the gap between the
mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172
pages, low quality figures, better quality figures upon request to the
authors, comments welcom
On the shape of a pure O-sequence
An order ideal is a finite poset X of (monic) monomials such that, whenever M
is in X and N divides M, then N is in X. If all, say t, maximal monomials of X
have the same degree, then X is pure (of type t). A pure O-sequence is the
vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree.
Equivalently, in the language of commutative algebra, pure O-sequences are the
h-vectors of monomial Artinian level algebras. Pure O-sequences had their
origin in one of Richard Stanley's early works in this area, and have since
played a significant role in at least three disciplines: the study of
simplicial complexes and their f-vectors, level algebras, and matroids. This
monograph is intended to be the first systematic study of the theory of pure
O-sequences. Our work, making an extensive use of algebraic and combinatorial
techniques, includes: (i) A characterization of the first half of a pure
O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel;
(ii) A study of (the failing of) the unimodality property; (iii) The problem of
enumerating pure O-sequences, including a proof that almost all O-sequences are
pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type
t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents
perhaps the strongest possible structural result short of an (impossible?)
characterization; (v) A pithy connection of the ICP with Stanley's matroid
h-vector conjecture; (vi) A specific study of pure O-sequences of type 2,
including a proof of the Weak Lefschetz Property in codimension 3 in
characteristic zero. As a corollary, pure O-sequences of codimension 3 and type
2 are unimodal (over any field); (vii) An analysis of the extent to which the
Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii)
Some observations about pure f-vectors, an important special case of pure
O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly
minor revisions with respect to last year's versio
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
Pascual Jordan's legacy and the ongoing research in quantum field theory
Pascual Jordan's path-breaking role as the protagonist of quantum field
theory (QFT) is recalled and his friendly dispute with Dirac's particle-based
relativistic quantum theory is presented as the start of the field-particle
conundrum which, though in modified form, persists up to this date. Jordan had
an intuitive understanding that the existence of a causal propagation with
finite propagation speed in a quantum theory led to radically different
physical phenomena than those of QM. The conceptional-mathematical
understanding for such an approach began to emerge only 30 years later. The
strongest link between Jordan's view of QFT and modern "local quantum physics"
is the central role of causal locality as the defining principle of QFT as
opposed to the Born localization in QM. The issue of causal localization is
also the arena where misunderstandings led to a serious derailment of large
part of particle theory e.g. the misinterpretation of an infinite component
pointlike field resulting from the quantization of the Nambu-Goto Lagrangian as
a spacetime quantum string. The new concept of modular localization, which
replaces Jordan's causal locality, is especially important to overcome the
imperfections of gauge theories for which Jordan was the first to note nonlocal
aspects of physical (not Lagrangian) charged fields. Two interesting subjects
in which Jordan was far ahead of his contemporaries will be presented in two
separate sections.Comment: improvement of formulation, addition of reference
From General Relativity to Quantum Gravity
In general relativity (GR), spacetime geometry is no longer just a background
arena but a physical and dynamical entity with its own degrees of freedom. We
present an overview of approaches to quantum gravity in which this central
feature of GR is at the forefront. However, the short distance dynamics in the
quantum theory are quite different from those of GR and classical spacetimes
and gravitons emerge only in a suitable limit. Our emphasis is on communicating
the key strategies, the main results and open issues. In the spirit of this
volume, we focus on a few avenues that have led to the most significant
advances over the past 2-3 decades.Comment: To appear in \emph{General Relativity and Gravitation: A Centennial
Survey}, commissioned by the International Society for General Relativity and
Gravitation and to be published by Cambridge University Press. Abhay Ashtekar
served as the `coordinating author' and combined the three contribution
- …