100 research outputs found
Algebraic conformal quantum field theory in perspective
Conformal quantum field theory is reviewed in the perspective of Axiomatic,
notably Algebraic QFT. This theory is particularly developped in two spacetime
dimensions, where many rigorous constructions are possible, as well as some
complete classifications. The structural insights, analytical methods and
constructive tools are expected to be useful also for four-dimensional QFT.Comment: Review paper, 40 pages. v2: minor changes and references added, so as
to match published versio
The Pivotal Role of Causality in Local Quantum Physics
In this article an attempt is made to present very recent conceptual and
computational developments in QFT as new manifestations of old and well
establihed physical principles. The vehicle for converting the
quantum-algebraic aspects of local quantum physics into more classical
geometric structures is the modular theory of Tomita. As the above named
laureate to whom I have dedicated has shown together with his collaborator for
the first time in sufficient generality, its use in physics goes through
Einstein causality. This line of research recently gained momentum when it was
realized that it is not only of structural and conceptual innovative power (see
section 4), but also promises to be a new computational road into
nonperturbative QFT (section 5) which, picturesquely speaking, enters the
subject on the extreme opposite (noncommutative) side.Comment: This is a updated version which has been submitted to Journal of
Physics A, tcilatex 62 pages. Adress: Institut fuer Theoretische Physik
FU-Berlin, Arnimallee 14, 14195 Berlin presently CBPF, Rua Dr. Xavier Sigaud
150, 22290-180 Rio de Janeiro, Brazi
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
Fell bundles and imprimitivity theorems
Our goal in this paper and two sequels is to apply the
Yamagami-Muhly-Williams equivalence theorem for Fell bundles over groupoids to
recover and extend all known imprimitivity theorems involving groups. Here we
extend Raeburn's symmetric imprimitivity theorem, and also, in an appendix, we
develop a number of tools for the theory of Fell bundles that have not
previously appeared in the literature.Comment: minor change
Modular Theory and Eyvind Wichmann's Contributions to modern Particle Physics Theory
Some of the consequences of Eyvind Wichmann's contributions to modular theory
and the QFT phase-space structure are presented. In order to show the power of
those ideas in contemporary problems, I selected the issue of algebraic
holography as well as a new nonperturbative constructive approach (based on the
modular structur of wedge-localized algebras and modular inclusions) and show
that these ideas are recent consequences of the pathbreaking work which
Wichmann together with his collaborator Bisognano initiated in the mid
70$^{ies}.Comment: A rogue address which entered chapter 2 has since been omitted. 21
pages, tcilatex, to be published in a special Festschrift volume, dedicated
to Prof. E. Wichmann on the occasion of his seventieth birthda
AQFT from n-functorial QFT
There are essentially two different approaches to the axiomatization of
quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and
functorial QFT, going back to Atiyah and Segal. More recently, based on ideas
by Baez and Dolan, the latter is being refined to "extended" functorial QFT by
Freed, Hopkins, Lurie and others. The first approach uses local nets of
operator algebras which assign to each patch an algebra "of observables", the
latter uses n-functors which assign to each patch a "propagator of states".
In this note we present an observation about how these two axiom systems are
naturally related: we demonstrate under mild assumptions that every
2-dimensional extended Minkowskian QFT 2-functor ("parallel surface transport")
naturally yields a local net. This is obtained by postcomposing the propagation
2-functor with an operation that mimics the passage from the Schroedinger
picture to the Heisenberg picture in quantum mechanics.
The argument has a straightforward generalization to general
pseudo-Riemannian structure and higher dimensions.Comment: 39 pages; further examples added: Hopf spin chains and asymptotic
inclusion of subfactors; references adde
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
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