2,061 research outputs found
Modular Theory, Non-Commutative Geometry and Quantum Gravity
This paper contains the first written exposition of some ideas (announced in
a previous survey) on an approach to quantum gravity based on Tomita-Takesaki
modular theory and A. Connes non-commutative geometry aiming at the
reconstruction of spectral geometries from an operational formalism of states
and categories of observables in a covariant theory. Care has been taken to
provide a coverage of the relevant background on modular theory, its
applications in non-commutative geometry and physics and to the detailed
discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields
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
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
Algebraic constructive quantum field theory: Integrable models and deformation techniques
Several related operator-algebraic constructions for quantum field theory
models on Minkowski spacetime are reviewed. The common theme of these
constructions is that of a Borchers triple, capturing the structure of
observables localized in a Rindler wedge. After reviewing the abstract setting,
we discuss in this framework i) the construction of free field theories from
standard pairs, ii) the inverse scattering construction of integrable QFT
models on two-dimensional Minkowski space, and iii) the warped convolution
deformation of QFT models in arbitrary dimension, inspired from non-commutative
Minkowski space.Comment: Review article, 57 pages, 3 figure
Open/Closed String Topology and Moduli Space Actions via Open/Closed Hochschild Actions
In this paper we extend our correlation functions to the open/closed case.
This gives rise to actions of an open/closed version of the Sullivan PROP as
well as an action of the relevant moduli space. There are several unexpected
structures and conditions that arise in this extension which are forced upon us
by considering the open sector. For string topology type operations, one cannot
just consider graphs, but has to take punctures into account and one has to
restrict the underlying Frobenius algebras. In the moduli space, one first has
to pass to a smaller moduli space which is closed under open/closed duality and
then consider covers in order to account for the punctures
Open/closed string topology and moduli space actions via open/closed Hochschild actions
In this paper we extend our correlation functions to the open/closed case.
This gives rise to actions of an open/closed version of the Sullivan PROP as
well as an action of the relevant moduli space. There are several unexpected
structures and conditions that arise in this extension which are forced upon us
by considering the open sector. For string topology type operations, one cannot
just consider graphs, but has to take punctures into account and one has to
restrict the underlying Frobenius algebras. In the moduli space, one first has
to pass to a smaller moduli space which is closed under open/closed duality and
then consider covers in order to account for the punctures
Characteristic classes of gauge systems
We define and study invariants which can be uniformly constructed for any
gauge system. By a gauge system we understand an (anti-)Poisson supermanifold
provided with an odd Hamiltonian self-commuting vector field called a
homological vector field. This definition encompasses all the cases usually
included into the notion of a gauge theory in physics as well as some other
similar (but different) structures like Lie or Courant algebroids. For
Lagrangian gauge theories or Hamiltonian first class constrained systems, the
homological vector field is identified with the classical BRST transformation
operator. We define characteristic classes of a gauge system as universal
cohomology classes of the homological vector field, which are uniformly
constructed in terms of this vector field itself. Not striving to exhaustively
classify all the characteristic classes in this work, we compute those
invariants which are built up in terms of the first derivatives of the
homological vector field. We also consider the cohomological operations in the
space of all the characteristic classes. In particular, we show that the
(anti-)Poisson bracket becomes trivial when applied to the space of all the
characteristic classes, instead the latter space can be endowed with another
Lie bracket operation. Making use of this Lie bracket one can generate new
characteristic classes involving higher derivatives of the homological vector
field. The simplest characteristic classes are illustrated by the examples
relating them to anomalies in the traditional BV or BFV-BRST theory and to
characteristic classes of (singular) foliations.Comment: 23 pages, references added, typos correcte
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