93 research outputs found
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
Invariant Distributionally Scrambled Manifolds for an Annihilation Operator
This note proves that the annihilation operator of a quantum harmonic oscillator admits an invariant distributionally ε-scrambled linear manifold for any 0<ε<2. This is a positive answer to Question 1 by Wu and Chen (2013)
Mini-Workshop: Product Systems and Independence in Quantum Dynamics
Quantum dynamics, both reversible (i.e., closed quantum systems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in particular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field
Fourth moment theorems on the Poisson space in any dimension
We extend to any dimension the quantitative fourth moment theorem on the
Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). In
particular, by adapting the exchangeable pairs couplings construction
introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove
our results under the weakest possible assumption of finite fourth moments.
This yields a Peccati-Tudor type theorem, as well as an optimal improvement in
the univariate case. Finally, a transfer principle "from-Poisson-to-Gaussian"
is derived, which is closely related to the universality phenomenon for
homogeneous multilinear sums.Comment: Minor revision. to appear in Electron. J. Proba
Shocks, Superconvergence, and a Stringy Equivalence Principle
We study propagation of a probe particle through a series of closely situated
gravitational shocks. We argue that in any UV-complete theory of gravity the
result does not depend on the shock ordering - in other words, coincident
gravitational shocks commute. Shock commutativity leads to nontrivial
constraints on low-energy effective theories. In particular, it excludes
non-minimal gravitational couplings unless extra degrees of freedom are
judiciously added. In flat space, these constraints are encoded in the
vanishing of a certain "superconvergence sum rule." In AdS, shock commutativity
becomes the statement that average null energy (ANEC) operators commute in the
dual CFT. We prove commutativity of ANEC operators in any unitary CFT and
establish sufficient conditions for commutativity of more general light-ray
operators. Superconvergence sum rules on CFT data can be obtained by inserting
complete sets of states between light-ray operators. In a planar 4d CFT, these
sum rules express (a-c)/c in terms of the OPE data of single-trace operators.Comment: 93 pages plus appendice
Introduction to Modern Canonical Quantum General Relativity
This is an introduction to the by now fifteen years old research field of
canonical quantum general relativity, sometimes called "loop quantum gravity".
The term "modern" in the title refers to the fact that the quantum theory is
based on formulating classical general relativity as a theory of connections
rather than metrics as compared to in original version due to Arnowitt, Deser
and Misner. Canonical quantum general relativity is an attempt to define a
mathematically rigorous, non-perturbative, background independent theory of
Lorentzian quantum gravity in four spacetime dimensions in the continuum. The
approach is minimal in that one simply analyzes the logical consequences of
combining the principles of general relativity with the principles of quantum
mechanics. The requirement to preserve background independence has lead to new,
fascinating mathematical structures which one does not see in perturbative
approaches, e.g. a fundamental discreteness of spacetime seems to be a
prediction of the theory providing a first substantial evidence for a theory in
which the gravitational field acts as a natural UV cut-off. An effort has been
made to provide a self-contained exposition of a restricted amount of material
at the appropriate level of rigour which at the same time is accessible to
graduate students with only basic knowledge of general relativity and quantum
field theory on Minkowski space.Comment: 301 pages, Latex; based in part on the author's Habilitation Thesis
"Mathematische Formulierung der Quanten-Einstein-Gleichungen", University of
Potsdam, Potsdam, Germany, January 2000; submitted to the on-line journal
Living Reviews; subject to being updated on at least a bi-annual basi
- …