200 research outputs found
Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds
We present a perturbative construction of interacting quantum field theories
on any smooth globally hyperbolic manifold. We develop a purely local version
of the Stueckelberg-Bogoliubov-Epstein-Glaser method of renormalization using
techniques from microlocal analysis. As byproducts, we describe a perturbative
construction of local algebras of observables, present a new definition of Wick
polynomials as operator-valued distributions on a natural domain, and we find a
general method for the extension of distributions which were defined on the
complement of some surfaces.Comment: 38 pages, LaTeX with AMSLaTeX style option, Micro.tex macrofil
Remarks on time-energy uncertainty relations
Using a recent construction of observables characterizing the time of
occurence of an effect in quantum theory, we present a rigorous derivation of
the standard time-energy uncertainty relation. In addition, we prove an
uncertainty relation for time measurements only.Comment: 9 pages, to be pubblished in Rev. Math. Phys. issue in honor of H.
Arak
Modular dynamics in diamonds
We investigate the relation between the actions of Tomita-Takesaki modular
operators for local von Neumann algebras in the vacuum for free massive and
massless bosons in four dimensional Minkowskian spacetime. In particular, we
prove a long-standing conjecture that says that the generators of the mentioned
actions differ by a pseudo-differential operator of order zero. To get that,
one needs a careful analysis of the interplay of the theories in the bulk and
at the boundary of double cones (a.k.a. diamonds). After introducing some
technicalities, we prove the crucial result that the vacuum state for massive
bosons in the bulk of a double cone restricts to a KMS state at its boundary,
and that the restriction of the algebra at the boundary does not depend anymore
on the mass. The origin of such result lies in a careful treatment of classical
Cauchy and Goursat problems for the Klein-Gordon equation as well as the
application of known general mathematical techniques, concerning the interplay
of algebraic structures related with the bulk and algebraic structures related
with the boundary of the double cone, arising from quantum field theories in
curved spacetime. Our procedure gives explicit formulas for the modular group
and its generator in terms of integral operators acting on symplectic space of
solutions of massive Klein-Gordon Cauchy problem.Comment: 48 page
General Covariance in Algebraic Quantum Field Theory
In this review we report on how the problem of general covariance is treated
within the algebraic approach to quantum field theory by use of concepts from
category theory. Some new results on net cohomology and superselection
structure attained in this framework are included.Comment: 61 pages, 3 figures, LaTe
Topological features of massive bosons on two dimensional Einstein space-time
In this paper we tackle the problem of constructing explicit examples of
topological cocycles of Roberts' net cohomology, as defined abstractly by
Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum
field theory on the two dimensional Einstein cylinder. After deriving some
crucial results of the algebraic framework of quantization, we address the
problem of the construction of the topological cocycles. All constructed
cocycles lead to unitarily equivalent representations of the fundamental group
of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces).
The construction is carried out using only Cauchy data and related net of local
algebras on the circle.Comment: 41 pages, title changed, minor changes, typos corrected, references
added. Accepted for publication in Ann. Henri Poincare
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