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124
Microlocal analysis and interacting quantum field theory: Renormalizability of ϕ 4
 Operator Algebras and Quantum Field Theory. Proceedings
, 1996
"... Dedicated to the memory of Professor Roberto Stroffolini Abstract. We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) spacetimes. We develop a purely local version of the StückelbergBogoliubovEpsteinGlaser method of renormalization ..."
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Cited by 92 (15 self)
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Dedicated to the memory of Professor Roberto Stroffolini Abstract. We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) spacetimes. We develop a purely local version of the StückelbergBogoliubovEpsteinGlaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, Köhler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved spacetimes, we construct timeordered operatorvalued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of timeordered products to all points of spacetime. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum
Asymptotic completeness in quantum field theory. Massive PauliFierz Hamiltonians
, 1997
"... Spectral and scattering theory of massive PauliFierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of ob ..."
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Cited by 54 (7 self)
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Spectral and scattering theory of massive PauliFierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of observables. Its main result is what we call geometric asymptotic completeness. Finally, the last part is a proof of asymptotic completeness itself. 1 Introduction Our paper is devoted to a class of Hamiltonians used in physics to describe a quantum system ("matter" or "an atom") interacting with a bosonic field ("radiation"). K and K are respectively the Hilbert space and the Hamiltonian describing the matter. The bosonic field is described by a Fock space \Gamma(h) with the oneparticle space eg. h = L 2 (IR d ; dk), where IR d is the momentum space, and a free Hamiltonian of the form d\Gamma(!(k)) = Z !(k)a (k)a(k)dk: The function !(k) is called the dispersion relation. The inte...
Some comments on the representation theory of the algebra underlying loop quantum gravity
"... Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance. Recentl ..."
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Cited by 22 (3 self)
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Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance. Recently, in the context of the quest for semiclassical states, states of the theory in which the quantum gravitational field is close to some classical geometry, it was realized that it might also be worthwhile to study different representations of the algebra A of observables. The content of the present note is the observation that under some mild assumptions, the mathematical structure of representations of A can be analyzed rather effortlessly, to a certain extent: Each representation can be labeled by sets of functions and measures on the space of (generalized) connections that fulfill certain conditions. These considerations are however mostly of mathematical nature. Their physical content remains to be clarified, and physically interesting examples are yet to be constructed. 1
A Rosetta stone for quantum mechanics with an introduction to quantum computation
, 2002
"... Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading ..."
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Cited by 21 (11 self)
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Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading
Adiabatic vacuum states on general spacetime manifolds: Defintion, construction, and physical properties
 ANN HENRI POINCARÉ
, 2002
"... Adiabatic vacuum states are a wellknown class of physical states for linear quantum fields on RobertsonWalker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting f ..."
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Cited by 21 (0 self)
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Adiabatic vacuum states are a wellknown class of physical states for linear quantum fields on RobertsonWalker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the KleinGordon field on globally hyperbolic spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.
MoralesTecotl, Cosmological applications of loop quantum gravity; grqc/0306008
"... According to general relativity, not only the gravitational field but also the structure of space and time, the stage for all the other fields, is governed by the dynamical laws of physics. The space we see is not a fixed background, but it evolves on large time scales, even to such extreme situatio ..."
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Cited by 18 (8 self)
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According to general relativity, not only the gravitational field but also the structure of space and time, the stage for all the other fields, is governed by the dynamical laws of physics. The space we see is not a fixed background, but it evolves on large time scales, even to such extreme situations as singularities
Perturbative algebraic field theory, and deformation quantization
 Fields Institute Communications
"... Abstract. A perturbative formulation of algebraic field theory is presented, both for the classical and for the quantum case, and it is shown that the relation between them may be understood in terms of deformation quantization. 1 ..."
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Cited by 16 (5 self)
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Abstract. A perturbative formulation of algebraic field theory is presented, both for the classical and for the quantum case, and it is shown that the relation between them may be understood in terms of deformation quantization. 1
The Hadamard condition for Dirac fields and adiabatic states on RobertsonWalker spacetimes
 Commun. Math. Phys
, 2001
"... We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on RobertsonWalker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. We then show that ..."
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Cited by 16 (2 self)
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We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on RobertsonWalker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. We then show that any two such states (of sufficiently high order) are locally quasiequivalent. We propose a microlocal version of the Hadamard condition for spinor fields on arbitrary spacetimes, which is shown to entail the usual short distance behaviour of the twopoint function. The polarisation set of these twopoint functions is determined from the Dencker connection of the spinorial KleinGordon operator which we show to be equals the (pullback) of the spin connection. Finally it is demonstrated that adiabatic states of infinite order are Hadamard, and that those of order n correspond, in some sense, to a truncated Hadamard series and will therefore allow for a point splitting renormalisation of the expected stressenergy tensor.