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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...
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.
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 (9 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
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 21 (2 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
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
The quantum moment problem and bounds on entangled multiprover games
 In Proceedings of 23rd IEEE Conference on Computational Complexity
, 2008
"... We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state ρ and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is p ..."
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Cited by 17 (2 self)
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We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state ρ and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on ρ is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of oneround multiprover games with entangled provers. Under the conjecture that the provers need only share states in finitedimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascués, Pironio and Acín [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. It follows that the class of languages recognized by a multiprover interactive proof system where the provers share entanglement is recursive.