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55
Axiomatic quantum field theory in curved spacetime
, 2008
"... The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincare invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globa ..."
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Cited by 689 (18 self)
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The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincare invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is locally and covariantly constructed from the spacetime metric), a microlocal spectrum condition, an "associativity" condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spinstatistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.
H.: Equilibrium statistical mechanics of Fermion lattice systems
"... We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle wit ..."
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Cited by 39 (9 self)
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We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C ∗dynamical systems for the Fermion (CAR) algebra A with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics αt commutes with the evenoddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δα of αt contains the set A ◦ of all strictly local elements of A. (III) The set A ◦ is the core of δα.
Mathematical Theory of NonEquilibrium Quantum Statistical Mechanics
, 2002
"... We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium steady states, entropy production and ..."
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Cited by 37 (3 self)
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We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.
Introduction to Representations of the Canonical Commutation and Anticommutation Relations
, 2005
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Another Return of “Return to Equilibrium”
, 2008
"... The property of “return to equilibrium” is established for a class of quantummechanical models describing interactions of a (toy) atom with blackbody radiation, or of a spin with a heat bath of scalar bosons, under the assumption that the interaction strength is sufficiently weak. For models descr ..."
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Cited by 33 (6 self)
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The property of “return to equilibrium” is established for a class of quantummechanical models describing interactions of a (toy) atom with blackbody radiation, or of a spin with a heat bath of scalar bosons, under the assumption that the interaction strength is sufficiently weak. For models describing the first class of systems, our upper bound on the interaction strength is independent of the temperature T, (with 0 < T ≤ T0 < ∞), while, for the spinboson model, it tends to zero logarithmically, as T → 0. Our result holds for interaction form factors with physically realistic infrared behaviour. Three key ingredients of our analysis are: a suitable concrete form of the ArakiWoods representation of the radiation field, Mourre’s positive commutator method combined with a recent virial theorem, and a norm bound on the difference between the equilibrium states of the interacting and the noninteracting system (which, for the system of an atom coupled to blackbody radiation, is valid for all temperatures T ≥ 0, assuming only that the interaction strength is sufficiently weak).
The GreenKubo formula and the Onsager reciprocity relations in quantum statistical mechanics
, 2005
"... We study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the GreenKubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Our derivation is axiomatic and the key assumptions concern ergodic pro ..."
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Cited by 27 (11 self)
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We study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the GreenKubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Our derivation is axiomatic and the key assumptions concern ergodic properties of nonequilibrium steady states.
An analogue of the KacWakimoto formula and black hole conditional entropy
 COMMUN. MATH. PHYS
, 1996
"... A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by KacWakimot ..."
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Cited by 25 (12 self)
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A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by KacWakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the ConnesStœrmer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.