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57
Quantum Field Theory in Curved Spacetime
, 1995
"... We review the mathematically rigorous formulation of the quantum theory of a linear field propagating in a globally hyperbolic spacetime. This formulation is accomplished via the algebraic approach, which, in essence, simultaneously admits all states in all possible (unitarily inequivalent) Hilbert ..."
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Cited by 687 (17 self)
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We review the mathematically rigorous formulation of the quantum theory of a linear field propagating in a globally hyperbolic spacetime. This formulation is accomplished via the algebraic approach, which, in essence, simultaneously admits all states in all possible (unitarily inequivalent) Hilbert space constructions. The physically nonsingular states are restricted by the requirement that their twopoint function satisfy the Hadamard condition, which insures that the ultraviolet behavior of the state be similar to that of the vacuum state in Minkowski spacetime, and that the expected stressenergy tensor in the state be finite. We briefly review the Unruh and Hawking effects from the perspective of the theoretical framework adopted here. A brief discussion also is given of several open issues and questions in quantum field theory in curved spacetime regarding the treatment of “backreaction”, the validity of some version of the “averaged null energy condition”, and the formulation and properties of quantum field theory in causality violating spacetimes.
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.
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 37 (8 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.
Introduction to Representations of the Canonical Commutation and Anticommutation Relations
, 2005
"... ..."
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 29 (13 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 26 (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.
Positive Commutators in NonEquilibrium Quantum Statistical Mechanics
 Commun. Math. Phys
, 2000
"... The method of positive commutators, developed for zero temperature problems over the last twenty years, has been powering progress in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to nonequilibrium quantum statistical mechanics. We ..."
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Cited by 22 (11 self)
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The method of positive commutators, developed for zero temperature problems over the last twenty years, has been powering progress in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to nonequilibrium quantum statistical mechanics. We use the positive commutator technique to give an alternative proof of a fundamental property of large quantum systems, called Return to Equilibrium. This property says that equilibrium states are (asymptotically) stable: if a system is slightly perturbed from its equilibrium state, then it converges back to that equilibrium state as time goes to infinity. Keywords: positive commutator, Mourre estimate, return to equilibrium, virial theorem, Fermi golden rule Mathematics Subject Classification (2000): 82C10, 81Q10 1 Introduction In this paper, we study a class of open quantum systems consisting of two interacting subsystems: a finite system, called the particle system coupled to a reservo...