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131
Selfadjointness of the PauliFierz Hamiltonian for arbitrary coupling constants
, 2001
"... The PauliFierz Hamiltonian describes a system of N electrons interacting with a quantized radiation field. The electrons have spin and an ultraviolet cutoff is imposed on the quantized radiation field. For arbitrary coupling constants, selfadjointness and essential selfadjointness of the PauliFi ..."
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Cited by 47 (11 self)
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The PauliFierz Hamiltonian describes a system of N electrons interacting with a quantized radiation field. The electrons have spin and an ultraviolet cutoff is imposed on the quantized radiation field. For arbitrary coupling constants, selfadjointness and essential selfadjointness of the PauliFierz Hamiltonian are proven under a class of ultraviolet cutoffs. 1 Introduction The purpose of this paper is to establish the selfadjointness and the essential selfadjointness of the PauliFierz Hamiltonian [24] for arbitrary coupling constants. The PauliFierz Hamiltonian governs a system of N electrons interacting with a quantized radiation field. The N electrons are assumed to have spin and the quantized radiation field is smeared by an ultraviolet cutoff. The dynamics of the system is determined by the oneparameter unitary timeevolution generated by the PauliFierz Hamiltonian. So, as a first step, it is necessary to establish the selfadjointness of the PauliFierz Hamiltonian. Gener...
Existence of atoms and molecules in nonrelativistic quantum electrodynamics
 Adv. Theor. Math. Phys
"... We show that the Hamiltonian describing N nonrelativistic electrons with spin, interacting with the quantized radiation field and several fixed nuclei with total charge Z has a ground state when N < Z + 1. The result holds for any value of the fine structure constant α and for any value of the ul ..."
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Cited by 42 (4 self)
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We show that the Hamiltonian describing N nonrelativistic electrons with spin, interacting with the quantized radiation field and several fixed nuclei with total charge Z has a ground state when N < Z + 1. The result holds for any value of the fine structure constant α and for any value of the ultraviolet cutoff Λ on the radiation field. There is no infrared cutoff. The basic mathematical ingredient in our proof is a novel way of localizing the electromagnetic field in such a way that the errors in the energy are of smaller order than 1/L, where L is the localization radius. 1
InfraredFinite Algorithms in QED  I. The Groundstate of . . .
, 2005
"... In this first of a series of papers, the groundstate of a nonrelativistic atom, minimally coupled to the quantized radiation field, and its groundstate energy are constructed by an iteration scheme inspired by [8]. This scheme successively removes an infrared cutoff in momentum space and yields a c ..."
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Cited by 38 (10 self)
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In this first of a series of papers, the groundstate of a nonrelativistic atom, minimally coupled to the quantized radiation field, and its groundstate energy are constructed by an iteration scheme inspired by [8]. This scheme successively removes an infrared cutoff in momentum space and yields a convergent algorithm enabling us to calculate the groundstate and the groundstate energy, to arbitrary order in the feinstructure constant ff, 1=137.
Ground State of the Massless Nelson Model Without Infrared Cutoff in a NonFock Representation
 Rev. Math. Phys
, 2000
"... We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a nonFock representation of the timezero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling const ..."
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Cited by 37 (1 self)
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We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a nonFock representation of the timezero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The nonFock representation used is inequivalent to the Fock one if no infrared cutoff is made. Key words: Nelson's model, massless quantum field, infrared divergence, ground state, canonical commutation relations, nonFock representation 1 Introduction We consider a system of N quantum particles (N 2 IN) moving on the ddimensional Euclidean space IR d (d 2 IN) under the influence of an external potential V : IR dN ! IR (Borel measurable) and coupled to a massless quantum scalar field. The model we discuss here is the socalled massless Nelson model [10]. The problem to which we address ourselves in this paper is that of existen...
The infrared behaviour in Nelson’s model of a quantum particle coupled to a massless scalar field
 Ann. Henri Poincare
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Between classical and quantum
, 2008
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
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Cited by 37 (5 self)
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The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
Exponential Decay and Ionization Thresholds in NonRelativistic Quantum Electrodynamics
 J. Funct. Anal
, 2002
"... Spatial localization for quantum mechanical particles (electrons) interacting with quantized radiation (photons) is studied at energies below the ionization threshold. We give two de nitions of the ionization threshold. One in terms of minimal energies of nonlocalized states, and a second one in te ..."
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Cited by 32 (6 self)
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Spatial localization for quantum mechanical particles (electrons) interacting with quantized radiation (photons) is studied at energies below the ionization threshold. We give two de nitions of the ionization threshold. One in terms of minimal energies of nonlocalized states, and a second one in terms of spectral data of cluster Hamiltonians. We show that these de nitions agree, and that all states in the spectral subspace of energies below the ionization threshold decay exponentially in the particle coordinates. The latter result is derived from a new, general result on exponential decay tailored to t our problem, but applicable to many nonrelativistic quantum systems outside quantum electrodynamics as well. 1
Ground state degeneracy of the PauliFierz Hamiltonian inlcuding spin
 Adv. Theor. Math. Phys
, 2001
"... We consider an electron, spin 1/2, minimally coupled to the quantized radiation field in the nonrelativistic approximation, a situation defined by the PauliFierz Hamiltonian H. There is no external potential and H fibers as R according to the total momentum p. We prove that the ground state subspa ..."
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Cited by 31 (13 self)
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We consider an electron, spin 1/2, minimally coupled to the quantized radiation field in the nonrelativistic approximation, a situation defined by the PauliFierz Hamiltonian H. There is no external potential and H fibers as R according to the total momentum p. We prove that the ground state subspace of H p is twofold degenerate provided the charge e and the total momentum p are sufficiently small. We also establish that the total angular momentum of the ground state subspace is 1=2 and study the case of a confining external potential.
Ground state properties of the Nelson Hamiltonian  A Gibbs measurebased approach
 Rev. Math. Phys
, 2001
"... The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in positi ..."
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Cited by 30 (17 self)
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The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.
Mass renormalization and energy level shift in nonrelativistic QED
 Adv. Theor. Math. Phys
"... Dedicated to Elliott Lieb on the occasion of his 70th birthday Abstract. Using the PauliFierz model of nonrelativistic quantum electrodynamics, we calculate the binding energy of an electron in the field of a nucleus of charge Z and in presence of the quantized radiation field. We consider the cas ..."
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Cited by 28 (0 self)
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Dedicated to Elliott Lieb on the occasion of his 70th birthday Abstract. Using the PauliFierz model of nonrelativistic quantum electrodynamics, we calculate the binding energy of an electron in the field of a nucleus of charge Z and in presence of the quantized radiation field. We consider the case of small coupling constant α, but fixed Zα and ultraviolet cutoff Λ. We prove that after renormalizing the mass the binding energy has, to leading order in α, a finite limit as Λ goes to infinity; i.e., the cutoff can be removed. The expression for the ground state energy shift thus obtained agrees with Bethe’s formula for small values of Zα, but shows a different behavior for bigger values. 1.