Toshimitsu Takaesu
In this paper, we investigate a system of quantum electrodynamics with cutoffs. The total Hamiltonian is defined on a tensor product of a fermion Fock space and a boson Fock. It is shown that, under spatially localized conditions and momentumregularity conditions, the total Hamiltonian has a ground state for all values of coupling constants. In particular, its multiplicity is finite.
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We consider an Hamiltonian with ultraviolet and infrared cutoffs, describing the interaction of relativistic electrons and positrons in the Coulomb potential with photons in Coulomb gauge. The interaction includes both interaction of the current density with transversal photons and the Coulomb interaction of charge density with itself. We prove that the Hamiltonian is self-adjoint and has a ground state for sufficiently small coupling constants.
In this paper the Hamiltonian of quantum electrodynamics with spatial cutoffs is investigated. We define a scaled total Hamiltonian and consider its asymptotic behavior. In the main theorem, it is shown that the scaled total Hamiltonian converges to a self-adjoint operator in the strong resolvent sense, and effective potentials are derived.
In this paper, we consider the spectrum of a model in quantum electrodynamics with a spatial cutoff. It is proven that (1) the Hamiltonian is self-adjoint; (2) under the infrared regularity condition, the Hamiltonian has a unique ground state for sufficiently small values of coupling constants. The spectral scattering theory is studied as well and it is shown that asymptotic fields exist and the spectral gap is closed.
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 alpha and for any value of the ultraviolet cutoff Lambda 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.
We show that relativistic quantum electrodynamics in the Coulomb gauge satisfies the following bound, which establishes stability: let $H(Λ,V)$ denote the Hamiltonian of $QED_{1+3}$ on the three-dimensional torus of volume $V$ and with ultraviolet cutoff $Λ$. Then there exists a constant $0<μ(Λ,V)<\infty$ (the vacuum energy renormalization) such that the renormalized Hamiltonian is positive: $H_{ren}(Λ,V) \equiv H_{Λ,V}+μ_{Λ, V}\cdot \mathbb{1} \ge 0 $.
In this paper, we investigate the $ϕ^4$ model with cutoffs. By introducing a spatial cutoff and a momentum cutoff, the total Hamiltonian is a self-adjoint operator on a boson Fock space. Under regularity conditions of the momentum cutoff, we obtain the first order expansion of a non-degenerate ground state energy of the total Hamiltonian.
The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state--one that minimizes the energy and satisfies the Schrödinger equation. We prove quite generally that this state exists for {\it all values} of the fine-structure constant and ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions.
We consider a system of N nonrelativistic particles of spin 1/2 interacting with the quantized Maxwell field (mass zero and spin one) in the limit when the particles have a small velocity, imposing to the interaction an ultraviolet cutoff, but no infrared cutoff. Two ways to implement the limit are considered: c going to infinity with the velocity v of the particles fixed, the case for which rigorous results have already been discussed in the literature, and v going to 0 with c fixed. The second case can be rephrased as the limit of heavy particles, m_{j} --> epsilon^{-2}m_{j}, observed over a…
Ground states of the so called Yukawa model is considered. The Yukawa model describes a Dirac field interacting with a Klein-Gordon field. By introducing both ultraviolet cutoffs and spatial cutoffs, the total Hamiltonian is defined as a self-adjoint operator on a boson-fermion Fock space. It is shown that the total Hamiltonian has a positive spectral gap for all values of coupling constants. In particular the existence of ground states is proven.
A translation invariant Hamiltonian $H$ in the nonrelativistic quantum electrodynamics is studied. This Hamiltonian is decomposed with respect to the total momentum $\tot$: $$H=\int_{\BR} ^\oplus \fri(P) dP,$$ where the self-adjoint fiber Hamiltonian $\fri(P)$ is defined for arbitrary values of coupling constants. It is discussed a relationship between rotation invariance of $H(P)$ and polarization vectors, and functional integral representations of $n$ point Euclidean Green functions of $H(P)$ is given. From these, some applications concerning with degeneracy of ground states, ground state en…
In this paper we consider the massless translation invariant Nelson model with ultraviolet cutoff. It is proven that the fiber operators have no ground state if there is no infrared cutoff.
The effective mass $\mass$ of the the Pauli-Fierz Hamiltonain with ultraviolet cutoff $Λ$ and the bare mass $m$ in nonrelativistic QED with spin 1/2 is investigated. Analytic properties of $\mass$ in coupling constant $e$ are shown and explicit forms of constants $a_1(Λ/m)$ and $a_2(Λ/m)$ depending on $Λ/m$ such that $$\mass/m =1 + a_1(Λ/m) e^2+ a_2(Λ/m) e^4+ {\mathcal O}(e^6)$$ are given. It is shown that the spin interaction enhances the effective mass and that there exist strictly positive constants $b_1,b_2, c_1$ and $c_2$ such that $$\d b_1\leq \lim_{Λ\to\infty} \frac{a_1(\Lamdda/m)}{\log…
We study three well known models of matter coupled to the ultraviolet cutoff, quantized radiation field and to the Coulomb potential of arbitrarily many nuclei. Two are nonrelativistic: the first uses the kinetic energy (p+eA(x))^2 and the second uses the Pauli-Fierz energy (p+eA(x))^2 +eσ\cdot B(x). The third, no-pair model, is relativistic and replaces the kinetic energy with the Dirac operator D(A), but restricted to its positive spectral subspace, which is the ``electron subspace''. In each case we are able to give an upper bound to the {\it binding} energy -- as distinct from the less dif…
The translation invariant model in quantum field theory is considered by functional integrations. Ultraviolet renormalization of the translation invariant Nelson model with a fixed total momentum is proven by functional integrations. As a corollary it can be shown that the Nelson Hamiltonian with zero total momentum has a ground state for arbitrary values of coupling constants in two dimension. Furthermore the ultraviolet renormalization of the polaron model is also studied.
We consider the semi-relativistic Pauli-Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo's iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain Hamiltonian unitarily equivalent to a renormalized fiber Hamiltonian acting in a coherent state representation space. The ground state energy is an exactly two-fold degene…
Hamiltonian models based on two different infrared approximations are studied in order to obtain an explicit comparison with the standard analysis of the infrared contributions, occurring in the relativistically covariant perturbative formulation of Quantum Electrodynamics. Moller operators, preserving respectively the Hilbert scalar product, for the Coulomb-gauge models, and an indefinite metric, for the models formulated in Feynman's gauge, are obtained in the presence of an infrared cutoff, after the removal of an adiabatic switching and with the aid of a suitable mass renormalization. In t…
We discuss recent results concerning the ground state of non-relativistic quantum electrodynamics as a function of a magnetic coupling constant or the fine structure constant, obtained by the authors in [12,13,14].
The Pauli-Fierz model with a variable mass $v$ is considered. An ultraviolet cutoff and an infrared regularity condition are imposed on a quantized radiation field. It is shown that the ground state exists for arbitrary values of coupling constants, when $v(x)\leq c\langle x\rangle^{-β}$, $β>3$.
The relativistic ``no pair'' model of quantum electrodynamics uses the Dirac operator, D(A), for the electron dynamics together with the usual self-energy of the quantized ultraviolet cutoff electromagnetic field A -- in the Coulomb gauge. There are no positrons because the electron wave functions are constrained to lie in the positive spectral subspace of some Dirac operator, D, but the model is defined for any number, N, of electrons, and hence describes a true many-body system. In addition to the electrons there are a number, K, of fixed nuclei with charges \leq Z. If the fields are not qua…
A Lagrangian for quantum electrodynamics is found which makes it explicit that the photon mass is eventually set to zero in the physical part on observational ground. It remains possible to obtain a counterterm Lagrangian where the only non-gauge-invariant term is proportional to the squared divergence of the potential, while the photon propagator in momentum space falls off like the inverse squared power of k at large k, which indeed agrees with perturbative renormalizability. The resulting radiative corrections to the Coulomb potential in QED are also shown to be gauge-independent. A fundame…
Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics
本文研究无光子量子电动力学平均场近似（BDF模型）中电荷约束下能量泛函极小元的存在性，证明在束缚条件下极小元存在，并给出弱耦合与非相对论两种满足条件的情形，且非相对论极限收敛于Hartree-Fock基态。
We study two Hamiltonian models, based on infrared approximations which render them solvable, in order to obtain an operator formulation of the soft-photon corrections to the scattering of a single electron, as given in Quantum Electrodynamics by the method of Feynman's diagrams. The first model is based on the same approximations of the Pauli-Fierz Hamiltonian, the second one stems from an expansion in powers of the four-momentum transfer, along the lines of Bloch and Nordsieck. For both models, the dynamics of the charge is accounted for by suitably chosen classical currents, interacting wit…
This work addresses the problem of infrared mass renormalization for a scalar electron in a translation-invariant model of non-relativistic QED. We assume that the interaction of the electron with the quantized electromagnetic field comprises a fixed ultraviolet regularization and an infrared regularization parametrized by $σ>0$. For the value $p=0$ of the conserved total momentum of electron and photon field, bounds on the renormalized mass are established which are uniform in $σ\to0$, and the existence of a ground state is proved. For $|p|>0$ sufficiently small, bounds on the renormalized ma…
Hamiltonians for polaron models with subcritical ultraviolet singularities
针对非相对论量子场论中极化子类模型的紫外问题，研究覆盖所有次临界超可重整相互作用，通过有限迭代得到显式自伴的无截断哈密顿量，且截断哈密顿量经基态能量微扰近似扣除后在强预解式意义下收敛于该算子。
We study the mean-field approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal-ordering or choice of bare electron/positron subspaces. Neglecting photons, we define properly this Hamiltonian in a finite box $[-L/2;L/2)^3$, with periodic boundary conditions and an ultraviolet cut-off $Λ$. We then study the limit of the ground state (i.e. the vacuum) energy and of the minimizers as $L$ goes to infinity, in the Hartree-Fock approximation. In case with no external field, we prove that the energy p…
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