We study the ultraviolet problem for quantum electrodynamics on a three dimensional torus. We start with the lattice gauge theory on a toroidal lattice and seek to control the singularities as the lattice spacing is taken to zero. This is done by following the flow of a sequence of renormalization group transformations. The analysis is facilitated by splitting the space of gauge fields into large and small fields at each step following Balaban. In this paper we explore the first step in detail.
We continue the study of quantum electrodynamics on a three dimensional torus as a limit of a lattice gauge theory. In this paper we give a preliminary treatment of the renormalization group flow. We study the propagators which arise under multiple block spin averaging, both in global and local versions. We also study low order perturbation theory. However we do not control remainders. This is left to the more complete treatment of the following paper.
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 $.
A group theoretical description of the Majorana-Oppenheimer quantum electrodynamics is considered. Different spinor realizations of the Maxwell and Dirac fields are discussed. A representation of the Majorana-Oppenheimer wave equations in terms of the Gel'fand-Yaglom formalism is given.
A physical subspace and physical Hilbert space associated with asymptotic fields of nonrelativistic quantum electrodynamics are constructed through the Gupta-Bleuler procedure. Asymptotic completeness is shown and a physical Hamiltonian is defined on the physical Hilbert space.
We propose a new method to derive certain higher order estimates in quantum electrodynamics. Our method is particularly convenient in the application to the non-local semi-relativistic models of quantum electrodynamics as it avoids the use of iterated commutator expansions. We re-derive higher order estimates obtained earlier by Fröhlich, Griesemer, and Schlein and prove new estimates for a non-local molecular no-pair operator.
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.
An algebraic description of basic physical fields (neutrino field, electron-positron field and electromagnetic field) is studied. It is sown that the electromagnetic field can be described within a quotient representation of the proper orthochronous Lorentz group. The relation of such a description with Majorana-Oppenheimer quantum electrodynamics and de Broglie-Jordan neutrino theory of light is discussed.
The Bogoliubov-Dirac-Fock (BDF) model is a no-photon, mean-field approxi- mation of quantum electrodynamics. It describes relativistic electrons in the Dirac sea. In this model, a state is fully characterized by its one-body density matrix, an infinite rank nonnegative operator. We prove the existence of the positronium, the bound state of an electron and a positron, represented by a critical point of the energy functional in the absence of external field. This state is interpreted as the ortho-positronium, where the two particles have parallel spins.
The Staruszkiewicz quantum model of the long-range structure in electrodynamics is reviewed in the form of a Weyl algebra. This is followed by a personal view on the asymptotic structure of quantum electrodynamics.
A causal scattering matrix of quantum electrodynamics is constructed by means of chronological product of Lagrangians where the fields have the different arguments. This scattering matrix is a convergent series and does not contain the diverging integrals.
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.
Phase-space techniques are generalized to nonlinear quantum electrodynamics beyond the rotating wave approximation, resulting in an essentially classical picture of radiation dynamics.
In this article, we are interested in a spin model including the quantized electromagnetic field (photons). With this model of quantum electrodynamics (QED) related to nuclear magnetic resonance (NMR) we give explicit quantum radiative corrections of the time evolution for the spin observables, for the electric and magnetic fields observables and for the photon number observable. As a by-product, this underlines that Bloch equations are the semiclassical limit of the model in QED considered here. In addition, transition probabilities for the same model are investigated.
We prove the limiting absorption principle for the standard model of the non-relativistic quantum electrodynamics (QED) and for the Nelson model describing interactions of electrons with phonons. To this end we use the spectral renormalization group technique on the continuous spectrum in conjunction with the Mourre theory.
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.
In 1999, A. Connes and D. Kreimer have discovered the Hopf algebra structure on the Feynman graphs of scalar field theory. They have found that the renormalization can be interpreted as a solving of some Riemann -- Hilbert problem. In this work a generalization of their scheme to the case of quantum electrodynamics is proposed. The action of the gauge group on the Hopf algebra of diagrams are defined and the proof that this action is consistent with the Hopf algebra structure is given.
We propose a new method to quantize gauge theories formulated on a canonical noncommutative spacetime with fields and gauge transformations taken in the enveloping algebra. We show that the theory is renormalizable at one loop and compute the beta function and show that the spin dependent contribution to the anomalous magnetic moment of the fermion at one loop has the same value as in the commutative quantum electrodynamics case.
This paper discusses an attempt to develop a mathematically rigorous theory of Quantum Electrodynamics (QED). It deviates from the standard version of QED mainly in two aspects: it is assumed that the Coulomb forces are carried by transversely polarized photons, and a reducible representation of the canonical commutation and anti-commutation relations is used. Both interventions together should suffice to eliminate the mathematical inconsistencies of standard QED.
The quantum field theory of extended objects is employed to address the hitherto nonrenormalizable Pauli interaction. This is achieved by quantizing the Dirac field using the infinite dimensional generalization of the extended object formulation. The order $α$ contribution to the anomalous magnetic moment of the electron (and of the muon) arising from the Pauli term is calculated.
In this paper, we construct the quantum Torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum Torus Lie algebra in the KP system by acting on its tau function. Comparing to the $W_{\infty}$ symmetry, this quantum Torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum Torus symmetries of the KdV and BKP hierarchies and further derive the quantum Torus constraints on their tau functions. These quantum Torus c…
