arXiv subjects
Publications and source records attributed to J. Dimock.
This is a non-perturbative treatment of correlation functions for the weakly coupled massless Gross-Neveu model in a finite volume. The main result is that all correlation functions, treated as distributions, are uniformly bounded in the ultraviolet cutoff.
We study flow of renormalization group (RG) transformations for the massless Gross-Neveu model in a non-perturbative formulation. The model is defined on a d=2 dimensional Euclidean space with a finite volume. The quadratic approximation to the flow stays bounded after suitable renormalization. We show that for weak coupling this property also is true for the complete flow. As an application we prove an ultraviolet stability bound for the model. Our treatment is an application of a method of Bauerschmidt, Brydges, and Slade. The method was developed for an infrared problem, and is now applied to an ultraviolet problem.
We report on a result on quantum electrodynamics on a three dimensional Euclidean spacetime. The model is formulated on a toroidal lattice with unit volume and variable lattice spacing. The result is that the renormalized partition function is bounded above and below uniformly in the lattice spacing. This is a first step toward showing that the partition function and correlation functions have limits as the lattice spacing goes to zero.
We study the radiation of photons from a classical charged particle. We particularly consider a situation where the particle has a constant velocity in the distant past, then is accelerated, and then has a constant velocity in the distant future. Starting with no photons in the distant past we seek to characterize the quantum state of the photon field in the distant future. Working in the Coulomb gauge and in a C* algebra formulation, we give sharp conditions on whether this state is or is not in Fock space.
We continue the study of the ultraviolet problem for QED in d=3 using Balaban's formulation of the renormalization group. The model is defined on a fine toroidal lattice and we seek control as the lattice spacing goes to zero. Drawing on earlier papers in the series the renormalization group flow is completely controlled for weak coupling. The main result is an ultraviolet stability bound in a fixed finite volume.
We consider the quantum mechanics of a charged particle in the presence of Dirac's magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the a Feynman-Kac formula expressing solutions of the imaginary time Schrodinger equation as stochastic integrals.
We construct wave operators and a scattering operator for the scattering of a charged particle on the Dirac magnetic monopole. The analysis features a two Hilbert space approach in which the identification operator matches states of the same angular momentum.
We continue the study of the ultraviolet problem for QED in d=3. The model is defined on a fine toroidal lattice and we seek control as the lattice spacing goes to zero. The problem is analyzed using Balaban's formulation of the renormalization group. This involves a sequence of transformations consisting of a split into large and small field regions, then block averaging, and then scaling. The the effective actions generated by this method depend strongly on certain multi-scale propagators and minimizers. The study of these objects both for fermions and for gauge fields is content of this paper. Earlier work on the subject is reviewed. In addition for fermions a polymer expansion is obtained for the determinants of the fermion propagators. For the gauge field a detailed local regularity result is obtained for the minimizers.
We study the ultraviolet problem for QED in d=3 using Balaban's formulation of the renormalization group. The model is defined on a fine toroidal lattice and we seek control as the lattice spacing goes to zero. As a first step we take a bounded field approximation and solve the renormalization problem. Namely we show that the bare energy density and the bare fermion mass can be chosen to depend on the lattice spacing, so that under the renormalization group flow they take preassigned values on unit scale. This is accomplished by a nonpertubative technique which is insensitive to whether the renormalizations are finite or infinite.
For scalar QED on a three-dimensional toroidal lattice with a fine lattice spacing we consider the renormalization problem of choosing counter terms depending on the lattice spacing, so that the theory stays finite as the spacing goes to zero. We employ a renormalization group method which analyzes the flow of the mass and the vacuum energy as a problem in discrete dynamical systems. The main result is that counter terms can be choosen so that at the end of the iteration these quantities take preassigned values. No use is made of perturbation theory. The renormalization group transformations are defined with bounded fields, an approximation which can be justified in Balaban's approach to the renormalization group.
We consider abelian gauge theories on a lattice and develop properties of an axial gauge that is covariant under lattice symmetries. Particular attention is paid to a version that behaves nicely under block averaging renormalization group transformations.
This is an expository account of Balaban's approach to the renormalization group. The method is illustrated with a treatment of the ultraviolet problem for the scalar phi^4 model on a toroidal lattice in dimension d=3. In this third paper we demonstrate convergence of the expansion and complete the proof of a stability bound.
This is an expository account of Balaban's approach to the renormalization group. The method is illustrated with a treatment of the the ultraviolet problem for the scalar phi^4 model on toroidal lattice in dimension d=3. In this second paper we control the large field contribution to the partition function.
This is an expository account of Balaban's approach to the renormalization group. The method is illustrated with a treatment of the the ultraviolet problem for the scalar phi^4 model on a toroidal lattice in dimension d=3. This yields another proof of the stability bound. In this first paper we analyze the small field contribution to the partition function.
We give an alternate definition of the free Dirac field featuring an explicit construction of the Dirac sea. The treatment employs a semi-infinite wedge product of Hilbert spaces. We also show that the construction is equivalent to the standard Fock space construction.
We consider a classical dipole gas in with low activity and show that the pressure has a limit as the volume goes to infinity. The result is obtained by a renormalization group analysis of the model.
We consider a conformal field theory for bosons on the Riemann sphere. Correlation functions are defined as singular limits of functional integrals. The main result is that these amplitudes define transition amplitudes, that is multilinear Hilbert-Schmidt functionals on a fixed Hilbert space.
We consider scalar quantum fields on the sphere, both massive and massless. In the massive case we show that the correlation functions define amplitudes which are trace class operators between tensor products of a fixed Hilbert space. We also establish certain sewing properties between these operators. In the massless case we consider exponential fields and have a conformal field theory. In this case the amplitudes are only bilinear forms but still we establish sewing properties. Our results are obtained in a functional integral framework.