The time evolution of quantized fields with bounded quasi-local interaction density

We extend to arbitrary dimension the proof by Guenin that the time-evolution is an automorphism group of the local algebras, if the interaction Hamiltonian is a space-integral of a bounded local density with finite range.Supported by the Science Research Council.     

Boson fields with nonlinear selfinteraction in two dimensions

Semiboundedness of the total Hamiltonian is proved for a selfinteracting Boson field in two dimensional space time. The interaction is given by a Wick polynomial:P():. The polynomialP is required to have even degree and its leading coefficient must be positive. A space cutoff is introduced in the interaction Hamiltonian.This work was supported in part by the National Science Foundation, NSF GP 7477     

A local relativistic Boson field with the λ|ϕ| interaction

We prove that the Heisenberg picture fields for a self interacting Boson field with the || interaction in four space time dimensions exists as weak limits of Heisenberg picture fields corresponding to the cut-off interactions.     

Quantized fields in interaction with external fields

We consider a massive, charged, scalar quantized field interacting with an external classical field. Guided by renormalized perturbation theory we show that whenever the integral equations defining the Feynman or retarded or advanced interaction kernel possess non perturbative solutions, there exists anS-operator which satisfies, up to a phase, the axioms of Bogoliubov, and is given for small external fields by a power series which converges on coherent states. Furthermore this construction is shown to be equivalent to the one based on the Yang-Källen-Feldman equation. This is a consequence of the relations between chronological and retarded Green's functions which are described in detail.     

Boson fields with the :Φ4: Interaction in three dimensions

The :⁴: interaction for boson fields is considered in three dimensional space time. A space cutoff is included in the interaction term. The main result is that the renormalized HamiltonianH _ren is a densely defined symmetric operator. In addition to the infinite vacuum energy and infinite mass renormalizations, this theory has an infinite wave function renormalization. Consequently the Hilbert space (of physical particles) in whichH _ren acts is disjoint from the bare particle Fock Hilbert space in which the unrenormalized Hamiltonian is defined.This work was supported in part by the National Science Foundation, NSF GP 7477.     

Boson fields under a general class of cut-off interactions

We consider a boson field (x) under an interaction of the form V( _k (x))dx, where _k (x) is the momentum cut-off field, andV() is a continuous bounded function. Under a weak regularity condition onV(), we prove that the total energy operator is self adjoint, that the asymptotic fields exist and that the scattering operator exists.     

Boson free fields as a limit of fields of a more general type

We study a class of W¹-algebras with a state, a group of automorphisms and a space of some selected self-adjoint elements. By means of a kind of averaging over the space we obtain a special class of operators that behave like boson fields in the large scale limit.     

Support properties of the free measure for Boson fields

Let μ be the measure onI′ (? ^d ) corresponding to the Gaussian process with mean zero and covariance (f,(?Δ+1)^?1 g) onI (? ^d ). It is proven that the set $$( - \Delta _{d - 1} + 1)^{d/4 - \tfrac{1}{2} + \alpha } (1 + x^2 )^{d/4} [\log (2 + x^2 )]^\beta L^2 (\mathbb{R}^d )$$ has μ measure one if α>0 and β>1/2 and μ measure zero if α>0 and β<1/2; here Δ _d?1 is the Laplacian in anyd?1 dimensions whend>1 and Δ₀=Δ.     

Coulomb final state interaction in identical Boson interferometry

Recently, the muonic three-body systemdtμ, has aroused considerable attention in relation to the realization of a useful muon-catalyzed fusion [1, 2]. This bound system must be solved up to 6 digits in energy to establish the muon catalyzed fusion process. We applied the ATMS method [3] and the coupled-channel method [4] to the Coulomb three-body system and obtained the detailed information on the energy levels and the wave functions of the bound states. Further we investigated the effect of the nuclear interaction on the fusion rate in the bound states [5] and developed a formalism to calculate the muon sticking to⁴He [6]. In this paper we report the results obtained in our collaborations: Figure 2 shows up our main results.     

Fermions in interaction with time-dependent fields

We solve a two-dimensional model describing the interaction of fermions with time-dependent external fields. We work out the second quantized formulation and obtain conditions for equivalence of representations at different times. This implies the existence of sectors which describe charged states. We obtain the time dependence of charges and observe that charge differences become integer for unitary equivalent states. For scattering we require the equivalence of in- and out-representations; nevertheless charged sectors may be reached by suitable interactions and ionization is possible.     

On the Gibbs states for one-dimensional lattice Boson systems with a long-range interaction

We consider an infinite chain of interacting quantum (anharmonic) oscillators. The pair potential for the oscillators at lattice distanced is proportional to {d ²[log(d+1)]F(d)}^–1 where _rZ [rF(r)]^–1 < . We prove that for any value of the inverse temperature> 0 there exists a limiting Gibbs state which is translationally invariant and ergodic. Furthermore, it is analytic in a natural sense. This shows the absence of phase transitions in the systems under consideration for any value of the thermodynamic parameters.     

Boson fields under a general class of local relativistic invariant interactions

We consider a boson field (x) under an interaction of the form V((x))dx, whereV() is a bounded continuous real function of a real variable . IfV() has a uniformly continuous and bounded first derivative, we prove that the Heisenberg picture field exists as weak limits of the Heisenberg picture fields corresponding to the cut-off interaction.     

Group classification of scalar fields with potential interaction

A system of scalar fields with Lagrangian L=(1/2)u_,iu^,i + V(u) in multi-dimensional space—time is considered. The structure of the permissible Lie group is studied for an arbitrary field multiplet. A complete group classification of field doublets and triplets is presented. All conservative fluxes of these systems are calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 116–121, March, 1995.     

Velocity fields and transition densities in nuclear collective modes

The shape of the deformations occurring in nuclear collective modes is investigated by means of a microscopic approach. Analytical solutions of the equations of motion are obtained by using simplified nuclear potentials. It is found that the structure of the velocity field and of the transition density of low-lying modes is considerably different from the predictions of irrotational hydrodynamic models. The low-lying octupole state is studied in particular detail by using the Skyrme force.     

Boson expansions and quantization of time-dependent self-consistent fields (I). Particle-hole excitations

Two concrete methods are presented for quantizing the time-dependent Hartree equations in terms of boson operators. The first is the well-known infinite boson expansion analogous to the Holstein-Primakoff representation of angular momentum operators. The second, a new development, consists of finite boson quadratic forms, and is analogous to the Schwinger representation of angular momenta. In each case, a physical boson subspace can easily be constructed within which the full fermion dynamics is exactly duplicated. It therefore follows that quantization of the time-dependent Hartree equations, including all degrees of freedom, retrieves the exact many-body problem. The discussion in this paper is limited to particle-hole excitations of an N-particle system. A generalization to one-nucleon transfer processes on the N-particle system is also given in terms of ideal odd nucleons, but this brings in infinite expansions.     

An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction

Using fixed point theorems for local contractions in Banach spaces, an existence and uniqueness proof for the Hartree-Fock time-dependent problem is given in the case of a finite Fermi system interacting via a bounded two-body potential. The existence proof for the “strong” solution of the evolution problem is obtained under suitable conditions on the initial state.