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.     

On a class of relativistic invariant distributions

Space-time properties of a class of relativistic invariant distributions of the type were considered for different sequences of coefficientsc _n .     

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.     

Local relativistic invariant flows for quantum fields

For quantum fields with trigonometric interaction in arbitrary space dimension we construct a representation of the Lorentz group by automorphisms on a Banach space generated by the Weyl algebra.     

Constancy of the local velocity of light in general relativistic gravitational fields

Using the natural, real units (proper time and proper length) for the definition of the spatial velocity, it is shown that the velocity of light during its propagation through gravitational fields is constant and exactly equal to its value in vacuum. Therefore, according to General Relativity, it is not correct to speak about a variable velocity of light in gravitational fields.     

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.     

Singularity formation in general relativistic dynamics of homogeneous scalar fields

Collapsing dynamics of a wide class of self-interacting, self-gravitating homogeneous scalar field models is analyzed. The assumptions made on the potential satisfy some general conditions allowing to show that the generic evolution is divergent in a finite time. Combining results shown here with the ones from [R. Giambó, F. Giannoni, G. Magli, J. Math. Phys. 49 (2008) 042504], dealing with sub-exponential growing potentials, allows us to obtain the same results of singularity formation for more general potentials. Moreover it turns out that these models can be completed to find radiating collapsing star models of the Vaidya type, where blackholes are generically formed.     

The Jost-Lehmann-Dyson representation for a wide class of local fields

The connection between solutions of the wave equation in +1 space-time dimensions and the support properties of the restriction to dimensions of their Fourier transforms is investigated for a class of functionals over function spaces which are supposed to contain some elements with compact support, and then, may describe a wide class of localizable quantized fields.     

Particle dynamics in a relativistic invariant stochastic medium

The dynamics of particles moving in a medium defined by its relativistically invariant stochastic properties is investigated. For this aim, the force exerted on the particles by the medium is defined by a stationary random variable as a function of the proper time of the particles. The equations of motion for a single one-dimensional particle are obtained and numerically solved. A conservation law for the drift momentum of the particle during its random motion is shown. Moreover, the conservation of the mean value of the total linear momentum for two particles repelling each other according to the Coulomb interaction also follows. Therefore, the results indicate the realization of a kind of stochastic Noether theorem in the system under study.     

Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n ^–1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.     

Lagrangian interactions within a special class of covariant mixed-symmetry type tensor gauge fields

Consistent non-trivial interactions within a special class of covariant mixed-symmetry type tensor gauge fields of degree three are constructed from the deformation of the solution to the master equation combined with specific cohomological techniques. In spacetime dimensions strictly greater than four, the only consistent interaction terms are those gauge invariant under the original symmetry. Only in four spacetime dimensions the gauge symmetry is found to be deformed. Received: 21 November 2002 / Revised version: 9 December 2002 / Published online: 14 February 2003     

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.     

Models of relativistic hamiltonian interactions

A model of a relativistically invariant Hamiltonian 2-particle interaction is given. It is classical in the sense of having 6 degrees of freedom. This model shows that the author's earlier general definition of such systems is not vacuous. In this model the forces of interaction die away as the particles are removed from each other.     

Review of invariant time formulations of relativistic quantum theories

The purpose of this paper is to review relativistic quantum theories with an invariant evolution parameter. Parametrized relativistic quantum theories (PRQT) have appeared under such names as constraint Hamiltonian dynamics, four-space formalism, indefinite mass, micrononcausal quantum theory, parametrized path integral formalism, relativistic dynamics, Schwinger proper time method, stochastic interpretation of quantum mechanics and stochastic quantization. The review focuses on the fundamental concepts underlying the theories. Similarities as well as differences are highlighted, and an extensive bibliography is provided.     

Shape invariant propagation of polychromatic fields

When a polychromatic field propagates the transverse distribution of the power spectrum generally changes from one plane to another. Conditions are found under which the spectrum distribution maintains the same shape in any cross-section. Examples of fields satisfying these conditions are given.     

Algebraic study of a class of relativistic wave equations

First-order relativistic wave equations are considered whose irreducible matrix coefficients satisfy the simplest (except for the Dirac algebra) unique mass condition, (β · p)³ = p²(β · p), which is also sufficient to guarantee causality in a minimally coupled external electromagnetic field. All of the associated representations of SL(2, ©) are classified and studied up to and including those which are the direct sum of four irreducible components, (n, m), with either n or m less than two. A large number of families of representations are found which permit the algebraic condition to be satisfied. These are tabulated according to whether a Hermitian choice for β⁰ is possible and their spin content is given. If a unique spin is described, then the only possible representations are

$\text{(1) (}\text{n}\text{,0) ⊕ (}\text{n}\text{? 1/2, 1/2)}$

$\text{(2) (}\text{n}\text{,0) ⊕ (}\text{n}\text{+ 1/2, 1/2)}$

$\text{(3) (}\text{n}\text{+ 1/2, 1/2) ⊕ (}\text{n}\text{,0) ⊕ (}\text{n}\text{? 1/2, 1/2)}$

$\text{(4) (1,0) ⊕ (1/2, 1/2) ⊕ (0,1)}$

and their conjugates. If, in addition, the representation is assumed to be self-conjugate, then only the Dirac and Petiau-Duffin-Kemmer equations survive.     

A class of nontrivial weakly local massive wightman fields with interpolating properties

It is shown that in a quantum field theory satisfying Wightman's axioms with locality replaced by weak locality and cyclicity by a weak irreducibility, every unitary Poincaré invariant and CPT-invariant operator is a scattering operator (in the LSZ-sense). The proof is given by explicit construction of a corresponding class of nontrivial weakly local massive Wightman fields. This result implies Jost's conjecture that only locality leads to nontrivial restrictions for the scattering operator and extends corresponding results of Schneider.     

A general class of quantum fields without cut-offs in two space-time dimensions

We consider a selfinteracting boson field in two space-time dimensions, with interaction densities of the form:V((x)): where (x) is a scalar boson field, andV() is a real positive function of exponential type. We define the space cut-off interaction by and prove thatH _r =H ₀+V _r , whereH ₀ is the free energy, is essentially self adjoint. This permits us to take away the space cut-off and we obtain a quantum field free of cut-offs.At leave from Mathematical Institute, Oslo University.This research partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.