Regular Series


Vol. 6 (1975), No. 7 – 8, pp. 467 – 650


Quantization of the Electromagnetic Field in Riemannian Spaces

abstract

The canonical quantization procedure for the free electromagnetic field is generalized on arbitrary space-times. The metric is considered as classical background field which influences the quantized Maxwell field. Using the Heisenberg picture we get a system of first-order linear ordinary differential equations describing the time dependence of the operators. In order to compute the coefficients of this system one has to solve a 3-dimensional eigenvalue problem. It is not necessary to choose a special gauge of the potentials.


Spherical Symmetry in Rainich’s Theory

abstract

A time-dependent, spherically symmetric solution of Rainich’s field equations is derived when the field is null. It is concluded that in this case the Already Unified Field Theory and the Einstein–Maxwell theory are not equivalent.


Non-Linear Electrodynamics in the Newman–Penrose Formalism

abstract

An approach to non-linear electrodynamics by means of the spin-coefficient formalism of Newman and Penrose is presented. The field equations are rewritten in the Newman–Penrose form, and their spherically symmetric solutions are discussed. An approximation procedure, suitable for treating radiation problems in non-linear electrodynamics, is then suggested on the basis of an analysis of the Maxwell electrodynamics with sources. Since the field equations are non-linear, wave tails will in general develop. This is illustrated in detail on an example of the approximate solution, representing a radiating dipole in the zero approximation. Conserved quantities, analogous to those discovered by Newman and Penrose in Maxwell’s and Einstein’s theories are found for a large class of non-linear theories of electrodynamics. Their number depends on the choice of a particular theory — it is greater than, or equal to 16 for theories satisfying the correspondence principle with Maxwell’s theory.


Null Canonical Formalism I, Maxwell Field

abstract

The purpose of this paper is to formulate the canonical formalism on null hypersurfaces for the Maxwell electrodynamics. The set of the Poisson brackets relations for null variables of the Maxwell field is obtained. The asymptotic properties of the theory are investigated. The Poisson bracket relations for the news-functions of the Maxwell field are computed. The Hamiltonian form of the asymptotic Maxwell equations in terms of these news-functions is obtained.


Null Canonical Formalism II, Einstein Field

abstract

The purpose of this paper is to formulate the canonical formalism on null hypersurfaces for the Einstein field. Using the analogy between Maxwell’s theory and gravitation theory, a set of the Poisson bracket relations for the null Weyl tensor components is obtained. The asymptotic properties of the theory are investigated. The Poisson bracket relations for the Bondi news-functions are computed. The Hamiltonian form of the asymptotic Einstein equations on terms of the news-functions is found.


The Gauss-Codazzi Equations and Field Equations for a Special Case of Null Hypersurfaces in the Theory of Gravitation

abstract

It is shown that on every null hypersurface a linear connection exists which is both metric and integrable, but not symmetric in general. Using this connection the Gauss–Codazzi equations are derived in the case when the null hypersurface admits this connection to be symmetric. Then these equations are very simple and are used to prove that the Einstein constraint equations impose no restrictions on the inner geometry of those particular null hypersurfaces.


A Class of Cosmological Models with Torsion and Spin

abstract

Homogeneous models of the Universe filled with a spinning fluid are studied in the framework of the Einstein–Cartan theory of gravitation. It is assumed that the models admit a group of motions simply transitive on three-surfaces orthogonal to the world lines of the substratum. For certain group types, the field equations are partially integrated. The models of the Bianchi types I, VII\(_0\), V are shown to be non-singular, provided the influence of spin exceeds that of shear, and an equation of state satisfies some physically reasonable conditions.


The Einstein–Cartan Equations in Astrophysically Interesting Situations. I. The Case of Spherical Symmetry

abstract

The Einstein–Cartan equations (general relativity plus spin in a manifold with curvature and torsion) are written explicitly for the case of spherical symmetry. In this case there may exist, at most, eight non-vanishing independent components of the torsion tensor when one does not assume the “classical description” of spin. It is shown explicitly, by giving various non-equivalent ways to do it, how exact solutions of spherical symmetry for matter-filled regions may be generalized from general relativity into the Einstein–Cartan theory. A classification of cosmological models with the Robertson–Walker metric in the Einstein–Cartan theory is given.


Wave Equations for Unstable Particles and Resonances: General Considerations and Soluble Models

abstract

We propose a quantum-mechanical description of an unstable object, which is characterized by an equation of motion with complex energy (mass) parameter. In order to satisfy conventional axioms of QM, we introduce the source term which is proportional to the root of the imaginary part of the complex energy, and in the stable limit disappears. The decay properties are therefore characterized by a comp]ex energy (mass) parameter and the source term which describes the deviation from “nonunitary” space-time development. We consider models with space variables, nonrelativistic and relativistic ones. The explicit formulas for simple models of wave functions are given. Finally, we present within our framework the description of an unstable \(V\)-particle in \(N{\mit \Theta }\)-sector of the Lee model.


High Energy Behaviour of Non-Planar and Planar Dual Multiloop Amplitudes

abstract

Multi-loop tour-point amplitudes constructed from iteration of non-planar or planar orientable self-energy operators are studied in the asymptotic limit of large \(s\). We find expected factorization properties to sum up the leading contributions of multi-loop graphs of arbitrary order. This leads to the definition of renormalized Pomeron and Regge trajectories.


Geometrical Scaling, Quarks and the Pomeron

abstract

From quark model additivity applied in the impact parameter plane to the inelastic overlap function we obtain a kind of scale invariant factorizable Pomeron. Quarks themselves are seen as behaving asymptotically like extended objects (quark pancakes), cross-sections and multiplicities being related to their overlap in a high energy collision. Predictions, which can be tested soon at NAL, are given for overlap functions, cross-sections, the ratio \(\sigma ^{\rm el}/\sigma ^{\rm tot}\) in the case of various reactions. Universality features of multiplicity distributions are explained in a natural way and an attempt is made to compute the modifications coming from the leading particle effect.


Geometrical Description of Hadronic Collisions

abstract

We discuss the geometrical picture of hadronic collisions. ISR elastic scattering data are analyzed. Some features of multiparticle production are studied, emphasizing qualitative ways of discriminating between the geometrical and the multiperipheral models.


Odd Neutron Hole \(N=81\) Nuclei in the Intermediate Coupling Unified Model

abstract

The intermediate coupling unified model is applied to the odd neutron hole \(N=81\) nuclei \(^{145}\)Gd, \(^{143}\)Sm, \(^{141}\)Nd, \(^{139}\)Ce and \(^{137}\)Ba. In the calculations all the single-neutron hole states of the major shell 50–82 are coupled to the \(N=82\) core excitations up to three phonons. The spectra of the levels with the spins up to 19/2 are in agreement with the experiment if the first 4\(^+\) states in the neighbouring \(N=82\) nuclei are approximated by two-phonon states.


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