Regular Series

Vol. 17 (1986), No. 6, pp. 481 – 565

Electromagnetic Field as a Non-Linear Connection

Acta Phys. Pol. B 17, 481 (1986)

page 481 •


The “unification” of the gravitation and the electromagnetism in a Randers’ space with a non-linear connection is presented. The equations of motion and the field equations appear in the natural manner.

Investigation of the Wave Equation for One Dirac and One Duffin–Kemmer Particle: A New Form of the Klein Paradox

Acta Phys. Pol. B 17, 485 (1986)

page 485 •


The wave equation for one Dirac and one Duffin–Kemmer particle proposed recently by Królikowski is investigated. The radial equation derived in a previous paper is written down in the component form and reduced by eliminating auxiliary components of the wave function. Then, the limiting behaviour at \(r \to 0\) is checked. In the case of the Duffin–Kemmer spin equal to 1 and the potential having the singularity \(r^{-a} (a \gt 0)\) it turns out that there is only one regular solution instead of three, two of them becoming oscillating solutions. It is shown that this phenomenon is a drastic form of the Klein paradox. A possibility is discussed how to apply the derived radial equations to quark–diquark systems, using the regular potential emerging from the finite size of diquarks.

Symmetry and Solution Set Singularities in Hamiltonian Field Theories

Acta Phys. Pol. B 17, 499 (1986)

page 499 •


This paper discusses singularities in the set of solutions to the field equations in Hamiltonian field theories with first class constraints. We assume that the evolution equations are well posed, so that such singularities are caused only by the constraint equations. The latter require that the moments associated to the gauge transformations vanish. The moment map is a concept in classical mechanics which generalizes the idea of angular momentum associated to the rotation group. It is shown that level sets of certain moment maps have quadratic singularities exactly at points in phase space that are fixed under subgroups of positive dimension. A normal form for the moment map in canonical coordinates is derived. Applying these results to gravitational and Yang–Mills fields shows that the solution sets for these field equations have quadratic singularities exactly at fields with (infinitesimal) symmetries. Thus at a symmetric solution, a linearized solution must satisfy not only the linearized equations but also a quadratic condition if it is to be tangent to a curve of solutions to the full field equations.

Prelimit Effective Potential from Lattice \(\phi ^4_4\) Theory

Acta Phys. Pol. B 17, 525 (1986)

page 525 •


An effective potential for lattice version of \(\phi ^4_4\) theory is calculated in one-loop approximation. Lattices, both finite, are imposed in momentum and in the position spaces. Dependence on the lattice constants is explicitly shown.

Fermions on a Lattice

Acta Phys. Pol. B 17, 531 (1986)

page 531 •


Peculiarities of fermions on a lattice are reviewed with emphasis on Wilson’s method and its relation to the decoupling problem of heavy fermions.

The Modified Version of the Centre-of-Mass Correction to the Bag Model

Acta Phys. Pol. B 17, 553 (1986)

page 553 •


We propose the improvement of the recently considered version of the centre-of-mass correction to the bag model. We identify a nucleon bag with a physical nucleon confined in an external fictitious spherical well potential with an additional external fictitious pressure characterized by the parameter \(b\). The introduction of such a pressure restores the conservation of the canonical energy-momentum tensor, which was lost in the former model. We propose several methods to determine the numerical value of \(b\). We calculate the Roper resonance mass, as well as static electroweak parameters of a nucleon with centre-of-mass corrections taken into account.

Algebraic Construction of the Effective Mass Matrix for Leptons and Quarks

Acta Phys. Pol. B 17, 557 (1986)

page 557 •


An algebra of some “pseudo-annihilation” and “pseudo-creation” operators is found which generates an exponentially rising spectrum bounded from below. In terms of these operators an effective mass matrix is constructed for leptons and quarks in consistency with their observed rising mass spectra and decreasing generation mixing in the case of quarks.


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