A hydrodynamic form of the wave equation for the neutrino wave function is derived. The formulation is similar to that given by Takabayasi for the nonrelativistic Pauli equation. The hydrodynamic variables comprise one scalar field — the density — and two vector fields — the velocity and momentum. The reduction in the number of variables to four requires a quantization condition — the same as in the nonrelativistic case — that relates the curl of the momentum field to an axial vector built from the velocity field.

We describe the generators of \(\kappa \)-conformal transformations, leaving invariant the \(\kappa \)-deformed d’Alembert equation. In such a way one obtains the conformal extension of the off-shell spin zero realization of \(\kappa \)-deformed Poincaré algebra. Finally the algebraic structure of \(\kappa \)-deformed conformal algebra is discussed.

Colored quarklike scalars, \(y\), appear both in the familiar supersymmetric model and in a recently proposed Dirac’s square-root model based on (in general reducible) representations of the Dirac algebra, defined by means of Clifford algebras. Five types of new (generally unstable) hadrons, \(q\bar y, y\bar y, qqy, qyy, yyy\), are briefly discussed. Two different flavor assignments for the scalars \(y\) are considered.

For the classical Volkov–Akulov Model the energy-momentum tensor, the supercurrent as well as the equations of motion are given. The quantal approach based on these quantities is preliminarily discussed with emphasis upon the phenomenon of spontaneously broken supersymmetry.

We derive explicit formulas for curvature and torsion of a line of the field of \(n\) electric charges. These formulas show that in general the torsion of a field line is not zero if \(n \geq 3\). We also propose a geometric interpretation of the derived formulas. In the second part of the paper we present an outline of a new description of equipotential surfaces of two and three electric charges. In this description the golden section appears in a natural way when two electric charges are equal. This approach also relates an equipotential surface of three charges to the classic cubic surface containing twenty seven straight lines.

Let \(R\) be \(\Bbbk \)-algebra. This paper is a preliminary step to study biderivations of braided Hopf \(R\)-algebras. We describe a free or cofree graded Hopf \(R\)-algebras as braid dependent deformations of bifree graded Hopf \(R\)-algebra. We introduce braid dependent \(\Bbbk \)-derivations of \(R\)-bimodules (and of \(R\)-algebras) and consider an application of derivations of zero grade for Dirac theory.

The electric charge is a Lorentz invariant quantity. Its canonically conjugate partner, however, called phase, cannot be Lorentz invariant. We calculate explicitly the Lorentz transformation of the phase. “This is the simplest example of a pathological representation of the Lorentz group. It may very well be that this pathological representation is essential for the physics of the future.” P.A.M. Dirac

Odd-dimensional Riemannian spaces that are non-orientable, but have a pin structure, require the consideration of the twisted adjoint representation of the corresponding pin group. It is shown here how the Dirac operator should be modified, also on even-dimensional spaces, to make it equivariant with respect to the action of that group when the twisted adjoint representation is used in the definition of the pin structure. An explicit description of a pin structure on a hypersurface, defined by its immersion in a Euclidean space, is used to derive a Schrödinger transform of the Dirac operator in that case. This is then applied to obtain — in a simple manner — the spectrum and eigenfunctions of the Dirac operator on spheres and real projective spaces.

The concise discussion of various aspects of dissipative quantum hydrodynamics is given. Particular attention is paid to the Gisin formulation of dissipative quantum mechanics.