This article is a pedagogical introduction to relativistic quantum mechanics of the free Majorana particle. This relatively simple theory differs from the well-known quantum mechanics of the Dirac particle in several important aspects. We present its
three equivalent formulations. Next, so called axial momentum observable is introduced, and general solution of the Dirac equation is discussed in terms of eigenfunctions of that operator. Pertinent irreducible representations of the Poincare group are discussed. Finally, we show that in the case of massless Majorana particle the quantum mechanics can be reformulated as a spinorial gauge theory.
Historically, Ehrenfests theorem (1927) is the first one which shows that classical physics can emerge from quantum physics as a kind of approximation. We recall the theorem in its original form. Next, we highlight its generalizations to the relativi
stic Dirac particle, and to a particle with spin and izospin. We argue that apparent classicality of the macroscopic world can probably be explained within the framework of standard quantum mechanics.
The Hilbert space of states of the relativistic Majorana particle consists of normalizable bispinors with real components, and the usual momentum operator $- i abla$ can not be defined in this space. For this reason, we introduce the axial momentum
operator, $ - i gamma_5 abla$ as a new observable for this particle. In the Heisenberg picture, the axial momentum contains a component which oscillates with the amplitude proportional to $m/E$, where $E$ is the energy and $m$ the mass of the particle. The presence of the oscillations discriminates between the massive and massless Majorana particle. We show how the eigenvectors of the axial momentum, called the axial plane waves, can be used as a basis for obtaining the general solution of the evolution equation, also in the case of free Majorana field. Here a novel feature is a coupling of modes with the opposite momenta, again present only in the case of massive particle or field.
Forces in the systems of two opposite sign and three identical charges coupled to the dynamical scalar field of the signum-Gordon model are investigated. Three-body force is present, and the exact formula for it is found. Flipping the sign of one of
the two charges changes not only the sign but also the magnitude of the force. Both effects are due to nonlinearity of the field equation.
Force between static point particles coupled to a classical ultra-massive scalar field is calculated. The field potential is proportional to the modulus of the field. It turns out that the force exactly vanishes when the distance between the particle
s exceeds certain finite value. Moreover, each isolated particle is surrounded by a compact cloud of the scalar field that completely screens its scalar charge.
Several classes of self-similar, spherically symmetric solutions of relativistic wave equation with nonlinear term of the form sign(phi) are presented. They are constructed from cubic polynomials in the scale invariant variable t/r. One class of solu
tions describes a process of wiping out the initial field, another an accumulation of field energy in a finite and growing region of space.
We present a new class of oscillons in the (1+1)-dimensional signum-Gordon model. The oscillons periodically move to and fro in the space. They have finite total energy, finite size, and are strictly periodic in time. The corresponding solutions of t
he scalar field equation are explicitly constructed from the second order polynomials in the time and position coordinates.
We present explicit solutions of the signum-Gordon scalar field equation which have finite energy and are periodic in time. Such oscillons have a strictly finite size. They do not emit radiation.
We investigate a (1+1)-dimensional nonlinear field theoretic model with the field potential $V(phi)| = |phi|.$ It can be obtained as the universal small amplitude limit in a class of models with potentials which are symmetrically V-shaped at their mi
nima, or as a continuum limit of certain mechanical system with infinite number of degrees of freedom. The model has an interesting scaling symmetry of the on shell type. We find self-similar as well as shock wave solutions of the field equation in that model.
In this lecture we outline the main results of our investigations of certain field-theoretic systems which have V-shaped field potential. After presenting physical examples of such systems, we show that in static problems the exact ground state value
of the field is achieved on a finite distance - there are no exponential tails. This applies in particular to soliton-like object called the topological compacton. Next, we discuss scaling invariance which appears when the fields are restricted to small amplitude perturbations of the ground state. Evolution of such perturbations is governed by nonlinear equation with a non-smooth term which can not be linearized even in the limit of very small amplitudes. Finally, we briefly describe self-similar and shock wave solutions of that equation.
