Relativistic dynamics on a null plane

Publisher Summary Galileo Galilei explicitly introduced the principle of relativity in physics. He was the first one to recognize the existence of inertial transformations, connecting various frames of reference in which the laws of physics take the same form. Starting with the recognition that the Galilean transformation laws were but an approximation to the more exact Lorentz formulas, one could distinguish the abstract principle of relativity from its concrete expressions, as various possible theories of relativity. At the same time, one could begin to see the basic role played by the relevant theory of relativity in structuring a given physical theory—Galilean relativity for classical mechanics, Einstein relativity for “relativistic” mechanics, and electromagnetism. Applications of the Galilei group are described for classical physics and for quantum physics. A “super-Galilei group” has been defined in relation to non-relativistic cosmology. This is a sixteen-parameter group acting on the five-dimensional manifold of space-time and gravitational potential.

Galilei Group and Galilean Invariance

This article is divided in five parts, the first one having nothing to do with tachyons. In fact, to prepare the ground, in part I (sect.2) we shall merely show that special relativity - even without tachyons - can be given a form such to describe both particles and antiparticles. Part II is the largest one: initially, after some historical remarks and having revisited the postulates of special relativity, we presnt a review of the elegant ''model theory'' of tachyons in two dimensions; passing then to four dimensions, we review the main results of the classical theory of tachyons that do not depend on the existence of Superluminal reference frames (or that are at least independent of the explicit form of the Superluminal Lorentz ''transformations''). In particular, we discuss how tachyons would look like, i.e. their apparent ''shape''. Last but not least, all the common causality problems are thoroughly solved, on the basis of the previously reviewed tachyon kinematics. Part III deals with tachyons in general relativity; in particular the question of the apparent Superluminal expansions in astrophysics is reviewed. Part IV shows the interesting, possible role of tachyons in elementary-particle physcis and in quantum theory. In part V, the last one, we face the (still open) problem of the Superluminal Lorentz ''transformations'' in four dimensions, by introducing, for instance, an auxiliary six-dimensional space-time, and finally present the electromagnetic theory of tachyons: a theory that can be relevant also from the ''experimental'' point of view.

Classical tachyons and possible applications

Two subgroups of the Galilei group are shown to play a particular role in the case of charged systems in an external electro-magnetic field which is constant and uniform. Projective representations of these subgroups involve the electric charge and mass as generators of phase factors. Additivity and superselection rules for charge and mass appear as direct consequences. A comparison is made with the relativistic case investigated in the first part of this article.

Group-theoretical analysis of elementary particles in an external electromagnetic field II.—The nonrelativistic particle in a constant and uniform field

A thorough unified treatment is given of the quantum-mechanical operators and wave functions for a molecular system (composed of N moving charged particles) in static uniform electric and magnetic fields E and B. The treatment is rigorous within the nonrelativistic approximation. The system may either be neutral or charged. The fields may have arbitrary intensities and orientations. Close correspondence is maintained between the classical and quantum-mechanical treatments. The wave functions are expressed both in time-independent energy representation and in time-dependent wave packets. Three types of momentum play important roles. For single-particle systems they are the canonical momentum P; the mechanical momentum Pi = P(e/c) A = MR; and the pseudomomentum K = Pi-(e/c)R x B-eEt.B. For neutral molecules, the Schroedinger equation is ''pseudoseparated'' and the internal degrees of freedom are coupled to the center of mass motion by only the ''motional Stark Effect,'' which involves the constant of motion scrK. For ionic systems, only one component of the center of mass is coupled to the internal motion.

Interaction of atoms, molecules, and ions with constant electric and magnetic fields

Group-theoretical analysis of elementary particles in an external electromagnetic field@@@Теоретико-групповой анализ элементарных частиц во внешнем электромагнитном поле.: I. The relativistic particle in a constant and uniform field@@@I. Релятивистская частица в постоянном и однородном поле

H. Bacry

Papers

Group-theoretical analysis of elementary particles in an external electromagnetic field II.—The nonrelativistic particle in a constant and uniform field

Connected subgroups of the poincare group II