A. O. Barut

SO(4, 2)‐Formulation of the Symmetry Breaking in Relativistic Kepler Problems with or without Magnetic Charges

The relativistic Kepler problems in Dirac and Klein‐Gordon forms are solved by dynamical group methods for particles having both electric and magnetic charges (dyons). The explicit forms of the O(4, 2)‐algebra and two special O(2, 1)‐algebras (which coincide in the symmetry limit) are given, and a new group‐theoretical form of the symmetry breaking is pointed out. The Klein‐Gordon O(2, 1)‐algebra also solves the dynamics in the case of very strong coupling constants (attractive singular potential), if the principal series of representations is used instead of the discrete series.

Reduction of a class of o(4,2) representations with respect to so(4,1) and so(3,2)

A complete classification of representations of SO(4, 2) with infinitesimal generators SAB, characterized by the representation relation {SAB, SAC} = −2agBC, and their extension by parity have been determined. The possible values of a are a = 1 − S2, S = 0, 12, 1, 32, 2, ⋯. These representations have then been reduced according to the two chains SO(4,2)⊃SO(4,1)⊃SO(4)⊃SO(3) and SO(4,2)⊃SO(3,2)⊃SO(3)⊗SO(2). The equivalence of these representations with the oscillator representations is established.

On the shape of tachyons

SummaryWe study some aspects of the experimental behaviour of tachyons, in particular by finding out their « apparent » shape. A Superluminal particle, which in its own rest frame is spherical or ellipsoidal (and with an infinite lifetime), would « appear » to a laboratory frame as occupying the whole region of space bound by a double cone and a twosheeted hyperboloid. Such a structure (the tachyon « shape ») rigidly travels with the speed of the tachyon. However, if the Superluminal particle has a finite lifetimein its rest frame, then in the laboratory frame it gets afinite space extension. As a by-product, we are able to interpret physically the imaginary units entering—as is well known—the transverse co-ordinates in the Superluminal Lorentz transformations. The various particular or limiting cases of the tachyon shape are thoroughly considered. Finally, some brief considerations concerning possible experiments to look for tachyons are added.RiassuntoSi studiano alcuni aspetti del comportamento dei taohioni, in particolare trovando quale « apparirebbe » la loro forma. Una particella Superluminale, che sia sferica o ellissoidale (e con vita di durata finita) nel proprio riferimento a riposo, a un osservatore nel laboratorio sembrerebbe occupare Tintera regione di spazio limitata da un doppio cono e da un iperboloide a due falde. Tale struttura (la « forma » del tachione) viaggerà rigidamente con la velocità del tachione. Si noti però che, se la particella Superluminale ha una vita finita (nel suo riferimento a riposo), allora nel laboratorio essa risulta avere un’estensione spazialefinita. Come conseguenza della precedente analisi, siamo in grado d’interpretare fisicamente le unità immaginarie che entrano — come noto — nelle coordinate trasversali per azione delle trasformazioni di Lorentz Superluminali. Si esaminano dettagliatamente i vari casi particolari o casi limite della forma dei tachioni. Infine, si aggiungono alcune considerazioni circa eventuali esperimenti atti alla ricerca effettiva dei tachioni.РезюмеМы исследуем некоторые аспекты экспериментального поведения тахионов, в частности, посредством нахождения их кажущейся формы. Суперлюминальная частица, которая в своей собственной системе соординат является сферической или эллипсоидальной (и с бесконечным временем жизни), в лабораторной системе координат представляется занимающей всю область пространства, ограниченную двойным конусом и гиперболоидом с двумя слоями. Такая структура (« форма » тахиона) движется со скоростью тахиона. Однако, ески суперлюминальная частица имеет конечное время жизни в своей собственной системе координат, то в лабораторной системе координат эта частица занимает конечное пространство. Как вспомогательный результат, мы можем физически интерпретировать мнимые единицы, входящие, как известно, в попереченые координаты в суперлюминальных преобразованиях Лоренца. Подробно исследуются различные частные и предельные случаи формы тахиона. В заключение, проводится обсуждение возможных экспериментов по наблюдению тахионов.

Path integral treatment of the hydrogen atom in a curved space of constant curvature

The path integral treatment of the hydrogen atom in a spherical space is discussed. The dynamical group SU(1,1) of the system is used for path integration. By mapping the radial path integral onto the SU(1,1) manifold, the energy spectrum and the normalised wavefunctions are obtained. In the flat space limit, the standard hydrogen spectrum and the corresponding normalised energy eigenfunctions are recovered. The scattering states are also found in the limit.

A new approach to bound-state quantum electrodynamics: I. Theory

Abstract A fully covariant two-body equation is applied to the theory of hydrogen, muonium, and positronium spectra. Both particles are treated fully relativistically and complete spin algebras for both particles are taken into account. The major part of the 16 × 16 wave equation is exactly soluble including recoil of both particles to all orders. The terms of order α 5 , α 5 are treated perturbatively (although the equation is in principle numerically solvable to all orders). Self-energy (loop) effects are partly considered by an (effective) anomalous magnetic moment, but in a dynamical way using a Pauli coupling from the beginning. The theory simplifies and improves the bound-state QED problems in a number of ways.

Path integral treatment of the hydrogen atom in a curved space of constant curvature. II. Hyperbolic space

For pt.I see ibid., vol.20, p.6271. The path integral treatment of the hydrogen atom in a hyperbolic space is discussed. The authors show by mapping the radial path integral into the SU(1,1) group manifold that the system has a dynamical SU(1,1) symmetry. The energy spectrum and normalised energy eigenfunctions are calculated. In the flat-space limit, the standard hydrogen spectrum and corresponding normalised wavefunctions are regained.

On conformally covariant spin-2 and spin-3/2 equations

The standard massless spin-2 and spin-3/2 equations are not conformally covariant. By varying the coefficients of various terms in these equations the authors derive conformally covariant equations. These equations are then used to construct consistent coupling terms in the original field equations representing the self-interaction and the interaction of massless spin-2 (or 3/2) field with matter.

The exponential map for the conformal group O(2,4)

We present a general method to obtain a closed, finite formula for the exponential map from the Lie algebra to the Lie group, for the defining representation of the orthogonal groups. Our method is based on the Hamilton-Cayley theorem and some special properties of the generators of the orthogonal group, and is also independent of the metric. We present an explicit formula for the exponential of generators of the

The generalized Morse oscillator in the SO(4,2) dynamical group scheme

SO_+(p,q)

Behavior of the Scattering Amplitude for Large Angular Momentum

groups, with