It is proposed that the phenomenological theory of particles be based on the source concept, which is abstracted from the physical possibility of creating or annihilating any particle in a suitable collision. The source representation displays both the momentum and the space-time characteristics of particle behavior. Topics discussed include: spin and statistics, charge and the Euclidean postulate, massless particles, and $S{U}_{3}$ and spin. It is emphasized that the source description is logically independent of hypotheses concerning the fundamental nature of particles.

Particles and sources.

The algebraic expressions for the matrix elements of symmetric tensor operators (the powers of infinitesimal operators) of the unitary groups in the Gel'fand basis have been studied. The expressions for the isoscalar factors of the related Clebsch‐Gordan coefficients, one of the two representations to be coupled being symmetric, as well as the elements of a special recoupling matrix have been found. The supplementary symmetry properties of the isoscalar factors corresponding to the Regge symmetries of the Wigner and 6j coefficients of SU2 have been examined.

On the Symmetric Tensor Operators of the Unitary Groups

The connection of conformal fields with the Mackey theory is discussed. The necessary and sufficient conditions for the finite or infinite component field equation

$$\left( {L_\mu \partial ^\mu + m} \right)\psi \left( x \right) = 0$$

to be conformally covariant, are derived. The conditions are then explicitly solved under very general assumptions and thus conformally covariant equations of the above type are explicitly found (Theorem 5). The circumstances under which the equation may be obtained from a Lagrangian are discussed.

Conformally covariant field equations I. first-order equations with non-vanishing mass

We have done a «mininal» change in the existing 4-dimensional relativity, by extending it to the 6-dimensional conformal (ηa)-space (flat or curved one) with the metric tensorgab (a, b=0, 1, 2, 3, 5, 6), where all components of the 6-vector ηa=(ημ=κxμ, κ, λ) are considered as independent physical degrees of freedom. All basic equations of (special and general) relativity in 6-dimensional (flat or curved) conformal (ηa)-space have the same form as the corresponding equations in the 4-dimensional space. The novel feature of such an extended theory (named «conformal relativity») is the introduction of thescale degree of freedom κ, which can be different from 1 and can change along the particle world-line. However, if κ=1 ($\dot \kappa = 0$), then the conformal relativity reduces to the usual, 4-dimensional relativity. Geodesics in the curved (ηa)-space correspond to the motion of electrically charged test partcles in gravitational and/or electromagnetic fields. The field equations for the metric tensorgab are Einstein equations, extended to the (ηa)-space; they describe a gravitational and electromagnetic field.

Unified theory of gravitation and electromagnetism, based on the conformal group SO 4,2

If we assume that scale is a degree of freedom which a physical object, such as a galaxy possesses besides translational and rotational degrees of freedom, we are able to incorporate such diverse phenomena as the systematic cosmological red-shifts, anomalous red-shifts, controversals of quasars, and the expansion of Earth into a single theory. At the same time we can consider the proposed existence of active dilatation as a physical realization of conformal transformations. Active conformal transformations transform the rest masses of particles. The objection that this implies a continuous rest mass spectrum which is not observed is avoided by assuming that particles can form bound systems such as nuclei, atoms, crystals, galaxies, etc., only if ‘discrete’ scale relations are established among the constituent particles. Different galaxies can be dilated relative to each other. (Note the analogy with ‘discrete’ positions of atoms in a crystal. Different crystals can be translated arbitrarily each to the other.) An observer in a contracting galaxy experiences that the universe is expanding.

An attempt to resolve the astrophysical puzzles by postulating scale degree of freedom

Relativistic equations in which the fields cotransform under the direct sum of ordinary indecomposable representations of the Lorentz group are derived and discussed.

Relativistic equations and indecomposable representations of the Lorentz groupSL(2, ℂ)

Matrix elements of finite transformations of the group U(n) in an arbitrary unitary irreducible representation of U(n) are discussed and a technique for their determination is presented. As a by-product a recursion relation for a special Racah coefficient of U(n) is obtained.

A note on matrix elements of finiteU(n) transformations

Free particle states andS-matrix are defined on carrier spaces of representations of the Weyl groupW (Poincare group extended by dilatations). By using Clebsch-Gordan coefficients ofW the extension of the relativistic partial wave analysis of the two-bodyS-matrix is obtained. Scattering of particles with arbitrary masses and spins is treated and a comparison with relativistic case is discussed.

J. Niederle

Papers

Relativistic equations and indecomposable representations of the Lorentz groupSL(2, ℂ)

A note on matrix elements of finiteU(n) transformations

Partial wave analysis of dilatation-invariant relativisticS-matrix: II. Two-body scattering matrix

The unusual algebras and their applications in particle physics