CONFORMAL GEOMETRY AND ELEMENTARY PARTICLES

Abstract

The kinematical consequences of basing (classical or quantum) field theory on the conformal geometry are examined in this paper. The space in question is that of all spheres inR4 (flat 4-space of signature (+++−)); the fundamental invariant, the angle under which two spheres intersect. In the mathematical preliminaries (§1) a convenient inhomogeneous formalism is developed, permitting the sphere to be treated as a point in a 5-dimensional Riemannian space of constant unit curvature whose length element is the infinitesimal angle between two neighboring spheres. The conformal group (proper Lorentz transformations, translations, uniform dilations, and inversions in spheres) is just the 15-parameter group of motions (metric-preserving transformations) of this space. In §2 the spheres are interpreted as finite (non localized) test particles. Physical fields are thus defined on test particles rather than (e.g. in position space) on the events occupied by these test particles alone. The spheres can be labelled either with the 5-position (test particle’s spacetime position andsize) in aq-frame, or with the 5-momentum (test particle’s 4-momentum and rest mass) in ap-frame. There exists a motion transformingq- intop-space and conversely (in other words,q- andp-observers are physically equivalent), in virtue of which any conformal theory is shown to exhibit an automaticBorn-reciprocity betweenq- andp-space. The 5-position and 5-momentum satisfy an uncertainty-relation type equation; i.e., the non-localizations inq- andp-space are in an inverse relation with Planck’s ħ measuring the intrinsic correlation. All the other motions can be built up from subgroups takingq- andp-space into themselves. These are systematically interpreted as changes of frame. Those involving relative motion representuniform relative accelerations of Lorentz observers (Lorentz group ↔ zero accelerations). Test particle mass and size are invariant under the Lorentz group but non-uniformly renormalized by the accelerative motions. In §3 the geodesics of sphere space (motion equations of (elementary) test particles) are shown to describe uniform motion in the present (force-free) case. Test particles tend to dislocalize inq space with increasing time and inp-space with increasing energy. The time-constancy of the 5-momenta, their inter-dependence as given by special relativity dynamics follow from these motion equations. Free elementary particles are shown to maintain a state of uniform velocity under all motions, in particular the accelerative ones. This contradicts ordinary relativity and suggests an experiment capable in principle of choosing between the conformal and Lorentz geometries for physics.