Force-free Case Research Articles

A wave equation for massive particles of arbitrary spin

A wave equation is given which, in the force-free case, describes a particle of unique spin and mass and which remains consistent when interactions are included. The equation is simple in the sense that it involves only matrices which satisfy the Pauli commutation rules. This simplicity is achieved at the expense of extending the particle field vector from one of 2j + 1 components to one of 4j. However, the extra 2j — 1 components are just what are needed to remove the inconsistency of the interacting system. In the force-free case these additional components vanish and the resulting equations are equivalent to the usual ones of Dirac, Fierz and Pauli. One the other hand, for the interacting system, the extra components do not vanish. The second order propagation equations are deduced in the case of external electromagnetic and gravitational fields.

Interactions of massless particles of arbitrary spin

The self-consistency problem s arising when massless particles of higher spin are coupled to the electromagnetic and gravitational fields are investigated using a group theoretic method as opposed to the Lagrangian approach employed by Fierz & Pauli (1939) in their discussion for massive particles. A massless particle of spin j is described by a 4j dimensional vector satisfying a generalization of Rumer’s equation (1930). 2j + 1 of these 4j components form the conventional quantity associated with a particle of spin j and the remaining 2j — 1 are supernumary quantities needed for the consistency of the in teracting system ; the whole object transforms according to the reduced representation (j, 0)⊕ (j — 1, 0) of the homogeneous Lorentz group. It is suggested that this formalismis a convenient way of writing higher-spin equations in particle-like form with attendant advantages (see, for example, Good 1959). In the force-free case the 2j — 1 auxiliary quantities can be set equal to zero but this is not possible in the interacting system with out inconsistency. One conclusion is that it is impossible for a (massless) particle of unique spin to be coupled minimally to gravitation unless either its spin is less than 1/2, space-time is conformally flat or its spin is two and it describes gravitation radiation itself.

Conformal geometry and elementary particles

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 inR 4 (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.

CONFORMAL GEOMETRY AND ELEMENTARY PARTICLES

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.