Interactions of Massless Particles of Arbitrary Spin

J. S. Dowker, Yih P. Dowker

Abstract

The self-consistency problems 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 interacting system; the whole object transforms according to the reduced representation (j, 0)$\bigoplus$(j - 1, 0) of the homogeneous Lorentz group. It is suggested that this formalism is 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 without 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 $\frac{3}{2}$, space-time is conformally flat or its spin is two and it describes gravitation radiation itself.