A geometrical theorem on the asymptotic space-time properties of conservation laws in a classical field theory

Abstract

A result, having to do with the asymptotic space-time properties of conservations laws and which we call the cone bands theorem, is proved for a classical field theory model. With the aid of the theorem it is possible to define a tensor reduction procedure on conserved and asymptotically conserved quantities. The reduction procedure applied to such objects permits identification of components of radiation by means of conical hypersurfaces, or in analogy to particle world lines, radiation world cones, whose axes lie along the four velocities U μ of the radiation. It is shown that for the classical theory of a massive real scalar field (retarded solution), produced by a collection of point sources experiencing arbitrary accelerations in a finite region of space-time, these conditions are met by the stress-energy tensor T μν and by a set of asymptotically conserved amplitude vectors A μ ( α) α ( x). A dimensionless quantity d 3 N( U) arises which measures the amount of radiation lying along the direction U μ and which is identified with the radiated particle number, or particle “number-probability.” Because the cone bands property possesses a physical interpretation which is rooted in measurement procedures, it is noted that a necessary requirement on a classical field theory, for a particle interpretation to be meaningful in the asymptotic region is that all asymptotically conserved quantities which are regarded as being carried along by the particles possess the cone bands property. Relationship of the present work to quantum field theory is discussed.