Interaction and Fusion of Elementary Systems

Abstract

AbstractThe notion due to R.G. Newton and E.P. Wigner (1949) of an elementary system ES is sharpened to a system on a Lie group G-manifold as configuration space and the unitary irreducible representations (IR) of G as states. We study pairs of elementary systems with configuration space taken as the direct product (G×G)-manifold and with an interaction invariant under the right action of the subgroup diag(G×G). The (G×G)-manifoldis split into a new external group manifold 〈X〉 transformed by left action, and a new internal group manifold 〈x〉 unchanged under diag(G×G). By use of Kronecker products we transform IR pair states to external/internal coordinates. The general concept of fusion due to de Broglie (1932–34) is expressed in the new coordinates as the limit where x goes to the identity element. For elementary Poincaré systems, the distinction between massive Mackey and covariant fields becomes crucial. The presence of a full Poincaré-manifold and of corresponding observables are illuminated by position operators. The space translation parameters of the Poincaré group are related to the relativistic position operators of Newton and Wigner. For two Dirac elementary systems of equal mass m we recover by fusion the field of Bargmann and Wigner (1948) of spin S=1 which can be rewritten in terms of a massive vector field. The total mass of the Bargmann-Wigner field is shown to be minimal, M=2m. Interaction schemes are sketched for pairs of Euclidean and Poincaré-manifolds and ES. By Frobenius reciprocity, the process of fusion allows for a counterpart termed scission. Scission is constructed by use of the theory of induced representations.KeywordsElementary systemsLie group G-manifoldsexternal/internal coordinatesde Broglie-fusion