Dominance theorem in crossing matrices in arbitrary compact group

Abstract

Previously, the present author proved the new dominance theorem in crossing matrices in SU~ and SU B groups and discussed its physical implications in highenergy diffraction scattering theory (1). In this note we shall briefly show that this theorem holds generally in an arbitrary compact group. That is, the absolute magnitude of the nonvanishing crossing-matrix element between s-channel amplitude in the singlet state and t- or u-channel amplitude in a given arbitrary multiplet state is never smaller than that ~f the erossing-matrix~ element between s-channel amplitude in an arbitrary multiplet state and t- or re-channel amplitude in a given aa\bitrary multiptet st~.e. For saving space we shall use the definitions of the various quantities by A~AT~ et at. (~) and their notations with only brief explanations when there is no mathematical ambiguity. We assign the four particles of the scattering process to the four unitary irreducible representation spa~es for an ~rbitrary compact group which we designate by A, B, C, D. In each space we select an orthonormal basis and represent the vectors of these basis by IAa}, IBb), ICe}, IDd}. The spaces conjugate to A, B, C, D will be designated A, B, C, D and the basis of the e~njugate representation A, B, C, D will be designated by !Zg}, !B~>, IC~>, ID3~), respectively. If a paxtiele belongs to a representation, its antiparticle belongs to the conjugate one. We can decompose the product space such aM A  B into irreducible subspaces under an ~rbitrary compact group. V~e shall use the symbol X to represent both such invariant subspaces and the irreducible representations induced in it. In each such subspaee we may introduce an orthonormal basis 1X(~)x(A where ~ is introduced to distinguish the equivalent invariant subspaees. It is easily shown that the transformation coefficient (C.G. coefficient) (X(~) x 1Aa, Bb) has the usual properties such as orthonormality and symmetries as the C.G. coefficient in SU~ and SU s groups (s.4). Therefore we shatt use theme properties of C.G. coefficients without detailed mathems~ical pr(~of. The necessary properties of C.G. coefficients are listed Jn the