2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/54855As an application of the renormalization method introduced by the second author we give a causal definition of the phase of the quantum scattering matrix for fermions in external Yang-Mills potentials. The phase is defined using parallel transport along the path of renormalized time evolution operators. The time evolution operators are elements of the restricted unitary group $U_{res}$ of Pressley and Segal. The central extension of $U_{res}$ plays a central role.5 pages in AMSTEX format. This is a talk given by Jouko Mickelsson at "XXI. Colloquium on Group Theoretical Methods in Physics" in Goslar, July 1996High Energy Physics - Theorytext