  As 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.
