  We revisit the computation of the phase of the Dirac fermion scattering
operator in external gauge fields. The computation is through a parallel
transport along the path of time evolution operators. The novelty of the
present paper compared with the earlier geometric approach by Langmann and
Mickelsson, [LM], is that we can avoid the somewhat arbitrary choice in the
regularization of the time evolution for intermediate times using a natural
choice of the connection form on the space of appropriate unitary operators.
