  We review the well-known polarization optics matrix methods, i.e., Jones and
Stokes-Mueller calculus, and show how they can be formulated in terms of
four-dimensional (4d) rotations of the four independent electromagnetic field
quadratures. Since 4d rotations is a richer description than the conventional
Jones and Stokes-Mueller calculi, having six rather than four degrees of
freedom (DOF), we propose an extension of those calculi to handle all six DOF.
  For the Stokes-Mueller analysis, this leads to a novel and potentially useful
extension that accounts for the absolute phase of the optical field, and which
can be valuable in the areas where the optical phase is of interest, e.g.
interferometry or coherent communications. As examples of the usefulness we use
the formalism to explain the Pancharatnam phase by parallel transport, and
shows its connection with the Berry phase.
  In addition we show that the two extra DOF in the 4d description represents
unphysical transformations, forbidden for propagating photons, since they will
not obey the fundamental quantum mechanical boson commutation relations.
