  We propose an effective theory of non-Abelian superconductivity, an
SU(2)xU(1) extension of the Abelian Landau-Ginzburg theory, which could be
viewed as an effective theory of ferromagnetic superconductivity made of
spin-up and spin-down doublet Cooper pair. Just like the Abelian
Landau-Ginzburg theory it has the U(1) electromagnetic interaction, but the new
ingredient is the non-Abelian SU(2) gauge interaction between the spin doublet
Cooper pair. A remarkable feature of the theory is the mixing between the
photon and the diagonal part of the SU(2) gauge boson. After the mixing it has
massless gauge boson (the massless non-Abelian spinon) and massive gauge boson
(the massive photon), in addition to the massive off-diagonal gauge bosons (the
massive non-Abelian spinons) which induces the spin-flip interaction between
the spin up and down components of the Cooper pair. So, unlike the ordinary
Landau-Ginzburg theory it has a long range interaction mediated by the massless
non-Abelian spinon, which could be responsible for the long range magnetic
order and spin waves observed in ferromagnetic superconductors. The theory is
characterized by three scales. In addition to the correlation length fixed by
the mass of the Higgs field it has two different penetration lengths, the one
fixed by the mass of the photon (which generates the well known Meissner
effect) and the other fixed by the mass of the off-diagonal spinons (which
determines the scale of the spin flip interaction). The non-Abelian structure
of the theory naturally accommodates new topological objects, the non-Abrikosov
quantized spin vortex (as well as the well known Abrikosov vortex) and
non-Abelian spin monopole. We discuss the physical implications of the
non-Abelian Landau-Ginzburg theory.
