The Bose-Einstein condensates of alkali atomic gases are spinor fields withlocal ``spin-gauge" symmetry. This symmetry is manifested by a superfluidvelocity ${\bf u}_{s}$ (or gauge field) generated by the Berry phase of thespin field. In ``static" traps, ${\bf u}_{s}$ splits the degeneracy of theharmonic energy levels, breaks the inversion symmetry of the vortex nucleationfrequency ${\bf \Omega}_{c1}$, and can lead to {\em vortex ground states}. Theinversion symmetry of ${\bf \Omega}_{c1}$, however, is not broken in ``dynamic"traps. Rotations of the atom cloud can be generated by adiabatic effectswithout physically rotating the entire trap.
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