  The Bogoliubov theory is extended to a Bose-Einstein condensation with
internal degrees of freedom, realized recently in $^{23}$Na gases where several
hyperfine states are simultaneously cooled optically. Starting with a
Hamiltonian constructed from general gauge and spin rotation symmetry
principles fundamental equations for condensate are derived. The ground state
where time reversal symmetry is broken in some case and low-lying collective
modes, e.g. spin and density wave modes, are discussed. Novel vortex as a
topological defect can be created experimentally.
