  Spinor Bose-Einstein condensates provide a unique example in which the
Bogoliubov theory fails to describe the metastability associated with
first-order quantum phase transitions. This problem is resolved by developing
the spinor Beliaev theory which takes account of quantum fluctuations of the
condensate. It is these fluctuations that generate terms of higher than the
fourth order in the order-parameter field which are needed for the first-order
phase transitions. Besides the conventional first-order phase transitions which
are accompanied by metastable states, we find a class of first-order phase
transitions which are not accompanied by metastable states. The absence of
metastability in these phase transitions holds to all orders of approximation
since the metastability is prohibited by the symmetry of the Hamiltonian at the
phase boundary. Finally, the possibility of macroscopic quantum tunneling from
a metastable state to the ground state is discussed.
