  In this paper we discuss Bose-Einstein spinor condensates for F=1 atoms in
the context of 87Rb, as studied experimentally by the Stamper-Kurn group
[Sadler et al. Nature (2006)]. The dynamical quantum fluctuations of a sample
that starts as a condensate of N atoms in a pure F=1, m_F = 0 state are
described in analogy to the `two-mode squeezing' of quantum optics in terms of
an su(1,1) algebra. In this system the initial m_F=0 condensate acts as a
source (`pump') for the creation pairs of m_F =+1,-1 atoms. We show that even
though the system as a whole is described by a pure state with zero entropy,
the reduced density matrix for the m_F = +1 degree of freedom, obtained by
tracing out the m_F = -1,0 degrees of freedom, corresponds to a thermal state.
Furthermore, these quantum fluctuations of the initial dynamics of the system
provide the seeds for the formation of domains of ferromagnetically aligned
spins.
