In the unitary gauge the abelian Higgs model is a second-class system
with non-polynomial field dependent Dirac brackets. We quantize this t
heory following Batalin and Fradkin, by first converting it into a fir
st-class system in an extended phase space, and then constructing the
unitarizing Hamiltonian and the BRST charge. In particular we show how
the partition function of the first-class and second-class (unitary g
auge) formulations is recovered for different choices of gauge conditi
ons in the extended phase space. In Faddeev-Popov-like gauges, the aux
iliary Batalin-Fradkin scalar field is identified with the Goldstone b
oson of the Higgs model.