Globular clusters harbor many millisecond pulsars and 10 bright low-ma
ss X-ray binary systems. It has been generally believed that almost al
l millisecond pulsars are spun up by accretion from a companion star d
uring a (progenitor) low-mass X-ray binary (LMXB) phase. However, most
of the short-period millisecond pulsars are found in the apparently y
ounger globular clusters, which have much lower core densities than th
ose characteristic of the apparently older globulars containing the ma
jority of the LMXBs. The vast majority of the globular LMXBs are X-ray
burst sources, and among them there are no bright sources similar to
those in the Galactic disk, such as Sco X-1, Cyg X-2, and GX 5-1. Furt
hermore, the radio pulse properties of the very short period milliseco
nd pulsars in globular clusters differ from those of pulsars spun up i
n the Galactic disk, such as PSR 1937 + 21 and PSR 1957 + 20. Only the
properties of the latter class are consistent with what is expected f
or the spin-up scenario. We argue that, in most globular cluster LMXBs
, the neutron stars were formed spinning rapidly, and are likely to be
the descendants, rather than the progenitors, of weakly magnetized mi
llisecond pulsars.