Almost 30 years ago, the European Muon Collaboration uncovered a crisis in nuclear physics: part of the proton's spin was missing. The measured spin of the three valence quarks could not account for the known spin of the proton, and even today, we do not fully understand the proton's "spin budget." This "proton spin puzzle" has driven a revolution in our understanding of the complex structure of the proton as we search for the missing angular momentum.
One possible source of angular momentum is the polarization of radiated quarks and gluons which carry a very small percentage of the proton's energy (small x) and can only be measured at particle accelerators with very high energies. At high enough energies, multiple bremsstrahlung builds up a cascade of these small-x particles which may contain a substantial portion of the missing proton spin.
In this talk, I will describe the structure of this quantum evolution process for polarized quarks at small x. In contrast to the more familiar cascade of unpolarized particles at small x, the polarized evolution generates double logarithms of energy and grows more quickly at high energies. Its structure is much more complex than the unpolarized case, but early hints seem to suggest that the polarization may indeed become large at small x, making them an essential missing contribution to the proton spin puzzle.