Polarization of the cosmic microwave background, though not yet detected, provides a source of information about cosmological parameters complementary to temperature fluctuations. This paper provides a complete theoretical treatment of polarization fluctuations. After a discussion of the physics of polarization, the Boltzmann equation governing the evolution of the photon density matrix is derived from quantum theory and applied to microwave background fluctuations, resulting in a complete set of transport equations for the Stokes parameters from both scalar and tensor metric perturbations. This approach is equivalent at lowest order in scattering kinematics to classical radiative transfer, and provides a general framework for treating the cosmological evolution of density matrices. The metric perturbations are treated in the physically appealing longitudinal gauge. Expressions for various temperature and polarization correlation functions are derived. Detection prospects and theoretical utility of microwave background polarization are briefly discussed.

Replaced version corrects factor of 2 error in the Liouville equation. 24 pages, Postscript