The Boltzmann equation for the CMB describes the transport
of the photons under Thomson scattering by the electrons. The
radiation is described by the intensity matrix: the time
average of the electric field tensor *E*_{i}^{*} *E*_{j} over a time
long compared to the frequency of the light or equivalently as the
components of the photon density matrix (see [19] for
reviews). For radiation
propagating radially , so that
the intensity matrix exists on the subspace. The matrix can further
be decomposed in terms of the Pauli matrices
and the unit matrix on this subspace.

For our purposes, it is convenient to describe the polarization in temperature fluctuation units rather than intensity, where the analogous matrix becomes,

(34) |

Under a counterclockwise rotation of the axes
through an angle the intensity transforms as
.
and *V* remain distinct
while *Q* and *U* transform into one another. Since the
Pauli matrices transform as a more convenient
description is

(35) |

Since circular polarization cannot be generated by
Thomson scattering alone, we shall hereafter ignore *V*. It is then
convenient to reexpress the matrix as a vector

(36) |

(37) |