  New degrees of freedom having the form of a unit vector are identified for
characterizing the spin polarization of free electron. It is shown that when
only the spin is considered, the non-commutativity of the Cartesian components
of the Pauli vector allows us to use the azimuthal angle of a second direction,
denoted by unit vector $\mathbf I$, with respect to the quantization direction
to characterize the spin polarization. The rotation of $\mathbf I$ through an
angle about the quantization axis leads to a rotation of the spin polarization
vector through twice the angle about the same axis. Discussions are also made
in Heisenberg picture as well. Upon utilizing this approach to a free electron
and letting the quantization direction for each plane wave be the wave vector,
we arrive at a representation in which the unit vector $\mathbf I$ functions as
an independent index to characterize the spin polarization.
