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In this section, we derive the basic properties of the angular
modes of the
temperature and polarization
distributions that will be useful in §III
to describe their evolution. In particular,
the ClebschGordan relation for the addition of angular
momentum plays a central role in exposing the
simplicity of the total angular momentum representation.
A scalar, or spin field on the sky such as the temperature
can be decomposed into spherical harmonics
. Likewise a spins field on the sky
can be decomposed into the spinweighted spherical
harmonics and a tensor
constructed out of the basis vectors
,[12]. The basis
for a spin2 field such as the polarization is
[3,4]
where
 
(1) 
since it transforms under rotations as a symmetric
traceless tensor. This property is more easily seen through
the relation to the
Pauli matrices, , in spherical coordinates .The spins harmonics are expressed in terms of rotation
matrices^{}
as
[12]
The rotation matrix represents rotations by the Euler angles .Since the spin2 harmonics will be useful in the following
sections, we give their explicit form in Table 1 for ;
the higher harmonics are related to the
ordinary spherical harmonics as
 
(2) 
By virtue of their relation to the rotation
matrices,
the spin harmonics
satisfy: the compatibility relation
with spherical harmonics,
;the conjugation relation ;
the orthonormality relation,
 
(3) 
the completeness relation,
 
(4) 
the parity relation,
 
(5) 
the generalized addition relation,
 
(6) 
which follows from the group multiplication property of
rotation matrices which relates a rotation from through the origin to with a direct rotation
in terms of the Euler angles
defined in Fig. 1;
and the ClebschGordan relation,
 

Figure 1:
Addition theorem and scattering geometry. The addition
theorem for spins harmonics Eqn. (7) is
established by their relation to rotations Eqn. (2)
and by noting that
a rotation from through
the origin (pole) to is equivalent to a direct
rotation by the Euler angles . For the
scattering problem of Eqn. (48), these angles
represent the rotation by from the
frame to the scattering frame, by the scattering angle , and
by back into the frame.
It is worthwhile to examine the implications of these properties.
Note that the orthogonality and completeness relations
Eqns. (4) and (5) do not
extend to different spin states. Orthogonality between states is established by the Pauli basis of
Eqn. (1) and .
The parity equation (6) tells us that the
spin flips under a parity transformation so that unlike
the s=0 spherical harmonics, the higher spin harmonics are not
parity eigenstates. Orthonormal parity states can be constructed
as [3,4]
 
(7) 
which have ``electric'' and ``magnetic''
type parity for the
states respectively.
We shall see in §IIIC that the polarization
evolution naturally
separates into parity eigenstates. The addition property will
be useful in relating the scattering angle to coordinates
on the sphere in §IIIB. Finally
the ClebschGordan relation Eqn. (8) is
central to the following discussion and will be used to derive
the total angular momentum representation in
§IIB and evolution
equations for angular moments of the radiation §IIIC.
Next: Radial Modes
Up: Normal Modes
Previous: Normal Modes
Wayne Hu
9/9/1997