Wigner D-matrix

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner. stands for "Darstellung", which means "representation" in German.
Definition of the Wigner D-matrix.
Let be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics, these three operators are the components of a vector operator known as "angular momentum". Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor. 
In all cases, the three operators satisfy the following commutation relations,
where "i" is the purely imaginary number and Planck's constant has been set equal to one. The Casimir operator 
commutes with all generators of the Lie algebra. Hence, it may be diagonalized together with . 
This defines the spherical basis used here. That is, in this basis, there is a "complete set" of kets with
where "j" = 0, 1/2, 1, 3/2, 2, ... for SU(2), and "j" = 0, 1, 2, ... for SO(3). In both cases, .
A 3-dimensional rotation operator can be written as
where "α", "β", "γ" are Euler angles (characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation).
The Wigner D-matrix is a unitary square matrix of dimension 2"j" + 1 in this spherical basis with elements
where
is an element of the orthogonal Wigner's (small) d-matrix.
That is, in this basis, 
is diagonal, like the "γ" matrix factor, but unlike the above "β" factor.
Wigner (small) d-matrix.
Wigner gave the following expression:
The sum over "s" is over such values that the factorials are nonnegative, i.e. formula_9, formula_10.
"Note:" The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor formula_11 in this formula is replaced by formula_12 causing half of the functions to be purely imaginary. The realness of the d-matrix elements is one of the reasons that the z-y-z convention, used in this article, is usually preferred in quantum mechanical applications.
The d-matrix elements are related to Jacobi polynomials formula_13 with nonnegative formula_14 and formula_15 Let 
If
Then, with formula_18 the relation is
where formula_20
Properties of the Wigner D-matrix.
The complex conjugate of the D-matrix satisfies a number of differential properties that can be formulated concisely by introducing the following operators with formula_21
which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.
Further,
which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.
The operators satisfy the commutation relations
and the corresponding relations with the indices permuted cyclically. The formula_25 satisfy "anomalous commutation relations" (have a minus sign on the right hand side).
The two sets mutually commute,
and the total operators squared are equal,
Their explicit form is,
The operators formula_29 act on the first (row) index of the D-matrix,
The operators formula_25 act on the second (column) index of the D-matrix
and because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,
Finally,
In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebras generated by formula_35 and formula_36.
An important property of the Wigner D-matrix follows from the commutation of 
formula_37 with the time reversal operator 
formula_38
or
Here we used that formula_41 is anti-unitary (hence the complex conjugation after moving formula_42 from ket to bra), formula_43 and formula_44.
Orthogonality relations.
The Wigner D-matrix elements formula_45 form a set of orthogonal functions of the Euler angles formula_46 and formula_47:
This is a special case of the Schur orthogonality relations.
Crucially, by the Peter–Weyl theorem, they further form a "complete" set.
The group characters for SU(2) only depend on the rotation angle "β", being class functions, so, then, independent of the axes of rotation,
and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,
The completeness relation (worked out in the same reference, (3.95)) is 
whence, for formula_52
Kronecker product of Wigner D-matrices, Clebsch-Gordan series.
The set of Kronecker product matrices
forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:
The symbol formula_56 is a
Clebsch-Gordan coefficient.
Relation to spherical harmonics and Legendre polynomials.
For integer values of formula_57, the D-matrix elements with second index equal to zero are proportional
to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:
This implies the following relationship for the d-matrix:
A rotation of spherical harmonics formula_60 then is effectively a composition of two rotations, 
When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:
In the present convention of Euler angles, formula_63 is 
a longitudinal angle and formula_64 is a colatitudinal angle (spherical polar angles
in the physical definition of such angles). This is one of the reasons that the "z"-"y"-"z"
convention is used frequently in molecular physics.
From the time-reversal property of the Wigner D-matrix follows immediately
There exists a more general relationship to the spin-weighted spherical harmonics:
for "j" = 1
for "j" = 3/2 
for "j" = 2
Wigner d-matrix elements with swapped lower indices are found with the relation: 