[1] While the two approaches address exactly the same physical problem, the 3-j symbols do so more symmetrically.
The 3-j symbols are given in terms of the Clebsch–Gordan coefficients by The j and m components are angular-momentum quantum numbers, i.e., every j (and every corresponding m) is either a nonnegative integer or half-odd-integer.
The exponent of the sign factor is always an integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution m3 → −m3: where
Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example,
The CG coefficients are defined so as to express the addition of two angular momenta in terms of a third: The 3-j symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero: Here
It is apparent that the 3-j symbol treats all three angular momenta involved in the addition problem on an equal footing and is therefore more symmetrical than the CG coefficient.
The Wigner 3-j symbol is zero unless all these conditions are satisfied: A 3-j symbol is invariant under an even permutation of its columns: An odd permutation of the columns gives a phase factor: Changing the sign of the
quantum numbers (time reversal) also gives a phase: The 3-j symbols also have so-called Regge symmetries, which are not due to permutations or time reversal.
These are best displayed by the definition of a Regge symbol, which is a one-to-one correspondence between it and a 3-j symbol and assumes the properties of a semi-magic square:[3] whereby the 72 symmetries now correspond to 3!
These facts can be used to devise an effective storage scheme.
[3] A system of two angular momenta with magnitudes j1 and j2 can be described either in terms of the uncoupled basis states (labeled by the quantum numbers m1 and m2), or the coupled basis states (labeled by j3 and m3).
The 3-j symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations The triangular delta {j1 j2 j3} is equal to 1 when the triad (j1, j2, j3) satisfies the triangle conditions, and is zero otherwise.
The 3-jm symbols give the integral of the products of three spherical harmonics[5] with
Similar relations exist for the spin-weighted spherical harmonics if
Generally a better approximation obeying the Regge symmetry is given by where
The following quantity acts as a metric tensor in angular-momentum theory and is also known as a Wigner 1-jm symbol:[1] It can be used to perform time reversal on angular momenta.
Wigner 3-j symbols are related to Racah V-coefficients[7] by a simple phase: This section essentially recasts the definitional relation in the language of group theory.
The linear transformations can be given by a group of matrices with respect to some basis of the vector space.
A reducible representation is one where a change of basis can be applied to bring all the matrices into block diagonal form.
For each value of j, the 2j+1 kets form a basis for an irreducible representation (irrep) of SO(3)/SU(2) over the complex numbers.
symbol has been most intensely studied in the context of the coupling of angular momentum.
The original paper by Wigner[1] was not restricted to SO(3)/SU(2) but instead focussed on simply reducible (SR) groups.
These are groups in which For SR groups, every irrep is equivalent to its complex conjugate, and under permutations of the columns the absolute value of the symbol is invariant and the phase of each can be chosen so that they at most change sign under odd permutations and remain unchanged under even permutations.
General compact groups will neither be ambivalent nor multiplicity free.
symbol for the general case using the relation to the Clebsch-Gordon coefficients of where
symbols for compact groups has been performed based on these principles.
symbols or Clebsch-Gordon coefficients for the finite crystallographic point groups and the double point groups The book by Butler [24] references these and details the theory along with tables.
They need to be dealt with using Wigner's theory of corepresentations of unitary and antiunitary groups.
A significant departure from standard representation theory is that the multiplicity of the irreducible corepresentation
is generally smaller than the multiplicity of the trivial corepresentation in the triple product