View source for Wigner-Eckart theorem

A theorem describing the form of the matrix elements of tensor operators transforming under some representation of a group or a [[Lie algebra|Lie algebra]]. Tensor operators are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100801.png" /> be a finite-dimensional [[Irreducible representation|irreducible representation]] of a [[Compact group|compact group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100802.png" /> acting on a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100803.png" /> with a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100805.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100807.png" />, be a set of operators acting on a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100808.png" />. One says that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w1100809.png" /> is a tensor operator, transforming under the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008011.png" />, if there exists a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008012.png" /> (infinite dimensional if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008013.png" /> is infinite dimensional) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008015.png" /> such that for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008016.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008017.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008018.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008019.png" /> are the matrix elements of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008020.png" /> with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008021.png" />. If the compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008022.png" /> is a [[Lie group|Lie group]], then the definition of tensor operator can be given also in infinitesimal form. Infinitesimal operators of representations of Lie algebras are important examples of tensor operators [[#References|[a1]]].

In general, a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008023.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008024.png" /> is reducible and decomposes into irreducible components: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008025.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008027.png" />, be orthonormal bases in the support spaces of the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008028.png" />.

The Wigner–Eckart theorem states that if no multiple irreducible representations appear, then the matrix elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008029.png" /> of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008030.png" /> with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008032.png" /> are of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008033.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008034.png" /> are the Clebsch–Gordan coefficients of the tensor product of the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008037.png" /> (if multiple irreducible representations appear in these tensor products, then additional indices must be included) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008038.png" /> are the so-called reduced matrix elements of the tensor operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008039.png" />. The reduced matrix elements are independent of indices of basis elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008042.png" />.

The Wigner–Eckart theorem represents matrix elements of tensor operators as a product of two quantities: the first one (Clebsch–Gordan coefficient) is determined by a group structure and the second one (reduced matrix element) is independent of the group. The first quantity is the same for all tensor operators. Taking arbitrary numbers as reduced matrix elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008043.png" /> one obtains, by the Wigner–Eckart theorem, matrix elements of some tensor operator, transforming under the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110080/w11008044.png" />.

The Wigner–Eckart theorem can be formulated also for finite-dimensional and unitary infinite-dimensional representations of locally compact Lie groups, [[#References|[a2]]]. The definition of tensor operators and the corresponding Wigner–Eckart theorem for quantum groups are more complicated.

The Wigner–Eckart theorem is a generalization of Schur's lemma on operators commuting with all representation operators (cf. [[Schur lemma|Schur lemma]]). The Wigner–Eckart theorem and tensor operators are extensively used in quantum physics.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.C. Biedenharn,   J.D. Louck,   "Angular momentum in quantum physics" , Addison-Wesley  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.U Klimyk,   "Matrix elements and Clebsch-Gordan coefficients of group representations" , Naukova Dumka  (1979)  (In Russian)</TD></TR></table>