San José State University Department of Economics 

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In order to understand what ClebschGordan coefficients are and what they are used for, as opposed to dealing with their numerical values and computation, it is necessary to look at the nature of vectors and vector spaces.
What is ordinarily meant by the term vector is called a polar vector. There is also something called an axial vector, which is the vector (cross) product of two polar vectors. The difference between polar and axial vectors is revealed when we consider a transformation of the coordinate system that changes the handedness of the coordinate system.
Consider two vectors whose respresentation are:
A = (1, 2, 3) and B = (3, 4, 7).
Their cross product C=AxB is: (2, 2, 4).
If we invert the coordinate system; i.e.,
then
the representation of A becomes
(1, 2, 3) and
likewise the representation
of B becomes (3, 4, 7). (If the original coordinate system was righthanded
then the new coordinate system is lefthanded.) If C were a true vector then
its representation should go to (2. 2, 4). But the
cross product of the representations of A and B in the
new coordinate system is
(2, 2, 4). Thus representation of C = AxB did not go
to (2, 2, 4) as would be
required if C were to be a true vector. For this
reason C=AxB is called
a pseudovector or axial vector.
An axial vector can be considered a representation of a second order antisymmetric tensor; i.e.,
B_{1} 
B_{2} 
B_{3} 
= 
0  B_{1}  B_{2} 
B_{1}  0  B_{3} 
B_{2}  B_{3}  0 
This information can be summarized in the form of a table:
Types of Scalars, Vectors and Tensors  

Name  Origin  Examples  Transformation Law 
Polar Vector  Position vectors  Displacement, Electric Field  E'_{i} = Σl_{ir}E_{r} 
Axial Vector  cross product of polar vectors  Magnetic Field  B'_{i}= det (l)Σl_{ir} B_{r} 
Polar Tensor  relation between two vectors of the same type  Magnetic Permeability  P'_{i..j}= Σl_{ir} ...l_{kw}P_{r..w} 
Axial Tensor  Relation between two vectors of different types  Optical Gyration  Q'_{i..j}=det(l)Σl_{ir} ...l_{kw}Q_{r..w} 
Psuedoscalar  polar vector dotted with axial vector  Rotary Power of Optically Active Crystal 
A basis for a vector space is a set of vectors such that any vector in the space can be expressed as a linear combination of elements of the basis. The coefficients in the linear combination of basis lelements for a vector are the components of that vector with respect to the basis. Those components are not the vector itself; they are just the representation of that vector with respect to a particular basis.
For another basis there will be a different set of components. The essential character of a vector is how the representation changes as the basis changes. The change in the components of an arbitrary vector can be represented in terms of the components of the elements of one basis expressed in terms of the elements of the other basis.
This relationship is expressed algebraically as follows. Let B_{1}, B_{2}, ...,B_{n} be a basis for a vector space V. Let X be a vector in V. Then there exist coefficients a_{1}, a_{2},..., a_{n} such that
These representations of the B_{i} can now be substituted into the representation of X in terms of the B_{i}'s. The result is an expression of X in terms of the B'_{i}'s; ile.,
The elements of a set of basis vectors and hence the components of vectors are usually labeled with the positive integers, but there is no reason that this has to be. Any countable set will do. Since the basis set is often the eigenvectors of the Schroedinger equation, they can be labeled with their associated eigenvalues or some composite label which includes their eigenvalues.
(To be continued.)