IVC1. Representation Theory

The Invariance Group.
Classifying and Labeling Solutions

It often happens that the linear operator in a linear equation does not change form under some set of transformations of the variables.  This provides an extremely useful means of classifying and labeling solutions of the equation, and it can also be helpful in finding solutions.  To illustrate, consider the equation

(IVC1-1)                                                 

where

are complex variables, and the bar denotes complex conjugation.  This equation is invariant under the set of all unitary transformations

(IVC1-2)    or   

That is,

so the operator has the same form, and therefore the same set of solutions, in any rotated coordinate system.  If we follow one unitary transformation, A, by a second, B, the result, C=BA, is also a unitary transformation, so the set of all 2x2 unitary transformations forms a group, U(2).

There are an infinite number of solutions to Eq. (IVC1-1).  These can be grouped together according to which solutions are rotated into each other by a unitary transformation.  For example, the three solutions

(IVC1-3)                                               

are rotated into each other according to

(IVC1-4)                                                            

where the R’s are quadratic functions of the a’s (  etc.).  In this way, we can associate a 3x3 matrix R(A) with every 2x2 matrix A, with R(A) being the representative of A.  The representatives obey the same multiplication rules as the original matrices, , so the group structure is preserved.  The three vectors (functions of ) of Eq. (IVC1-3) thus form a basis for a three dimensional representation of U(2).

Generators of Groups

In group representation theory, the generators of infinitesimal transformations are very important.  They are the first order differential operators that generate transformations infinitesimally close to the identity transformation.  If a linear operator is invariant under a group, it will commute with all the generators.  In our case, if we ignore the overall phase and look at the generators of SU(2), they are (where h.a. means hermitian adjoint)

(IVC1-5)                  

Group Invariants.

For each group, there are invariant operators—that is, operators which commute with all the generators—that can be formed as sums of powers of the generators.  The single invariant for SU(2) is

(IVC1-6)   . 

Labeling of Solutions.

Solutions to linear equations with invariance groups are usually denoted by kets, ,with the label being the values of the group invariants and the eigenvalues of one or more of the generators.  For the solutions of Eq. (IVC1-4), the group invariant, , has a value of 2, and we use the eigenvalues of .  So the properly labeled kets for the functional basis of Eq. (IVC1-3) are

 (IVC1-7)         

It is important for Sec. III to note that when we see kets with group theoretic labels, they are most likely referring to, or representing, functions of some set of variables.

Representations of Operators and Functions.

In addition to having representations of group operations, one can also have representations of more general operators.  If we assume the basis vectors  are a complete set of states with respect to the operator O, where the ket represents or stands for a function of the variables in the problem and the l’s are the labels on the functions, then  and  is the (matrix) representative of the operator O. 

One also has representations of functions.  If a function is expanded in terms of the complete set , , then  is the (vector) representative of the function.  The equation  becomes, in the representation space, .  The representative  of the function is what we know as the wave function in quantum mechanics.