San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
U.S.A.

The Weight Diagrams for the SU(n) Lie
Groups and Their Accompanying Algebras

One of the most spectacular aspects of particle physics are the weight diagrams that come out of the analyses of the Special Unitary Groups of Rank n, SU(n), the groups of matrices of determinant equal to unity. The weight diagrams for SU(2) and SU(3) are illustrated in Lie Particles. This material is to establish overall relationships and the nature of weight diagrams.

Lie Groups

Lie groups are mathematical groups derived from transformations which are continuous functions of finite sets of parameters. This turns out to be equivalent to groups whose elements can be represented as matrices which can be generated as

L = exp(ΣαjMj)

where Mj is an m×m complex matrix which is Hermitian and αj is a real number. The summation is from j=1 to j=m and m is known as the rank of the Lie group. The Mjk matrices are linearly independent.

Lie Algebra;

The algebra associated with a Lie group is defined by the commutator relations; i.e.,

[Lj, Lk] = iΣcjkpLp

where the coefficients cjkp are known as the structure constants.

For now the analysis will concentrate on Lie groups and their generating matrices Mj and their eigenvalues and eigenvectors. It is sometimes convenient to work in terms of Lie groups and other times in terms of Lie algebras.

Weight Diagrams

The Special Unitary Group of 2x2 matrices, SU(2), can be generated by the 2×2 identity matrix and the following three matrices;

1/20
0-1/2
             
0  1
0  0
             
0  0
1  0

Note that the trace, the sum of the elements on the principal diagonal, of each of these matrices is zero.

These matrices need to be given names and they will be referred to as I3, I+ and I, respectively.

Consider now the vectors

p =
1
0

and

n =
0
1
.

The product of I3 with p is:

 
1/20
0-1/2
 
1
0
=  
1/2
0
      
= (1/2)
1
0
= (1/2)p.

Therefore p is an eigenvector of I3 and (1/2) is its eigenvalue. It is also true that n is an eigenvector of I3 but its eigenvalue is (-1/2) as shown below:

I3n =
1/20
0-1/2
 
0
1
=  
0
-1/2
      
= (-1/2)
0
1
= (-1/2)n.

In contrast,

Ip =
00
10
 
1
0
=   
0
1
 
= n.

Therefore p is not an eigenvector of I.

However,

I+p =
01
00
 
1
0
=
0
0
  = 0  
1
0
      
= 0p

so p is an eigenvector of I+ with an eigenvalue of 0.

The vectors p and n are eigenvectors of the identity matrix I, as are all vectors, because Ip = p and In = n and their eigenvalues are both +1.

When the pairs of eigenvalues, (+1/2,+1) for p and (-1/2,+1) are plotted the result is called a weight diagram. It is simple and symmetric as shown below.

The diagram is neat but not profound. SU(2) does not go very far in explaining subatomic particles.

The Special Unitary Group of 3x3 matrices, SU(3) is more interesting. It can be generated by the 3×3 identity matrix and the following eight matrices;

I3 =
1/2   0   0
0   -1/2   0
0   0   0
             
Y =
1/3   0   0
0   1/3   0
0   0   -2/3

I+ =
0   1   0
0   0   0
0   0   0

I- =
0   0   0
1   0   0
0   0   0

U+ =
0   0   0
0   0   1
0   0   0

U- =
0   0   0
0   0   0
0   1   0

V+ =
0   0   1
0   0   0
0   0   0

V- =
0   0   0
0   0   0
1   0   0

In the matrix pairs I+ and I-, U+ and U- and V+ and V- one is the transposes of the other. Note again that the trace, the sum of the elements on the principal diagonal, of each of these matrices is zero.

Consider the products of I3 and Y; i.e., I3Y and YI3. The computation shows that I3Y and YI3 are equal to the same matrix. This is an important result. It means that I3 and Y have the same eigenvectors.

Consider now the vectors

u =
1
0
0

and

d =
0
1
0
.

and

s =
0
0
1
.

Some quick computations show that I3u = (1/2)u and Yu = (1/3)u. Therefore u is an eigenvector of both I3 and Y and the eigenvalues are 1/2 and 1/3. Another quick computation shows that I3d = (-1/2)d and Yd = (1/3)d. Likewise I3s = (0)s and Ys = (-2/3)s. So the pairs of eigenvalues of I3 and Y associated with their mutual eigenvectors u, d and s are (1/2, 1/3), (-1/2, 1/3) and (0, -2/3). These eigenvalue pairs may be plotted in a weight diagram as shown below.

Weight Diagrams in General

Now the analysis will be limited to the generators of simple Lie algebras. Let Xa for a equal to 1 to q be the generators of an irreducible representation of a Lie algebra L. From this set of generators for a set of matrices Hj for j=1 to m with the following properties.

Each Hj is Hermitian.

Each pair of these matrices commute; i.e., [Hj, Hks] = 0.

The traces of the products of unlike pairs are zero; i.e., tr(HjHsk) =0

The traces of the squares of all the Hj's are constant; i.e., tr(H²j) = T, a constant for all j.

The size m of the set of Hj matrices, the generators of the representation, should be as large as possible. The value of m is known as the rank of the Lie algebra and this set of Hj matrices is known as the Cartan subalgebra of the Lie algebra. The rank is usually much less than the order of the algebra. Often it is no more than two.

The Hj matrices are diagonalized and their eigenvalues obtained. The eigenvalue largest in magnitude is known as the weight for its matrix. The weight vector is the vector of the eigenvalues of the set of Hj matrices. A weight vector represents a vertex of the weight diagram.

When the data for all the representations of the Lie algebra are plotted there is a geometric figure. Sets of subatomic particles are identified with the vertices of the geometric figure or with its center, as shown below. More than one weight vector may map into the same point of the weight diagram, again as shown below.

The Roots of a Representation

The eigenvectors dealt with previously are the basis vectors for the column vector space. There is also a dual space of row vectors which has as basis vectors the transposes of those eigenvectors.

For this dual space the representations are the negative of the matrices in the primal space. Thus to find the result of the action of I3 on the vector u we take the product of u with -I3. Thus ρ(I3))u = u(-I3) = (-1/2)u. Therefore the eigenvalue of I3 associated with u is -1/2. Likewise the eigenvalue of Y associated with u is (-1/3). The eigenvalues of I3 and Y associated with d are (1/2,-1/3). For s the eigenvalues are (2/3,0). When these eigenvalue pairs are plotted in a weight diagram with those for (u,d) and s the results are as shown below.

(To be continued.)


Source:

Howard Georgi, Lie Algebras in Particle Physics, Benjamin/Cummings Publishing Co., Reading, Massachusetts, 1982.

HOME PAGE OF applet-magic