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(Σα_{j}M_{j})
where M_{j} is an m×m complex matrix which is Hermitian and α_{j} 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 M_{jk} matrices are linearly independent.
Lie Algebra;
The algebra associated with a Lie group is defined by the commutator relations; i.e.,
[L_{j}, L_{k}] = iΣc_{jkp}L_{p}
where the coefficients c_{jkp} are known as the structure constants.
For now the analysis will concentrate on Lie groups and their generating matrices M_{j} 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;
1/2
0
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
I_{3}, I_{+} and I_{−}, respectively.
Consider now the vectors
p =
1
0
and
n =
0
1
.
The product of I_{3} with p is:
1/2
0
0
-1/2
1
0
=
1/2
0
= (1/2)
1
0
= (1/2)p.
Therefore p is an eigenvector of I_{3} and (1/2) is its eigenvalue.
It is also true that n is an eigenvector of I_{3} but its eigenvalue
is (-1/2) as shown below:
I_{3}n =
1/2
0
0
-1/2
0
1
=
0
-1/2
= (-1/2)
0
1
= (-1/2)n.
In contrast,
I_{−}p =
0
0
1
0
1
0
=
0
1
=
n.
Therefore p is not an eigenvector of I_{−}.
However,
I_{+}p =
0
1
0
0
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;
I_{3} =
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 I_{3} and Y; i.e., I_{3}Y and
YI_{3}. The computation shows that I_{3}Y and
YI_{3} are equal to the same matrix. This is an important result.
It means that I_{3} 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 I_{3}u = (1/2)u and
Yu = (1/3)u. Therefore u is an eigenvector of both I_{3}
and Y and the eigenvalues are 1/2 and 1/3. Another quick computation
shows that I_{3}d = (-1/2)d and Yd = (1/3)d. Likewise
I_{3}s = (0)s and Ys = (-2/3)s. So the pairs of eigenvalues
of I_{3} 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 X_{a} 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 H_{j} for j=1 to m with the following properties.
Each H_{j} is Hermitian.
Each pair of these matrices commute; i.e., [H_{j}, H_{k}s] = 0.
The traces of the products of unlike pairs are zero; i.e., tr(H_{j}H_{sk}) =0
The traces of the squares of all the H_{j}'s are constant; i.e., tr(H²_{j}) = T,
a constant for all j.
The size m of the set of H_{j} 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 H_{j} 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 H_{j} 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 H_{j} 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 I_{3}
on the vector u we take the product of u with
-I_{3}. Thus ρ(I_{3}))u = u(-I_{3}) = (-1/2)u.
Therefore the eigenvalue of I_{3} associated with u is -1/2.
Likewise the eigenvalue of Y associated with u is (-1/3). The eigenvalues
of I_{3} 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.