M.5. Matrix representations of generators, and of Lie groups
445
Having characterized a given representation by the eigenvalues of the
Casimir operator(s), a further labelling is then required to characterize the
ˆ
states within a given representation (the analogue of the eigenvalue of J 3 for
angular momentum). For SO(4) these further labels may be taken to be the
ˆ
ˆ
ˆ
ˆ
eigenvalues of M 3 and N 3 ; for SU(3) they are the eigenvalues of G 3 and G 8 –
i.e. those corresponding to the third component of isospin and hypercharge,
in the flavour case (see figures 12.3 and 12.4).
In the case of groups whose elements are themselves matrices, such as
SO(3), SO(4), SU(2), SU(3), and the Lorentz group, one particular representation of the generators may always be obtained by considering the general
form of a matrix in the group which is infinitesimally close to the unit element.
In a suitable parametrization, we may write such a matrix as
∑ r
1 + i
∈ ν X
(G)
ν ,
(M.68)
ν=1
( )
( )
( )
where (∈ 1 , ∈ 2 , . . . , ∈ r ) are infinitesimal parameters, and (X
G
1 , X
G
G
2 , . . . , X r )
are matrices representing the generators of the (matrix) group G. This is
exactly the same procedure we followed for SU(2) in section 12.1.1, where
(SU(2))
we found from (12.26) that the three X ν
’s were just τ /2, satisfying the
(SU(3))
SU(2) algebra. Similarly, in section 12.2 we saw that the eight SU(3) X ν
’s
were just λ/2, satisfying the SU(3) algebra. These particular two representations are called the fundamental representations of the SU(2) and SU(3)
algebras, respectively; they are the representations of lowest dimensionality.
(SO(3))
For SO(3), the three X ν
’s are (from (M.17))
(
)
0 0 0
(SO(3))
X 1
= ( 0 0 −i )
0 i 0
(
)
0 0 i
(SO(3))
X
= (
2
0 0 0 )
i 0 0
(
−
)
0
(SO(3))
X 3
= (
−i 0
i 0 0 )
(M.69)
0 0 0
(1)
which are the same as the 3 × 3 matrices T i of (12.48):
(
)
(1)
T i
= −i∈ ijk .
(M.70)
jk
(1)
The matrices τ i /2 and T i correspond to the values J = 1/2, J = 1, respectively, in angular momentum terms.
It is not a coincidence that the coefficients on the right-hand side of (M.70)
are (minus) the SO(3) structure constants. One representation of a Lie algebra
445
Having characterized a given representation by the eigenvalues of the
Casimir operator(s), a further labelling is then required to characterize the
ˆ
states within a given representation (the analogue of the eigenvalue of J 3 for
angular momentum). For SO(4) these further labels may be taken to be the
ˆ
ˆ
ˆ
ˆ
eigenvalues of M 3 and N 3 ; for SU(3) they are the eigenvalues of G 3 and G 8 –
i.e. those corresponding to the third component of isospin and hypercharge,
in the flavour case (see figures 12.3 and 12.4).
In the case of groups whose elements are themselves matrices, such as
SO(3), SO(4), SU(2), SU(3), and the Lorentz group, one particular representation of the generators may always be obtained by considering the general
form of a matrix in the group which is infinitesimally close to the unit element.
In a suitable parametrization, we may write such a matrix as
∑ r
1 + i
∈ ν X
(G)
ν ,
(M.68)
ν=1
( )
( )
( )
where (∈ 1 , ∈ 2 , . . . , ∈ r ) are infinitesimal parameters, and (X
G
1 , X
G
G
2 , . . . , X r )
are matrices representing the generators of the (matrix) group G. This is
exactly the same procedure we followed for SU(2) in section 12.1.1, where
(SU(2))
we found from (12.26) that the three X ν
’s were just τ /2, satisfying the
(SU(3))
SU(2) algebra. Similarly, in section 12.2 we saw that the eight SU(3) X ν
’s
were just λ/2, satisfying the SU(3) algebra. These particular two representations are called the fundamental representations of the SU(2) and SU(3)
algebras, respectively; they are the representations of lowest dimensionality.
(SO(3))
For SO(3), the three X ν
’s are (from (M.17))
(
)
0 0 0
(SO(3))
X 1
= ( 0 0 −i )
0 i 0
(
)
0 0 i
(SO(3))
X
= (
2
0 0 0 )
i 0 0
(
−
)
0
(SO(3))
X 3
= (
−i 0
i 0 0 )
(M.69)
0 0 0
(1)
which are the same as the 3 × 3 matrices T i of (12.48):
(
)
(1)
T i
= −i∈ ijk .
(M.70)
jk
(1)
The matrices τ i /2 and T i correspond to the values J = 1/2, J = 1, respectively, in angular momentum terms.
It is not a coincidence that the coefficients on the right-hand side of (M.70)
are (minus) the SO(3) structure constants. One representation of a Lie algebra