443
M.5. Matrix representations of generators, and of Lie groups
where a, b and c each run from 1 to 8. The structure constants are if abc , and
the non-vanishing f
′ s are as follows:
f 123 = 1, f 147 = 1/2, f 156 = −1/2, f 246 = 1/2, f 257 = 1/2
(M.59)
√
√
f 345 = 1/2, f 367 = −1/2, f 458 = 3/2, f 678 = 3/2.
(M.60)
Note that the f ’s are antisymmetric in all pairs of indices (Carruthers (1966)
chapter 2).
M.5 Matrix representations of generators, and of Lie
groups
We have shown how the generators X ˆ 1 , X ˆ 2 , . . . , X ˆ r of a Lie group can be constructed as differential operators, understood to be acting on functions of the
‘coordinates’ to which the transformations of the group refer. These generators satisfy certain commutation relations, the Lie algebra of the group. For
any given Lie algebra, it is also possible to find sets of matrices X 1 , X 2 , . . . , X r
(without hats) which satisfy the same commutation relations as the X ˆ ν ’s –
that is, they have the same algebra. Such matrices are said to form a (matrix) representation of the Lie algebra, or equivalently of the generators. The
idea is familiar from the study of angular momentum in quantum mechanics
(Schiff 1968, section 27), where the entire theory may be developed from the
commutation relations (with ħ = 1)
[J ˆ i , J ˆ j ] = i∈ ijk J ˆ k
(M.61)
for the angular momentum operators J ˆ i , together with the physical requirement that the J ˆ i ’s (and the matrices representing them) must be Hermitian.
In this case the matrices are of the form (in quantum-mechanical notation)
(
)
(J)
J
≡ ‹JM J
′
|J ˆ i |JM J ›,
(M.62)
i
′
M J MJ
2
where |JM J › is an eigenstate of J ˆ and of J ˆ 3 with eigenvalues J(J + 1) and
M J respectively. Since M J and M
′ each run over the 2J + 1 values defined by
J
−J ≤ M J , M J
′
≤ J, the matrices J
(J) are of dimension (2J + 1) × (2J + 1).
i
Clearly, since the generators of SU(2) have the same algebra as (M.61), an
identical matrix representation may be obtained for them; these matrices were
(T )
denoted by T
in section 12.1.2. It is important to note that J (or T )
i
can take an infinite sequence of values J = 0, 1/2, 1, 3/2, . . ., corresponding
physically to various ‘spin’ magnitudes. Thus there are infinitely many sets
(J)
(J)
(J)
of three matrices (J , J , J ) all with the same commutation relations as
1
2
3
(M.61).
M.5. Matrix representations of generators, and of Lie groups
where a, b and c each run from 1 to 8. The structure constants are if abc , and
the non-vanishing f
′ s are as follows:
f 123 = 1, f 147 = 1/2, f 156 = −1/2, f 246 = 1/2, f 257 = 1/2
(M.59)
√
√
f 345 = 1/2, f 367 = −1/2, f 458 = 3/2, f 678 = 3/2.
(M.60)
Note that the f ’s are antisymmetric in all pairs of indices (Carruthers (1966)
chapter 2).
M.5 Matrix representations of generators, and of Lie
groups
We have shown how the generators X ˆ 1 , X ˆ 2 , . . . , X ˆ r of a Lie group can be constructed as differential operators, understood to be acting on functions of the
‘coordinates’ to which the transformations of the group refer. These generators satisfy certain commutation relations, the Lie algebra of the group. For
any given Lie algebra, it is also possible to find sets of matrices X 1 , X 2 , . . . , X r
(without hats) which satisfy the same commutation relations as the X ˆ ν ’s –
that is, they have the same algebra. Such matrices are said to form a (matrix) representation of the Lie algebra, or equivalently of the generators. The
idea is familiar from the study of angular momentum in quantum mechanics
(Schiff 1968, section 27), where the entire theory may be developed from the
commutation relations (with ħ = 1)
[J ˆ i , J ˆ j ] = i∈ ijk J ˆ k
(M.61)
for the angular momentum operators J ˆ i , together with the physical requirement that the J ˆ i ’s (and the matrices representing them) must be Hermitian.
In this case the matrices are of the form (in quantum-mechanical notation)
(
)
(J)
J
≡ ‹JM J
′
|J ˆ i |JM J ›,
(M.62)
i
′
M J MJ
2
where |JM J › is an eigenstate of J ˆ and of J ˆ 3 with eigenvalues J(J + 1) and
M J respectively. Since M J and M
′ each run over the 2J + 1 values defined by
J
−J ≤ M J , M J
′
≤ J, the matrices J
(J) are of dimension (2J + 1) × (2J + 1).
i
Clearly, since the generators of SU(2) have the same algebra as (M.61), an
identical matrix representation may be obtained for them; these matrices were
(T )
denoted by T
in section 12.1.2. It is important to note that J (or T )
i
can take an infinite sequence of values J = 0, 1/2, 1, 3/2, . . ., corresponding
physically to various ‘spin’ magnitudes. Thus there are infinitely many sets
(J)
(J)
(J)
of three matrices (J , J , J ) all with the same commutation relations as
1
2
3
(M.61).