446
M. Group Theory
(R)
is always provided by a set of matrices {X ν } whose elements are defined by
(
)
(R)
ν
X
= −c
(M.71)
λ
λμ ,
μν
where the c’s are the structure constants of (M.12), and each of μ, ν, λ runs
from 1 to r. Thus these matrices are of dimensionality r × r, where r is the
number of generators. That this prescription works is due to the fact that the
generators satisfy the Jacobi identity
[X ˆ λ , [X ˆ μ , X ˆ ν ]] + [ X ˆ μ , [X ˆ ν , X ˆ λ ]] + [ X ˆ ν , [X ˆ λ , X ˆ μ ]] = 0.
(M.72)
Using (M.12) to evaluate the commutators, and the fact that the generators
are independent, we obtain
α β
α β
α β
c
+ c
c
(M.73)
μν c λα
νλ c μα + λμ c να = 0.
The reader may fill in the steps leading from here to the desired result:
(
) (
)
(
) (
)
(
)
(R)
X
(R)
X
(R)
(R)
α
X
(R)
X
−
X
= c
.
(M.74)
λ
μ
μ
λ
λμ
α
να
αβ
να
αβ
νβ
(M.74) is precisely the (νβ) matrix element of
(R) , X
(R)
α
λμ X
(R)
[X
] = c
,
(M.75)
λ
μ
α
(R)
showing that the X μ ’s satisfy the group algebra (M.12), as required. The
representation in which the generators are represented by (minus) the structure constants, in the sense of (M.71), is called the regular or adjoint representation.
Having obtained any particular matrix representation X
(P) of the generators of a group G, a corresponding matrix representation of the group elements
can be obtained by exponentiation, via
X
(P)
D
(P) (α) = exp{iα ·
},
(M.76)
where α = (α 1 , α 2 , . . . , α r ) (see (12.31) and (12.49) for SU(2), and (12.74)
and (12.81) for SU(3)). In the case of the groups whose elements are matrices,
exponentiating the generators X
(G) just recreates the general matrices of the
group, so we may call this the ‘self-representation’: the one in which the group
elements are represented by themselves. In the more general case (M.76), the
crucial property of the matrices D
(P) (α) is that they obey the same group
combination law as the elements of the group G they are representing: that
is, if the group elements obey
g(α)g(β) = g(γ(α, β)),
(M.77)
then
D
(P) (α)D
(P) (β) = D
(P) (γ(α, β)).
(M.78)
It is a rather remarkable fact that there are certain, say, 10 × 10 matrices
which multiply together in exactly the same way as the rotation matrices of
SO(3).
M. Group Theory
(R)
is always provided by a set of matrices {X ν } whose elements are defined by
(
)
(R)
ν
X
= −c
(M.71)
λ
λμ ,
μν
where the c’s are the structure constants of (M.12), and each of μ, ν, λ runs
from 1 to r. Thus these matrices are of dimensionality r × r, where r is the
number of generators. That this prescription works is due to the fact that the
generators satisfy the Jacobi identity
[X ˆ λ , [X ˆ μ , X ˆ ν ]] + [ X ˆ μ , [X ˆ ν , X ˆ λ ]] + [ X ˆ ν , [X ˆ λ , X ˆ μ ]] = 0.
(M.72)
Using (M.12) to evaluate the commutators, and the fact that the generators
are independent, we obtain
α β
α β
α β
c
+ c
c
(M.73)
μν c λα
νλ c μα + λμ c να = 0.
The reader may fill in the steps leading from here to the desired result:
(
) (
)
(
) (
)
(
)
(R)
X
(R)
X
(R)
(R)
α
X
(R)
X
−
X
= c
.
(M.74)
λ
μ
μ
λ
λμ
α
να
αβ
να
αβ
νβ
(M.74) is precisely the (νβ) matrix element of
(R) , X
(R)
α
λμ X
(R)
[X
] = c
,
(M.75)
λ
μ
α
(R)
showing that the X μ ’s satisfy the group algebra (M.12), as required. The
representation in which the generators are represented by (minus) the structure constants, in the sense of (M.71), is called the regular or adjoint representation.
Having obtained any particular matrix representation X
(P) of the generators of a group G, a corresponding matrix representation of the group elements
can be obtained by exponentiation, via
X
(P)
D
(P) (α) = exp{iα ·
},
(M.76)
where α = (α 1 , α 2 , . . . , α r ) (see (12.31) and (12.49) for SU(2), and (12.74)
and (12.81) for SU(3)). In the case of the groups whose elements are matrices,
exponentiating the generators X
(G) just recreates the general matrices of the
group, so we may call this the ‘self-representation’: the one in which the group
elements are represented by themselves. In the more general case (M.76), the
crucial property of the matrices D
(P) (α) is that they obey the same group
combination law as the elements of the group G they are representing: that
is, if the group elements obey
g(α)g(β) = g(γ(α, β)),
(M.77)
then
D
(P) (α)D
(P) (β) = D
(P) (γ(α, β)).
(M.78)
It is a rather remarkable fact that there are certain, say, 10 × 10 matrices
which multiply together in exactly the same way as the rotation matrices of
SO(3).