19
12.3. Flavour SU(3) f
generalization of isospin SU(2) f to include strangeness. Like SU(2), SU(3)
is a group whose elements are matrices – in this case, unitary 3 × 3 ones,
of unit determinant. The general group-theoretic analysis of SU(3) is quite
complicated, but is fortunately not necessary for the physical applications we
require. We can, in fact, develop all the results needed by mimicking the steps
followed for SU(2).
We start by finding the general form of an SU(3) matrix. Such matrices
obviously act on 3-component column vectors, the generalization of the 2component isospinors of SU(2). In more physical terms, we regard the three
quark wavefunctions u, d and s as being approximately degenerate, and we
consider unitary 3 × 3 transformations among them via
′
q = Wq
(12.69)
where q now stands for the 3-component column vector
(
)
u
q =
d
(12.70)
(
)
s
and W is a 3 × 3 unitary matrix of determinant 1 (again, an overall phase
has been extracted). The representation provided by this triplet of states
is called the ‘fundamental’ representation of SU(3) f (just as the isospinor
representation is the fundamental one of SU(2) f ).
To determine the general form of an SU(3) matrix W, we follow exactly
the same steps as in the SU(2) case. An infinitesimal SU(3) matrix has the
form
W infl = 1 3 + iχ
(12.71)
where χ is a 3 × 3 traceless Hermitian matrix. Such a matrix involves eight
independent parameters (problem (12.7)) and can be written as
χ = η · λ/2
(12.72)
where η = (η 1 , . . . , η 8 ) and the λ’s are eight matrices generalizing the τ matrices of (12.25). They are the generators of SU(3) in the three-dimensional
fundamental representation, and their commutation relations define the algebra of SU(3) (compare (12.28) for SU(2)):
[λ a /2, λ b /2] = if abc λ c /2,
(12.73)
where a, b and c run from 1 to 8.
The λ-matrices (often called the Gell-Mann matrices), are given in appendix M, along with the SU(3) structure constants if abc ; the constants f abc
are all real.
A finite SU(3) transformation on the quark triplet is then (cf (12.32))
′
q = exp(iα · λ/2)q,
(12.74)
12.3. Flavour SU(3) f
generalization of isospin SU(2) f to include strangeness. Like SU(2), SU(3)
is a group whose elements are matrices – in this case, unitary 3 × 3 ones,
of unit determinant. The general group-theoretic analysis of SU(3) is quite
complicated, but is fortunately not necessary for the physical applications we
require. We can, in fact, develop all the results needed by mimicking the steps
followed for SU(2).
We start by finding the general form of an SU(3) matrix. Such matrices
obviously act on 3-component column vectors, the generalization of the 2component isospinors of SU(2). In more physical terms, we regard the three
quark wavefunctions u, d and s as being approximately degenerate, and we
consider unitary 3 × 3 transformations among them via
′
q = Wq
(12.69)
where q now stands for the 3-component column vector
(
)
u
q =
d
(12.70)
(
)
s
and W is a 3 × 3 unitary matrix of determinant 1 (again, an overall phase
has been extracted). The representation provided by this triplet of states
is called the ‘fundamental’ representation of SU(3) f (just as the isospinor
representation is the fundamental one of SU(2) f ).
To determine the general form of an SU(3) matrix W, we follow exactly
the same steps as in the SU(2) case. An infinitesimal SU(3) matrix has the
form
W infl = 1 3 + iχ
(12.71)
where χ is a 3 × 3 traceless Hermitian matrix. Such a matrix involves eight
independent parameters (problem (12.7)) and can be written as
χ = η · λ/2
(12.72)
where η = (η 1 , . . . , η 8 ) and the λ’s are eight matrices generalizing the τ matrices of (12.25). They are the generators of SU(3) in the three-dimensional
fundamental representation, and their commutation relations define the algebra of SU(3) (compare (12.28) for SU(2)):
[λ a /2, λ b /2] = if abc λ c /2,
(12.73)
where a, b and c run from 1 to 8.
The λ-matrices (often called the Gell-Mann matrices), are given in appendix M, along with the SU(3) structure constants if abc ; the constants f abc
are all real.
A finite SU(3) transformation on the quark triplet is then (cf (12.32))
′
q = exp(iα · λ/2)q,
(12.74)