49
13.2. Local SU(3) Symmetry
This tensor is of fundamental importance in a (non-Abelian) gauge theory.
Since it arises from the commutator of two gauge-covariant derivatives, we are
guaranteed that it itself is gauge covariant – that is to say, ‘it transforms under
local SU(2) transformations in the way its SU(2) structure would indicate’.
Now F
μν has clearly three SU(2) components and must be an SU(2) triplet:
indeed, it is true that under an infinitesimal local SU(2) transformation
′μν
F
μν
− g∊(x) × F
μν
F
=
(13.51)
which is the expected law (cf (12.64)) for an SU(2) triplet. Problem 13.4
verifies that (13.51) follows from (13.49) and the transformation law (13.23)
for the W
μ fields. Note particularly that F
μν transforms ‘properly’, as an
SU(2) triplet should, without the ∂
μ part which appears in δW
μ .
This non-Abelian F
μν is a much more interesting object than the Abelian
F
μν
′μν
F
μν ). F
μν
(which is actually U(1)-gauge invariant, of course: F
=
contains the gauge coupling constant g, confirming (cf comment(c) in section 13.1.1) that the gauge fields themselves carry SU(2) ‘charge’, and act
as sources for the field strength. Appendix N shows how these field strength
tensors may be regarded as analogous to geometrical curvatures.
It is now straightforward to move to the quantum field case and construct
the SU(2) Yang-Mills analogue of the Maxwell Lagrangian −
1 F ˆ μν F ˆ μν . It is
4
μν
simply −
1
F ˆ μν · F ˆ , the SU(2) ‘dot product’ ensuring SU(2) invariance (see
4
problem 13.5), even under local transformation, in view of the transformation
law (13.51). But before proceeding in this way we first need to introduce local
SU(3) symmetry.
13.2 Local SU(3) Symmetry
Using what has been done for global SU(3) symmetry in section 12.2, and
the preceding discussion of how to make a global SU(2) into a local one, it
is straightforward to develop the corresponding theory of local SU(3). This
is the gauge group of QCD, the three degrees of freedom of the fundamental
quark triplet now referring to ‘colour’, as will be further discussed in chapter
14. We denote the basic triplet by ψ, which transforms under a local SU(3)
transformation according to
ψ
′ = exp[ig s λ · α(x)/2]ψ,
(13.52)
which is the same as the global transformation (12.74) but with the 8 constant
parameters α replaced by x-dependent ones, and with a coupling strength g s
inserted. The SU(3)-covariant derivative, when acting on an SU(3) triplet ψ,
is given by the indicated generalization of (13.10), namely
D
μ (acting on SU(3) triplet) = ∂
μ + ig s λ/2 · A
μ
(13.53)
13.2. Local SU(3) Symmetry
This tensor is of fundamental importance in a (non-Abelian) gauge theory.
Since it arises from the commutator of two gauge-covariant derivatives, we are
guaranteed that it itself is gauge covariant – that is to say, ‘it transforms under
local SU(2) transformations in the way its SU(2) structure would indicate’.
Now F
μν has clearly three SU(2) components and must be an SU(2) triplet:
indeed, it is true that under an infinitesimal local SU(2) transformation
′μν
F
μν
− g∊(x) × F
μν
F
=
(13.51)
which is the expected law (cf (12.64)) for an SU(2) triplet. Problem 13.4
verifies that (13.51) follows from (13.49) and the transformation law (13.23)
for the W
μ fields. Note particularly that F
μν transforms ‘properly’, as an
SU(2) triplet should, without the ∂
μ part which appears in δW
μ .
This non-Abelian F
μν is a much more interesting object than the Abelian
F
μν
′μν
F
μν ). F
μν
(which is actually U(1)-gauge invariant, of course: F
=
contains the gauge coupling constant g, confirming (cf comment(c) in section 13.1.1) that the gauge fields themselves carry SU(2) ‘charge’, and act
as sources for the field strength. Appendix N shows how these field strength
tensors may be regarded as analogous to geometrical curvatures.
It is now straightforward to move to the quantum field case and construct
the SU(2) Yang-Mills analogue of the Maxwell Lagrangian −
1 F ˆ μν F ˆ μν . It is
4
μν
simply −
1
F ˆ μν · F ˆ , the SU(2) ‘dot product’ ensuring SU(2) invariance (see
4
problem 13.5), even under local transformation, in view of the transformation
law (13.51). But before proceeding in this way we first need to introduce local
SU(3) symmetry.
13.2 Local SU(3) Symmetry
Using what has been done for global SU(3) symmetry in section 12.2, and
the preceding discussion of how to make a global SU(2) into a local one, it
is straightforward to develop the corresponding theory of local SU(3). This
is the gauge group of QCD, the three degrees of freedom of the fundamental
quark triplet now referring to ‘colour’, as will be further discussed in chapter
14. We denote the basic triplet by ψ, which transforms under a local SU(3)
transformation according to
ψ
′ = exp[ig s λ · α(x)/2]ψ,
(13.52)
which is the same as the global transformation (12.74) but with the 8 constant
parameters α replaced by x-dependent ones, and with a coupling strength g s
inserted. The SU(3)-covariant derivative, when acting on an SU(3) triplet ψ,
is given by the indicated generalization of (13.10), namely
D
μ (acting on SU(3) triplet) = ∂
μ + ig s λ/2 · A
μ
(13.53)