44
13. Local Non-Abelian (Gauge) Symmetries
Equating components of τ on both sides, we deduce
δW
μ = −∂
μ
∊(x) − g[∊(x) × W
μ ].
(13.23)
The reader may note the close similarity between these manipulations and
those encountered in section 12.1.3.
Equation (13.23) defines the way in which the SU(2) gauge fields W
μ
transform under an infinitesimal SU(2) gauge transformation. If it were not
for the presence of the first term ∂
μ
∊(x) on the right-hand side, (13.23) would
be simply the (infinitesimal) transformation law for the T = 1 triplet representation of SU(2) – see (12.64) and (12.65) in section 12.1.3. As mentioned at
the end of section 12.2, the T = 1 representation is the ‘adjoint’, or ‘regular’,
representation of SU(2), and this is the one to which gauge fields belong, in
general. But there is the extra term −∂
μ
∊(x). Clearly this is directly analogous to the −∂
μ χ(x) term in the transformation of the U(1) gauge field A
μ ;
here, an independent infinitesimal function ∈ i (x) is required for each compoμ
nent W (x). If the ∈’s were independent of x, then ∂
μ
∊(x) would of course
i
vanish and the transformation law (13.23) would indeed be just that of an
SU(2) triplet. Thus we can say that under global SU(2) transformations, the
W
μ behave as a normal triplet. But under local SU(2) transformations they
acquire the additional −∂
μ
∊(x) piece, and thus no longer transform ‘prop1
2
erly’, as an SU(2) triplet. In exactly the same way, ∂
μ ψ
( ) did not transform
‘properly’ as an SU(2) doublet, under a local SU(2) transformation, because
of the second term in (13.14), which also involves ∂
μ
∊(x). The remarkable re1
2
sult behind the fact that D
μ ψ
( ) does transform ‘properly’ under local SU(2)
transformations, is that the extra term in (13.23) precisely cancels that in
(13.14)!
To summarize progress so far: we have shown that, for infinitesimal transformations, the relation
1
2
1
2
(D
′μ ψ
( )′ ) = [1 + igτ · ∊(x)/2](D
μ ψ
( ) )
(13.24)
(where D
μ is given by (13.10)) holds true if in addition to the infinitesimal
1
2
on ψ
( )
local SU(2) phase transformation
1
2
1
2
ψ
( )′
∊(x)/2]ψ
( )
= [1 + igτ
(13.25)
·
the gauge fields transform according to
′μ
W = W
μ
− ∂
μ
∊(x) − g[∊(x) × W
μ ].
(13.26)
In obtaining these results, the form (13.10) for the covariant derivative has
been assumed, and only the infinitesimal version of (13.2) has been treated
explicitly. It turns out that (13.10) is still appropriate for the finite (noninfinitesimal) transformation (13.2), but the associated transformation law
for the gauge fields is then slightly more complicated than (13.26). Let us
write
U(α(x)) ≡ exp[igτ · α(x)/2]
(13.27)
13. Local Non-Abelian (Gauge) Symmetries
Equating components of τ on both sides, we deduce
δW
μ = −∂
μ
∊(x) − g[∊(x) × W
μ ].
(13.23)
The reader may note the close similarity between these manipulations and
those encountered in section 12.1.3.
Equation (13.23) defines the way in which the SU(2) gauge fields W
μ
transform under an infinitesimal SU(2) gauge transformation. If it were not
for the presence of the first term ∂
μ
∊(x) on the right-hand side, (13.23) would
be simply the (infinitesimal) transformation law for the T = 1 triplet representation of SU(2) – see (12.64) and (12.65) in section 12.1.3. As mentioned at
the end of section 12.2, the T = 1 representation is the ‘adjoint’, or ‘regular’,
representation of SU(2), and this is the one to which gauge fields belong, in
general. But there is the extra term −∂
μ
∊(x). Clearly this is directly analogous to the −∂
μ χ(x) term in the transformation of the U(1) gauge field A
μ ;
here, an independent infinitesimal function ∈ i (x) is required for each compoμ
nent W (x). If the ∈’s were independent of x, then ∂
μ
∊(x) would of course
i
vanish and the transformation law (13.23) would indeed be just that of an
SU(2) triplet. Thus we can say that under global SU(2) transformations, the
W
μ behave as a normal triplet. But under local SU(2) transformations they
acquire the additional −∂
μ
∊(x) piece, and thus no longer transform ‘prop1
2
erly’, as an SU(2) triplet. In exactly the same way, ∂
μ ψ
( ) did not transform
‘properly’ as an SU(2) doublet, under a local SU(2) transformation, because
of the second term in (13.14), which also involves ∂
μ
∊(x). The remarkable re1
2
sult behind the fact that D
μ ψ
( ) does transform ‘properly’ under local SU(2)
transformations, is that the extra term in (13.23) precisely cancels that in
(13.14)!
To summarize progress so far: we have shown that, for infinitesimal transformations, the relation
1
2
1
2
(D
′μ ψ
( )′ ) = [1 + igτ · ∊(x)/2](D
μ ψ
( ) )
(13.24)
(where D
μ is given by (13.10)) holds true if in addition to the infinitesimal
1
2
on ψ
( )
local SU(2) phase transformation
1
2
1
2
ψ
( )′
∊(x)/2]ψ
( )
= [1 + igτ
(13.25)
·
the gauge fields transform according to
′μ
W = W
μ
− ∂
μ
∊(x) − g[∊(x) × W
μ ].
(13.26)
In obtaining these results, the form (13.10) for the covariant derivative has
been assumed, and only the infinitesimal version of (13.2) has been treated
explicitly. It turns out that (13.10) is still appropriate for the finite (noninfinitesimal) transformation (13.2), but the associated transformation law
for the gauge fields is then slightly more complicated than (13.26). Let us
write
U(α(x)) ≡ exp[igτ · α(x)/2]
(13.27)