456
N. Geometrical Aspects of Gauge Fields
α and γ gives finally
(
)
∂V →
∂V
α
=
+ Γ
α
γβ V
γ → e α .
(N.11)
∂q β
∂q β
This is a very important result: it shows that, whereas the components of V → in
the basis → e α are just V
α , the components of the derivative of V → are not simply
∂V
α /∂q
β , but contain an additional term: the ‘components of the derivative
of a vector’ are not just the ‘derivatives of the components of the vector’.
Let us abbreviate ∂/∂q
β to ∂ β ; then (N.11) tells us that in the → e α basis,
as used in (N.11), the components of the ∂ β derivative of V → are
∂ β V
α + Γ
α
γβ V
γ
≡ D β V
α .
(N.12)
The expression (N.12) is called the ‘covariant derivative’ of V
α within the
context of the mathematics of general coordinate systems: it is denoted (as
in (N.12)) by D β V
α or, often, by V
α (in the latter notation, ∂ β V
α is V
α ).
;β
,β
The most important property of D β V
α is its transformation character under
general coordinate transformations. Crucially, it transforms as a tensor T
α
β
(see appendix D of volume 1) with the indicated ‘one up, one down’ indices;
we shall not prove this here, referring instead to Schutz (1988), for example.
This property is the reason for the name ‘covariant derivative’, meaning in
this case essentially that it transforms the way its indices would have you
believe it should. By contrast, and despite appearances, ∂ β V
α by itself does
not transform as a ‘T
α ’ tensor, and in a similar way Γ
α
is not a ‘T
α ’-type
β
γβ
γβ
tensor; only the combined object D β V
α is a ‘T
α ’.
β
This circumstance is highly reminiscent of the situation we found in the
case of gauge transformations. Consider the simplest case, that of U(1), for
which D μ ψ = ∂ μ ψ + iqA μ ψ. The quantity D μ ψ transforms under a gauge
transformation in the same way as ψ itself, but ∂ μ ψ does not. There is thus
a close analogy between the ‘good’ transformation properties of D β V
α and of
D μ ψ. Further, the structure of D μ ψ is very similar to that of D β V
α . There
are two pieces, the first of which is the straightforward derivative, while the
second involves a new field (Γ or A) and is also proportional to the original
field. The ‘i’ of course is a big difference, showing that in the gauge symmetry
case the transformations mix the real and imaginary parts of the wavefunction,
rather than actual spatial components of a vector.
Indeed, the analogy is even closer in the non-Abelian – e.g. local SU(2)
– case. As we have seen, ∂
μ ψ
(
1
2 ) does not transform as an SU(2) isospinor
because of the extra piece involving ∂
μ
∊; nor do the gauge fields W
μ transform
as pure T = 1 states, also because of a ∂
μ
∊ term. But the gauge covariant
combination (∂
μ + igτ · W
μ /2)ψ
(
1
2 ) does transform as an isospinor under local
SU(2) transformations, the two ‘extra’ ∂
μ
∊ pieces cancelling each other out.
There is a useful way of thinking about the two contributions to D β V
α
(or D μ ψ). Let us multiply (N.12) by dq
β and sum over β so as to obtain
β
DV
α
≡ ∂ β V
α dq
β + Γ
α
γβ V
γ dq .
(N.13)
N. Geometrical Aspects of Gauge Fields
α and γ gives finally
(
)
∂V →
∂V
α
=
+ Γ
α
γβ V
γ → e α .
(N.11)
∂q β
∂q β
This is a very important result: it shows that, whereas the components of V → in
the basis → e α are just V
α , the components of the derivative of V → are not simply
∂V
α /∂q
β , but contain an additional term: the ‘components of the derivative
of a vector’ are not just the ‘derivatives of the components of the vector’.
Let us abbreviate ∂/∂q
β to ∂ β ; then (N.11) tells us that in the → e α basis,
as used in (N.11), the components of the ∂ β derivative of V → are
∂ β V
α + Γ
α
γβ V
γ
≡ D β V
α .
(N.12)
The expression (N.12) is called the ‘covariant derivative’ of V
α within the
context of the mathematics of general coordinate systems: it is denoted (as
in (N.12)) by D β V
α or, often, by V
α (in the latter notation, ∂ β V
α is V
α ).
;β
,β
The most important property of D β V
α is its transformation character under
general coordinate transformations. Crucially, it transforms as a tensor T
α
β
(see appendix D of volume 1) with the indicated ‘one up, one down’ indices;
we shall not prove this here, referring instead to Schutz (1988), for example.
This property is the reason for the name ‘covariant derivative’, meaning in
this case essentially that it transforms the way its indices would have you
believe it should. By contrast, and despite appearances, ∂ β V
α by itself does
not transform as a ‘T
α ’ tensor, and in a similar way Γ
α
is not a ‘T
α ’-type
β
γβ
γβ
tensor; only the combined object D β V
α is a ‘T
α ’.
β
This circumstance is highly reminiscent of the situation we found in the
case of gauge transformations. Consider the simplest case, that of U(1), for
which D μ ψ = ∂ μ ψ + iqA μ ψ. The quantity D μ ψ transforms under a gauge
transformation in the same way as ψ itself, but ∂ μ ψ does not. There is thus
a close analogy between the ‘good’ transformation properties of D β V
α and of
D μ ψ. Further, the structure of D μ ψ is very similar to that of D β V
α . There
are two pieces, the first of which is the straightforward derivative, while the
second involves a new field (Γ or A) and is also proportional to the original
field. The ‘i’ of course is a big difference, showing that in the gauge symmetry
case the transformations mix the real and imaginary parts of the wavefunction,
rather than actual spatial components of a vector.
Indeed, the analogy is even closer in the non-Abelian – e.g. local SU(2)
– case. As we have seen, ∂
μ ψ
(
1
2 ) does not transform as an SU(2) isospinor
because of the extra piece involving ∂
μ
∊; nor do the gauge fields W
μ transform
as pure T = 1 states, also because of a ∂
μ
∊ term. But the gauge covariant
combination (∂
μ + igτ · W
μ /2)ψ
(
1
2 ) does transform as an isospinor under local
SU(2) transformations, the two ‘extra’ ∂
μ
∊ pieces cancelling each other out.
There is a useful way of thinking about the two contributions to D β V
α
(or D μ ψ). Let us multiply (N.12) by dq
β and sum over β so as to obtain
β
DV
α
≡ ∂ β V
α dq
β + Γ
α
γβ V
γ dq .
(N.13)