41
13.1. Local SU(2) symmetry
under transformations such as (13.2). We shall generally use the language of
isospin when referring to the physical states and operators, bearing in mind
that this will eventually mean weak isospin.
We shall mimic as literally as possible the discussion of electromagnetic
gauge covariance in sections 2.4 and 2.5 of volume 1. As in that case, no
free particle wave equation can be covariant under the transformation (13.2)
(taking the isospinor example for definiteness), since the gradient terms in the
equation will act on the phase factor α(x). However, wave equations with a
suitably defined covariant derivative can be covariant under (13.2); physically
this means that, just as for electromagnetism, covariance under local nonAbelian phase transformations requires the introduction of a definite force
field.
In the electromagnetic case the covariant derivative is
D
μ = ∂
μ + iqA
μ (x).
(13.4)
For convenience we recall here the crucial property of D
μ . Under a local U(1)
phase transformation, a wavefunction transforms as (cf (13.3))
ψ(x) → ψ
′ (x) = exp(iqχ(x))ψ(x),
(13.5)
from which it easily follows that the derivative (gradient) of ψ transforms as
∂
μ ψ(x) → ∂
μ ψ
′ (x) = exp(iqχ(x))∂
μ ψ(x) + iq∂
μ χ(x)exp(iqχ(x))ψ(x). (13.6)
Comparing (13.6) with (13.5), we see that, in addition to the expected first
term on the right-hand side of (13.6), which has the same form as the righthand side of (13.5), there is an extra term in (13.6). By contrast, the covariant
derivative of ψ transforms as (see section 2.4 of volume 1)
D
μ ψ(x) → D
′μ ψ
′ (x) = exp(iqχ(x))D
μ ψ(x)
(13.7)
exactly as in (13.5), with no additional term on the right-hand side. Note
that D
μ has to carry a prime also, since it contains A
μ which transforms to
A
′μ = A
μ
− ∂
μ χ(x) when ψ transforms by (13.5). The property (13.7) ensured
the gauge covariance of wave equations in the U(1) case; the similar property
in the quantum field case meant that a globally U(1)-invariant Lagrangian
could be converted immediately to a locally U(1)-invariant one by replacing
∂
μ by D ˆ μ (section 7.4).
In appendix D of volume 1 we introduced the idea of ‘covariance’ in the
context of coordinate transformations of 3- and 4-vectors. The essential notion
was of something ‘maintaining the same form’, or ‘transforming the same
way’. The transformations being considered here are gauge transformations
rather than coordinate ones; nevertheless it is true that, under them, D
μ ψ
transforms in the same way as ψ, while ∂
μ ψ does not. Thus the term covariant
derivative seems appropriate. In fact, there is a much closer analogy between
the ‘coordinate’ and the ‘gauge’ cases, which we did not present in volume 1,
but give now in appendix N, for the interested reader.
13.1. Local SU(2) symmetry
under transformations such as (13.2). We shall generally use the language of
isospin when referring to the physical states and operators, bearing in mind
that this will eventually mean weak isospin.
We shall mimic as literally as possible the discussion of electromagnetic
gauge covariance in sections 2.4 and 2.5 of volume 1. As in that case, no
free particle wave equation can be covariant under the transformation (13.2)
(taking the isospinor example for definiteness), since the gradient terms in the
equation will act on the phase factor α(x). However, wave equations with a
suitably defined covariant derivative can be covariant under (13.2); physically
this means that, just as for electromagnetism, covariance under local nonAbelian phase transformations requires the introduction of a definite force
field.
In the electromagnetic case the covariant derivative is
D
μ = ∂
μ + iqA
μ (x).
(13.4)
For convenience we recall here the crucial property of D
μ . Under a local U(1)
phase transformation, a wavefunction transforms as (cf (13.3))
ψ(x) → ψ
′ (x) = exp(iqχ(x))ψ(x),
(13.5)
from which it easily follows that the derivative (gradient) of ψ transforms as
∂
μ ψ(x) → ∂
μ ψ
′ (x) = exp(iqχ(x))∂
μ ψ(x) + iq∂
μ χ(x)exp(iqχ(x))ψ(x). (13.6)
Comparing (13.6) with (13.5), we see that, in addition to the expected first
term on the right-hand side of (13.6), which has the same form as the righthand side of (13.5), there is an extra term in (13.6). By contrast, the covariant
derivative of ψ transforms as (see section 2.4 of volume 1)
D
μ ψ(x) → D
′μ ψ
′ (x) = exp(iqχ(x))D
μ ψ(x)
(13.7)
exactly as in (13.5), with no additional term on the right-hand side. Note
that D
μ has to carry a prime also, since it contains A
μ which transforms to
A
′μ = A
μ
− ∂
μ χ(x) when ψ transforms by (13.5). The property (13.7) ensured
the gauge covariance of wave equations in the U(1) case; the similar property
in the quantum field case meant that a globally U(1)-invariant Lagrangian
could be converted immediately to a locally U(1)-invariant one by replacing
∂
μ by D ˆ μ (section 7.4).
In appendix D of volume 1 we introduced the idea of ‘covariance’ in the
context of coordinate transformations of 3- and 4-vectors. The essential notion
was of something ‘maintaining the same form’, or ‘transforming the same
way’. The transformations being considered here are gauge transformations
rather than coordinate ones; nevertheless it is true that, under them, D
μ ψ
transforms in the same way as ψ, while ∂
μ ψ does not. Thus the term covariant
derivative seems appropriate. In fact, there is a much closer analogy between
the ‘coordinate’ and the ‘gauge’ cases, which we did not present in volume 1,
but give now in appendix N, for the interested reader.