52
2. Electromagnetism as a Gauge Theory
the case. We shall introduce the spin-0 and spin1 relativistic equations in
2
chapter 3. For the present we note that (2.30) can be written in manifestly
Lorentz covariant form as
D
μ
≡ ∂
μ + iqA
μ
(2.42)
in terms of which (2.37) and (2.38) become
−iD
′μ ψ
′ = exp(iqχ) · (−iD
μ ψ).
(2.43)
It follows that any equation involving the operator ∂
μ can be made gauge
invariant under the combined transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ
ψ →
= exp(iqχ)ψ
ψ
′
if ∂
μ is replaced by D
μ . In fact, we seem to have a very simple prescription
for obtaining the wave equation for a particle in the presence of an electromagnetic field from the corresponding free particle wave equation: make the
replacement
∂
μ
→ D
μ
≡ ∂
μ + iqA
μ .
(2.44)
In the following section this will be seen to be the basis of the so-called ‘gauge
principle’ whereby, in accordance with the idea advanced in the previous sections, the form of the interaction is determined by the insistence on (local)
gauge invariance.
One final remark: this new kind of derivative
D
μ
≡ ∂
μ + iqA
μ
(2.45)
turns out to be of fundamental importance – it will be the operator which
generalizes from the (Abelian) phase symmetry of QED (see comment (iii)
of section 2.6) to the (non-Abelian) phase symmetry of our weak and strong
interaction theories. It is called the ‘gauge covariant derivative’, the term
being usually shortened to ‘covariant derivative’ in the present context. The
geometrical significance of this term will be explained in volume 2.
2.5 The argument reversed: the gauge principle
In the preceding section, we took it as known that the Schr¨ odinger equation,
for example, for a charged particle in an electromagnetic field, has the form
[
]
1 (−i∇ − qA)
2 + qV ψ = i∂ψ/∂t.
(2.46)
2m
2. Electromagnetism as a Gauge Theory
the case. We shall introduce the spin-0 and spin1 relativistic equations in
2
chapter 3. For the present we note that (2.30) can be written in manifestly
Lorentz covariant form as
D
μ
≡ ∂
μ + iqA
μ
(2.42)
in terms of which (2.37) and (2.38) become
−iD
′μ ψ
′ = exp(iqχ) · (−iD
μ ψ).
(2.43)
It follows that any equation involving the operator ∂
μ can be made gauge
invariant under the combined transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ
ψ →
= exp(iqχ)ψ
ψ
′
if ∂
μ is replaced by D
μ . In fact, we seem to have a very simple prescription
for obtaining the wave equation for a particle in the presence of an electromagnetic field from the corresponding free particle wave equation: make the
replacement
∂
μ
→ D
μ
≡ ∂
μ + iqA
μ .
(2.44)
In the following section this will be seen to be the basis of the so-called ‘gauge
principle’ whereby, in accordance with the idea advanced in the previous sections, the form of the interaction is determined by the insistence on (local)
gauge invariance.
One final remark: this new kind of derivative
D
μ
≡ ∂
μ + iqA
μ
(2.45)
turns out to be of fundamental importance – it will be the operator which
generalizes from the (Abelian) phase symmetry of QED (see comment (iii)
of section 2.6) to the (non-Abelian) phase symmetry of our weak and strong
interaction theories. It is called the ‘gauge covariant derivative’, the term
being usually shortened to ‘covariant derivative’ in the present context. The
geometrical significance of this term will be explained in volume 2.
2.5 The argument reversed: the gauge principle
In the preceding section, we took it as known that the Schr¨ odinger equation,
for example, for a charged particle in an electromagnetic field, has the form
[
]
1 (−i∇ − qA)
2 + qV ψ = i∂ψ/∂t.
(2.46)
2m