46
2. Electromagnetism as a Gauge Theory
What are these transformations? Clearly A can be changed by
A → A
′ = A + ∇χ
(2.12)
where χ is an arbitrary function, with no change in B since ∇ × ∇f = 0, for
any scalar function f . To preserve E, V must then change simultaneously by
∂χ
′
V → V = V −
.
(2.13)
∂t
These transformations can be combined into a single compact equation by
introducing the 4-vector potential
1 :
A
μ = (V, A)
(2.14)
and noting (from problem 2.1) that the differential operators (∂/∂t, −∇) form
the components of a 4-vector operator ∂
μ . A gauge transformation is then
specified by
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(2.15)
The Maxwell equations can also be written in a manifestly Lorentz covariant
form (see appendix D) using the 4-current j
μ given by
em
j
μ = (ρ em , j )
(2.16)
em
em
in terms of which the continuity equation takes the form (problem 2.1):
∂ μ j
μ = 0.
(2.17)
em
The Maxwell equations (2.1) and (2.8) then become (problem 2.2):
∂ μ F
μν
j
ν
=
(2.18)
em
where we have defined the field strength tensor:
F
μν
≡ ∂
μ A
ν
− ∂
ν A
μ .
(2.19)
Under the gauge transformation
A
μ
→ A
′μ
A
μ
− ∂
μ χ
=
(2.20)
F
μν remains unchanged:
′ μν
F
μν
→ F
F
μν
=
(2.21)
so F
μν is gauge invariant and so, therefore, are the Maxwell equations in
1 See appendix D for relativistic notation and for an explanation of the very important
concept of covariance, which we are about to invoke in the context of Lorentz transformations, and will use again in the next section in the context of gauge transformations; we
shall also use it in other contexts in later chapters.
2. Electromagnetism as a Gauge Theory
What are these transformations? Clearly A can be changed by
A → A
′ = A + ∇χ
(2.12)
where χ is an arbitrary function, with no change in B since ∇ × ∇f = 0, for
any scalar function f . To preserve E, V must then change simultaneously by
∂χ
′
V → V = V −
.
(2.13)
∂t
These transformations can be combined into a single compact equation by
introducing the 4-vector potential
1 :
A
μ = (V, A)
(2.14)
and noting (from problem 2.1) that the differential operators (∂/∂t, −∇) form
the components of a 4-vector operator ∂
μ . A gauge transformation is then
specified by
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(2.15)
The Maxwell equations can also be written in a manifestly Lorentz covariant
form (see appendix D) using the 4-current j
μ given by
em
j
μ = (ρ em , j )
(2.16)
em
em
in terms of which the continuity equation takes the form (problem 2.1):
∂ μ j
μ = 0.
(2.17)
em
The Maxwell equations (2.1) and (2.8) then become (problem 2.2):
∂ μ F
μν
j
ν
=
(2.18)
em
where we have defined the field strength tensor:
F
μν
≡ ∂
μ A
ν
− ∂
ν A
μ .
(2.19)
Under the gauge transformation
A
μ
→ A
′μ
A
μ
− ∂
μ χ
=
(2.20)
F
μν remains unchanged:
′ μν
F
μν
→ F
F
μν
=
(2.21)
so F
μν is gauge invariant and so, therefore, are the Maxwell equations in
1 See appendix D for relativistic notation and for an explanation of the very important
concept of covariance, which we are about to invoke in the context of Lorentz transformations, and will use again in the next section in the context of gauge transformations; we
shall also use it in other contexts in later chapters.