206
7. Quantum Field Theory III
7.4 Introduction of electromagnetic interactions
After all these preliminaries, the job of introducing the first of our gauge
field interactions, namely electromagnetism, into our non-interacting theory
of complex scalar fields, and of Dirac fields, is very easy. From our discussion
in chapter 2, we have a strong indication of how to introduce electromagnetic
interactions into our theories. The ‘gauge principle’ in quantum mechanics
consisted in elevating a global (space–time-independent) U(1) phase invariance
into a local (space–time-dependent) U(1) invariance – the compensating fields
being then identified with the electromagnetic ones. In quantum field theory,
exactly the same principle exists and leads to the form of the electromagnetic
interactions. Indeed, in the field theory formalism we have a true local U(1)
phase (gauge) invariance of the Lagrangian (rather than a gauge covariance
of a wave equation) and we shall be able to exhibit explicitly the symmetry
current, and symmetry operator, associated with the U(1) invariance – and
identify them precisely with the electromagnetic current and charge.
We have seen that for both the complex scalar and the Dirac fields the
free Lagrangian is invariant under U(1) transformations (see (7.22) and (7.48))
which, we once again emphasize, are global. Let us therefore promote these
global invariances into local ones in the way learned in chapter 2 – namely by
invoking the ‘gauge principle’ replacement
∂
μ
→ D ˆ μ = ∂
μ + iqA ˆ μ
(7.123)
for a particle of charge q, this time written in terms of the quantum field A ˆ μ .
In the case of the Dirac Lagrangian
¯
L ˆ D = ψ ˆ (iγ
μ ∂ μ − m)ψ ˆ
(7.124)
we expect to be able to ‘promote’ it to one which is invariant under the local
U(1) phase transformation
1
qχ ˆ(x,t) ˆ
ψ ˆ (x, t) → ψ ˆ ′ (x, t) = e
−i
ψ(x, t)
(7.125)
provided we make the replacement (7.123) and demand that the (quantized)
4-vector potential transforms as (cf (2.15) with the sign change for ˆ
χ)
′μ
χ.
(7.126)
A ˆ μ → A ˆ = A ˆ μ + ∂
μ ˆ
Thus the locally U(1)-invariant Dirac Lagrangian is expected to be
¯
L ˆ D lo a
ψ ˆ (iγ
μ D ˆ μ − m) ˆ
(7.127)
c l =
ψ.
1 Note that the classical field χ(x, t) of (2.34) has become a quantum field χ ˆ(x, t) in
(7.125); the sign change of ˆ
χ compared with χ is conventional in qft.
7. Quantum Field Theory III
7.4 Introduction of electromagnetic interactions
After all these preliminaries, the job of introducing the first of our gauge
field interactions, namely electromagnetism, into our non-interacting theory
of complex scalar fields, and of Dirac fields, is very easy. From our discussion
in chapter 2, we have a strong indication of how to introduce electromagnetic
interactions into our theories. The ‘gauge principle’ in quantum mechanics
consisted in elevating a global (space–time-independent) U(1) phase invariance
into a local (space–time-dependent) U(1) invariance – the compensating fields
being then identified with the electromagnetic ones. In quantum field theory,
exactly the same principle exists and leads to the form of the electromagnetic
interactions. Indeed, in the field theory formalism we have a true local U(1)
phase (gauge) invariance of the Lagrangian (rather than a gauge covariance
of a wave equation) and we shall be able to exhibit explicitly the symmetry
current, and symmetry operator, associated with the U(1) invariance – and
identify them precisely with the electromagnetic current and charge.
We have seen that for both the complex scalar and the Dirac fields the
free Lagrangian is invariant under U(1) transformations (see (7.22) and (7.48))
which, we once again emphasize, are global. Let us therefore promote these
global invariances into local ones in the way learned in chapter 2 – namely by
invoking the ‘gauge principle’ replacement
∂
μ
→ D ˆ μ = ∂
μ + iqA ˆ μ
(7.123)
for a particle of charge q, this time written in terms of the quantum field A ˆ μ .
In the case of the Dirac Lagrangian
¯
L ˆ D = ψ ˆ (iγ
μ ∂ μ − m)ψ ˆ
(7.124)
we expect to be able to ‘promote’ it to one which is invariant under the local
U(1) phase transformation
1
qχ ˆ(x,t) ˆ
ψ ˆ (x, t) → ψ ˆ ′ (x, t) = e
−i
ψ(x, t)
(7.125)
provided we make the replacement (7.123) and demand that the (quantized)
4-vector potential transforms as (cf (2.15) with the sign change for ˆ
χ)
′μ
χ.
(7.126)
A ˆ μ → A ˆ = A ˆ μ + ∂
μ ˆ
Thus the locally U(1)-invariant Dirac Lagrangian is expected to be
¯
L ˆ D lo a
ψ ˆ (iγ
μ D ˆ μ − m) ˆ
(7.127)
c l =
ψ.
1 Note that the classical field χ(x, t) of (2.34) has become a quantum field χ ˆ(x, t) in
(7.125); the sign change of ˆ
χ compared with χ is conventional in qft.