207
7.4. Introduction of electromagnetic interactions
The invariance of (7.127) under (7.125) is easy to check, using the crucial
property (2.43), which clearly carries over to the quantum field case:
−iq ˆ
D ˆ ′ ψ ˆ ′ = e
χ (D ˆ μ ψ ˆ ).
(7.128)
μ
Equation (7.128) implies at once that
−iq ˆ
(iγ
μ D ˆ
μ
′
− ψ
′ = e
D μ − ψ,
m) ˆ
χ (iγ
μ ˆ m) ˆ
(7.129)
while taking the conjugate of (7.125) yields
′
¯ ˆ
¯ ˆ iqχ ˆ
ψ = ψe .
(7.130)
Thus we have
′
¯ ˆ
D
′
ψ
′
¯ ˆ iqχ ˆ −iq ˆ
ψ (iγ
μ ˆ
μ − m) ˆ = ψe e
χ (iγ
μ D ˆ μ − m)ψ ˆ
(7.131)
¯
= ψ ˆ (iγ
μ D ˆ μ − m)ψ ˆ
(7.132)
and the invariance is proved.
The Lagrangian has therefore gained an interaction term
L ˆ D → L ˆ D local = L ˆ D + L ˆ int
(7.133)
where
¯
L ˆ int = −q ˆ ψA ˆ μ .
ψγ
μ ˆ
(7.134)
Since the addition of L int has not changed the canonical momenta, the Hamilˆ
H
′
tonian then becomes H ˆ = H D + ˆ , where
D
¯
ˆ
H ˆ ′
ˆ
ψ
† ˆ
ˆ
ˆ
= −L ˆ int = qψγ
μ ψ ˆ A ˆ μ = q ψA ˆ 0 − qψ
†
αψ ˆ · A
(7.135)
D
which is the field theory analogue of the potential in (3.102). It has the
expected form ‘ρA 0 − j·A’ if we identify the electromagnetic charge density
ψ ˆ† ˆ
operator with q ψ (the charge times the number density operator) and the
electromagnetic current density operator with qψ ˆ † αψ ˆ . The electromagnetic
4-vector current operator ˆ j
μ is thus identified as
em
¯
ˆ j
μ = q ˆ ψ,
ψγ
μ ˆ
(7.136)
em
which is gauge invariant and a Lorentz 4-vector. The Lagrangian (7.134) is
manifestly Lorentz invariant.
μ
We now note that ˆ j
μ is just q times the symmetry current N ˆ of secem
ψ
tion 7.2 (see equation (7.50)). Conservation of ˆ j
μ would follow from global
em
U(1) invariance alone (i.e. ˆ
χ a constant in equation (7.125)); but many Lagrangians, including interactions, could be constructed obeying this global
U(1) invariance. The force of the local U(1) invariance requirement is that it
7.4. Introduction of electromagnetic interactions
The invariance of (7.127) under (7.125) is easy to check, using the crucial
property (2.43), which clearly carries over to the quantum field case:
−iq ˆ
D ˆ ′ ψ ˆ ′ = e
χ (D ˆ μ ψ ˆ ).
(7.128)
μ
Equation (7.128) implies at once that
−iq ˆ
(iγ
μ D ˆ
μ
′
− ψ
′ = e
D μ − ψ,
m) ˆ
χ (iγ
μ ˆ m) ˆ
(7.129)
while taking the conjugate of (7.125) yields
′
¯ ˆ
¯ ˆ iqχ ˆ
ψ = ψe .
(7.130)
Thus we have
′
¯ ˆ
D
′
ψ
′
¯ ˆ iqχ ˆ −iq ˆ
ψ (iγ
μ ˆ
μ − m) ˆ = ψe e
χ (iγ
μ D ˆ μ − m)ψ ˆ
(7.131)
¯
= ψ ˆ (iγ
μ D ˆ μ − m)ψ ˆ
(7.132)
and the invariance is proved.
The Lagrangian has therefore gained an interaction term
L ˆ D → L ˆ D local = L ˆ D + L ˆ int
(7.133)
where
¯
L ˆ int = −q ˆ ψA ˆ μ .
ψγ
μ ˆ
(7.134)
Since the addition of L int has not changed the canonical momenta, the Hamilˆ
H
′
tonian then becomes H ˆ = H D + ˆ , where
D
¯
ˆ
H ˆ ′
ˆ
ψ
† ˆ
ˆ
ˆ
= −L ˆ int = qψγ
μ ψ ˆ A ˆ μ = q ψA ˆ 0 − qψ
†
αψ ˆ · A
(7.135)
D
which is the field theory analogue of the potential in (3.102). It has the
expected form ‘ρA 0 − j·A’ if we identify the electromagnetic charge density
ψ ˆ† ˆ
operator with q ψ (the charge times the number density operator) and the
electromagnetic current density operator with qψ ˆ † αψ ˆ . The electromagnetic
4-vector current operator ˆ j
μ is thus identified as
em
¯
ˆ j
μ = q ˆ ψ,
ψγ
μ ˆ
(7.136)
em
which is gauge invariant and a Lorentz 4-vector. The Lagrangian (7.134) is
manifestly Lorentz invariant.
μ
We now note that ˆ j
μ is just q times the symmetry current N ˆ of secem
ψ
tion 7.2 (see equation (7.50)). Conservation of ˆ j
μ would follow from global
em
U(1) invariance alone (i.e. ˆ
χ a constant in equation (7.125)); but many Lagrangians, including interactions, could be constructed obeying this global
U(1) invariance. The force of the local U(1) invariance requirement is that it