154
6. Quantum Field Theory II: Interacting Scalar Fields
has been singled out. In the end, physical quantities should come out correct,
but it is much more convenient to have everything looking nice and consistent
with relativity as we go along. This is one of the reasons for choosing to
work in yet a third ‘picture’, an ingenious kind of half-way-house between
the other two, called the ‘interaction picture’ (IP). We shall see other good
reasons shortly.
In the HP, all the time dependence is carried by the operators and none by
the state, while in the SP it is exactly the other way around. In the IP, both
states and operators are time-dependent but in a way that is well adapted
to perturbation theory, especially in quantum field theory. The operators
have a time dependence generated by the free Hamiltonian H ˆ 0 , say, and so a
‘free-particle’ mode expansion like (5.155) survives intact (here H ˆ 0 = H ˆ KG ).
The states have a time dependence generated by the interaction H ˆ ′ . Thus as
ˆ ′
H → 0 we return to the free-particle HP.
The way this works formally is as follows. In terms of the time-independent
SP operator A ˆ (cf appendix I), we define the corresponding IP operator A ˆ I (t)
by
iH ˆ 0 t ˆ −iH ˆ 0 t
ˆ
A I (t) = e
Ae
.
(6.17)
This is just like the definition of the HP operator A ˆ (t) in appendix I, except
that H ˆ 0 appears instead of the full H ˆ . It follows that the time dependence of
A ˆ I (t) is given by (I.8) with H ˆ → H ˆ 0 :
dA ˆ I (t) = −i[A ˆ I (t), H ˆ 0 ].
(6.18)
dt
Equation (6.18) can also, of course, be derived by carefully differentiating
(6.17). Thus – as mentioned already – the time dependence of A ˆ I (t) is generated by the free part of the Hamiltonian, by construction.
As applied to our model theory (6.12), then, our field φ ˆ will now be specified as being in the IP, φ ˆ I (x, t). What about the field canonically conjugate
to φ ˆ I (t), in the case when the interaction is included? In the HP, as long as
the interaction does not contain time derivatives, as is the case here, the field
canonically conjugate to the interacting field remains the same as the free-field
case:
∂ ˆ
∂ ˆ
L
L KG
˙
ˆ
π ˆ(x, t) =
=
= φ(x, t)
(6.19)
˙
˙
ˆ
ˆ
∂φ(x, t)
∂φ(x, t)
so that we continue to adopt the equal-time commutation relation
[φ ˆ (x, t), π ˆ(y, t)] = iδ
3 (x − y)
(6.20)
for the Heisenberg fields. But the IP fields are related to the HP fields by a
unitary transformation U ˆ , as we can see by combining (6.17) with (I.7):
i ˆ
−i ˆ
−i ˆ
Ht ˆ
Ht
ˆ
H0t
H0t
A I (t) = e
e
A(t)e
i ˆ e
= U ˆ A ˆ (t)U ˆ −1
(6.21)
6. Quantum Field Theory II: Interacting Scalar Fields
has been singled out. In the end, physical quantities should come out correct,
but it is much more convenient to have everything looking nice and consistent
with relativity as we go along. This is one of the reasons for choosing to
work in yet a third ‘picture’, an ingenious kind of half-way-house between
the other two, called the ‘interaction picture’ (IP). We shall see other good
reasons shortly.
In the HP, all the time dependence is carried by the operators and none by
the state, while in the SP it is exactly the other way around. In the IP, both
states and operators are time-dependent but in a way that is well adapted
to perturbation theory, especially in quantum field theory. The operators
have a time dependence generated by the free Hamiltonian H ˆ 0 , say, and so a
‘free-particle’ mode expansion like (5.155) survives intact (here H ˆ 0 = H ˆ KG ).
The states have a time dependence generated by the interaction H ˆ ′ . Thus as
ˆ ′
H → 0 we return to the free-particle HP.
The way this works formally is as follows. In terms of the time-independent
SP operator A ˆ (cf appendix I), we define the corresponding IP operator A ˆ I (t)
by
iH ˆ 0 t ˆ −iH ˆ 0 t
ˆ
A I (t) = e
Ae
.
(6.17)
This is just like the definition of the HP operator A ˆ (t) in appendix I, except
that H ˆ 0 appears instead of the full H ˆ . It follows that the time dependence of
A ˆ I (t) is given by (I.8) with H ˆ → H ˆ 0 :
dA ˆ I (t) = −i[A ˆ I (t), H ˆ 0 ].
(6.18)
dt
Equation (6.18) can also, of course, be derived by carefully differentiating
(6.17). Thus – as mentioned already – the time dependence of A ˆ I (t) is generated by the free part of the Hamiltonian, by construction.
As applied to our model theory (6.12), then, our field φ ˆ will now be specified as being in the IP, φ ˆ I (x, t). What about the field canonically conjugate
to φ ˆ I (t), in the case when the interaction is included? In the HP, as long as
the interaction does not contain time derivatives, as is the case here, the field
canonically conjugate to the interacting field remains the same as the free-field
case:
∂ ˆ
∂ ˆ
L
L KG
˙
ˆ
π ˆ(x, t) =
=
= φ(x, t)
(6.19)
˙
˙
ˆ
ˆ
∂φ(x, t)
∂φ(x, t)
so that we continue to adopt the equal-time commutation relation
[φ ˆ (x, t), π ˆ(y, t)] = iδ
3 (x − y)
(6.20)
for the Heisenberg fields. But the IP fields are related to the HP fields by a
unitary transformation U ˆ , as we can see by combining (6.17) with (I.7):
i ˆ
−i ˆ
−i ˆ
Ht ˆ
Ht
ˆ
H0t
H0t
A I (t) = e
e
A(t)e
i ˆ e
= U ˆ A ˆ (t)U ˆ −1
(6.21)