155
6.2. Perturbation theory for interacting fields
Ht
U ˆ
U
† ˆ
where U ˆ = e
iH ˆ 0t e
−i ˆ , and it is easy to check that ˆ U
† = ˆ U = I ˆ . So taking
U
−1
equation (6.20) and pre-multiplying by U ˆ and post-multiplying by ˆ
on
both sides, we obtain
[φ ˆ I (x, t), π ˆ I (y, t)] = iδ
3 (x − y)
(6.22)
showing that, in the interacting case, the IP fields φ ˆ I and ˆ
π I obey the free
field commutation relation. Thus in the IP case the interacting fields obey the
same equations of motion and the same commutation relations as the free-field
operators. It follows that the mode expansion (5.155), and the commutation
relations (5.158) for the mode creation and annihilation operators, can be
taken straight over for the IP operators.
We now turn to the states in the IP. To preserve consistency between the
matrix elements in the Schr¨ odinger and interaction pictures (cf the step from
(I.6) to (I.7)) we define the corresponding IP state vector by
|ψ(t)> I = e
iH ˆ 0 t
|ψ(t)>
(6.23)
in terms of the SP state |ψ(t)>. We now use (6.23) to find the equation of
motion of |ψ(t)> I . We have
(
)
d
d
i ˆ
H0 t
i |ψ(t)> I = e
−H ˆ 0 |ψ(t)> + i |ψ(t)>
dt
dt
i ˆ
= e
H0 t
{−H ˆ 0 |ψ(t)> + (H ˆ 0 + H ˆ ′ )|ψ(t)>}
i ˆ
H0 t ˆ
= e
H
′
|ψ(t)>
iH ˆ 0 t ˆ ′ −i ˆ
= e
H e
H0t
|ψ(t)> I
(6.24)
or
d
ˆ ′
i |ψ(t)> I = H I (t)|ψ(t)> I
(6.25)
dt
where
ˆ ′
iH ˆ 0t ˆ ′ −iH ˆ 0 t
H I = e
H e
(6.26)
is the interaction Hamiltonian in the interaction picture. The italicised words
′
are important: they mean that all operators in H ˆ have the (known) free-field
I
′
time dependence, which would not be the case for H ˆ in the HP. Thus, as
mentioned earlier, the states in the IP have a time dependence generated by
the interaction Hamiltonian, and this derivation has shown us that it is, in
fact, the interaction Hamiltonian in the IP which is the appropriate generator
of time change in this picture.
Equation (6.25) is a slightly simplified form of the Tomonaga–Schwinger
equation, which formed the starting point of the approach to QED followed by
Schwinger (Schwinger 1948b, 1949a, b) and independently by Tomonaga and
his group (Tomonaga 1946, Koba, Tati and Tomonaga 1947a, b, Kanesawa
and Tomonaga 1948a, b, Koba and Tomonaga 1948, Koba and Takeda 1948,
1949).
6.2. Perturbation theory for interacting fields
Ht
U ˆ
U
† ˆ
where U ˆ = e
iH ˆ 0t e
−i ˆ , and it is easy to check that ˆ U
† = ˆ U = I ˆ . So taking
U
−1
equation (6.20) and pre-multiplying by U ˆ and post-multiplying by ˆ
on
both sides, we obtain
[φ ˆ I (x, t), π ˆ I (y, t)] = iδ
3 (x − y)
(6.22)
showing that, in the interacting case, the IP fields φ ˆ I and ˆ
π I obey the free
field commutation relation. Thus in the IP case the interacting fields obey the
same equations of motion and the same commutation relations as the free-field
operators. It follows that the mode expansion (5.155), and the commutation
relations (5.158) for the mode creation and annihilation operators, can be
taken straight over for the IP operators.
We now turn to the states in the IP. To preserve consistency between the
matrix elements in the Schr¨ odinger and interaction pictures (cf the step from
(I.6) to (I.7)) we define the corresponding IP state vector by
|ψ(t)> I = e
iH ˆ 0 t
|ψ(t)>
(6.23)
in terms of the SP state |ψ(t)>. We now use (6.23) to find the equation of
motion of |ψ(t)> I . We have
(
)
d
d
i ˆ
H0 t
i |ψ(t)> I = e
−H ˆ 0 |ψ(t)> + i |ψ(t)>
dt
dt
i ˆ
= e
H0 t
{−H ˆ 0 |ψ(t)> + (H ˆ 0 + H ˆ ′ )|ψ(t)>}
i ˆ
H0 t ˆ
= e
H
′
|ψ(t)>
iH ˆ 0 t ˆ ′ −i ˆ
= e
H e
H0t
|ψ(t)> I
(6.24)
or
d
ˆ ′
i |ψ(t)> I = H I (t)|ψ(t)> I
(6.25)
dt
where
ˆ ′
iH ˆ 0t ˆ ′ −iH ˆ 0 t
H I = e
H e
(6.26)
is the interaction Hamiltonian in the interaction picture. The italicised words
′
are important: they mean that all operators in H ˆ have the (known) free-field
I
′
time dependence, which would not be the case for H ˆ in the HP. Thus, as
mentioned earlier, the states in the IP have a time dependence generated by
the interaction Hamiltonian, and this derivation has shown us that it is, in
fact, the interaction Hamiltonian in the IP which is the appropriate generator
of time change in this picture.
Equation (6.25) is a slightly simplified form of the Tomonaga–Schwinger
equation, which formed the starting point of the approach to QED followed by
Schwinger (Schwinger 1948b, 1949a, b) and independently by Tomonaga and
his group (Tomonaga 1946, Koba, Tati and Tomonaga 1947a, b, Kanesawa
and Tomonaga 1948a, b, Koba and Tomonaga 1948, Koba and Takeda 1948,
1949).