157
6.2. Perturbation theory for interacting fields
This is an integral equation in which the unknown |ψ(t)> I is buried under
the integral on the right-hand side, rather similar to the one we encounter in
non-relativistic scattering theory (equation (H.12) of appendix H). As in that
′
case, we solve it iteratively. If H ˆ is neglected altogether, then the solution is
I
(0)
|ψ(t)> = |i>.
(6.33)
I
′
To get the first order in H ˆ correction to this, insert (6.33) in place of |ψ(t
′ )> I
I
on the right-hand side of (6.32) to obtain
∫ t
(1)
′
|ψ(t)> = |i> +
(−iH ˆ
I (t 1 ))dt 1 |i>
(6.34)
I
−∞
recalling that |i> is a constant state vector. Putting this back into (6.32) yields
′
|ψ(t)> correct to second order in H ˆ
I :
(
∫ t
(2)
′
|ψ(t)>
=
1 +
(−iH ˆ
I (t 1 )) dt 1
I
−∞
∫
∫
)
t
t1
′
′
+
dt 1
dt 2 (−iH ˆ
I (t 1 ))(−iH ˆ
I (t 2 )) |i> (6.35)
−∞
−∞
which is as far as we intend to go. Letting t → ∞ then gives us our perturbative
series for the S ˆ -operator :
∫
∫
∫
∞
∞
t1
′
′
′
S ˆ = 1 +
(−iH ˆ
I (t 1 )) dt 1 +
dt 1
dt 2 (−iH ˆ
I (t 1 ))(−iH ˆ
I (t 2 )) + · · ·
−∞
−∞
−∞
(6.36)
with the dots indicating the higher-order terms, which are in fact summarized
by the full formula
∞
∫
∫
∫
∑
∞
t1
tn−1
ˆ
ˆ ′
′
ˆ ′
S =
(−i)
n
dt 1
dt 2 · · ·
dt n H I (t 1 )H ˆ
I (t 2 ) . . . H I (t n ). (6.37)
−∞
−∞
−∞
n=0
We could immediately start getting to work with (6.37), but there is one
more useful technical adjustment to make. Remembering that
∫
′
′
H ˆ
I (t) = H ˆ
I (x, t) d
3
x
(6.38)
we can write the second term of (6.36) as
∫ ∫
′
′
d
4 x 1 d
4 x 2 (−iH ˆ
I (x 1 ))(−iH ˆ
I (x 2 ))
(6.39)
t1 >t2
which looks much more symmetrical in x − t. However, there is still an awkward asymmetry between the x-integrals and the t-integrals because of the
t 1 > t 2 condition. The t-integrals can be converted to run from −∞ to ∞
6.2. Perturbation theory for interacting fields
This is an integral equation in which the unknown |ψ(t)> I is buried under
the integral on the right-hand side, rather similar to the one we encounter in
non-relativistic scattering theory (equation (H.12) of appendix H). As in that
′
case, we solve it iteratively. If H ˆ is neglected altogether, then the solution is
I
(0)
|ψ(t)> = |i>.
(6.33)
I
′
To get the first order in H ˆ correction to this, insert (6.33) in place of |ψ(t
′ )> I
I
on the right-hand side of (6.32) to obtain
∫ t
(1)
′
|ψ(t)> = |i> +
(−iH ˆ
I (t 1 ))dt 1 |i>
(6.34)
I
−∞
recalling that |i> is a constant state vector. Putting this back into (6.32) yields
′
|ψ(t)> correct to second order in H ˆ
I :
(
∫ t
(2)
′
|ψ(t)>
=
1 +
(−iH ˆ
I (t 1 )) dt 1
I
−∞
∫
∫
)
t
t1
′
′
+
dt 1
dt 2 (−iH ˆ
I (t 1 ))(−iH ˆ
I (t 2 )) |i> (6.35)
−∞
−∞
which is as far as we intend to go. Letting t → ∞ then gives us our perturbative
series for the S ˆ -operator :
∫
∫
∫
∞
∞
t1
′
′
′
S ˆ = 1 +
(−iH ˆ
I (t 1 )) dt 1 +
dt 1
dt 2 (−iH ˆ
I (t 1 ))(−iH ˆ
I (t 2 )) + · · ·
−∞
−∞
−∞
(6.36)
with the dots indicating the higher-order terms, which are in fact summarized
by the full formula
∞
∫
∫
∫
∑
∞
t1
tn−1
ˆ
ˆ ′
′
ˆ ′
S =
(−i)
n
dt 1
dt 2 · · ·
dt n H I (t 1 )H ˆ
I (t 2 ) . . . H I (t n ). (6.37)
−∞
−∞
−∞
n=0
We could immediately start getting to work with (6.37), but there is one
more useful technical adjustment to make. Remembering that
∫
′
′
H ˆ
I (t) = H ˆ
I (x, t) d
3
x
(6.38)
we can write the second term of (6.36) as
∫ ∫
′
′
d
4 x 1 d
4 x 2 (−iH ˆ
I (x 1 ))(−iH ˆ
I (x 2 ))
(6.39)
t1 >t2
which looks much more symmetrical in x − t. However, there is still an awkward asymmetry between the x-integrals and the t-integrals because of the
t 1 > t 2 condition. The t-integrals can be converted to run from −∞ to ∞