158
6. Quantum Field Theory II: Interacting Scalar Fields
without constraint, like the x ones, by a clever trick. Note that the ordering of
′
the operators H ˆ is significant (since they will contain non-commuting bits),
I
and that it is actually given by the order of their time arguments, ‘earlier’
operators appearing to the right of ‘later’ ones. This feature must be preserved, obviously, when we let the t-integrals run over the full infinite domain.
We can arrange for this by introducing the time-ordering symbol T , which is
defined by
′
′
ˆ ′
′
T (H ˆ
I (x 1 )H ˆ
I (x 2 )) = H I (x 1 )H ˆ
I (x 2 )
for t 1 > t 2
= H ˆ
I
′ (x 2 )H ˆ
I
′ (x 1 )
for t 1 < t 2
(6.40)
and similarly for more products, and for arbitrary operators. Then (see problem 6.1) (6.39) can be written as
∫ ∫
1
′
′
d
4 x 1 d
4 x 2 T [(−iH ˆ
I (x 1 ))(−iH ˆ
I (x 2 ))]
(6.41)
2
where the integrals are now unrestricted. Applying a similar analysis to the
general term gives us the Dyson expansion of the S ˆ operator :
∞
∫
∫
∑ (−i)
n
ˆ
′
′
ˆ ′
S =
. . . d
4 x 1 d
4 x 2 . . . d
4 x n T {H ˆ
I (x 1 )H ˆ
I (x 2 ) · · · H I (x n )}.
n!
n=0
(6.42)
This fundamental formula provides the bridge leading from the Tomonaga–
Schwinger equation (6.25) to the Feynman amplitudes (Feynman 1949a, b),
as we shall see in detail in section 7.3.2 for the ‘ABC’ case.
6.3 Applications to the ‘ABC’ theory
As previously explained, the simple self-interacting φ ˆ3 theory is not respectable.
Following Griffiths (2008) we shall instead apply the foregoing covariant perturbation theory to a hypothetical world consisting of three distinct types of
scalar particles A, B and C, with masses m A , m B , m C . Each is described by
a real scalar field which, if free, would obey the appropriate KG equation; the
ˆ ˆ ˆ
interaction term is gφ A φ B φ C . We shall from now on omit the IP subscript ‘I’,
since all operators are taken to be in the IP. Thus the Hamiltonian is
ˆ
ˆ
′
H = H 0 + H ˆ
(6.43)
where
∫
∑
ˆ
1
2
2 φ ˆ 2
H 0 =
[ ˆ
π i + (∇φ ˆ i )
2 + m i ] d
3
x
(6.44)
2
i
i=A,B,C
6. Quantum Field Theory II: Interacting Scalar Fields
without constraint, like the x ones, by a clever trick. Note that the ordering of
′
the operators H ˆ is significant (since they will contain non-commuting bits),
I
and that it is actually given by the order of their time arguments, ‘earlier’
operators appearing to the right of ‘later’ ones. This feature must be preserved, obviously, when we let the t-integrals run over the full infinite domain.
We can arrange for this by introducing the time-ordering symbol T , which is
defined by
′
′
ˆ ′
′
T (H ˆ
I (x 1 )H ˆ
I (x 2 )) = H I (x 1 )H ˆ
I (x 2 )
for t 1 > t 2
= H ˆ
I
′ (x 2 )H ˆ
I
′ (x 1 )
for t 1 < t 2
(6.40)
and similarly for more products, and for arbitrary operators. Then (see problem 6.1) (6.39) can be written as
∫ ∫
1
′
′
d
4 x 1 d
4 x 2 T [(−iH ˆ
I (x 1 ))(−iH ˆ
I (x 2 ))]
(6.41)
2
where the integrals are now unrestricted. Applying a similar analysis to the
general term gives us the Dyson expansion of the S ˆ operator :
∞
∫
∫
∑ (−i)
n
ˆ
′
′
ˆ ′
S =
. . . d
4 x 1 d
4 x 2 . . . d
4 x n T {H ˆ
I (x 1 )H ˆ
I (x 2 ) · · · H I (x n )}.
n!
n=0
(6.42)
This fundamental formula provides the bridge leading from the Tomonaga–
Schwinger equation (6.25) to the Feynman amplitudes (Feynman 1949a, b),
as we shall see in detail in section 7.3.2 for the ‘ABC’ case.
6.3 Applications to the ‘ABC’ theory
As previously explained, the simple self-interacting φ ˆ3 theory is not respectable.
Following Griffiths (2008) we shall instead apply the foregoing covariant perturbation theory to a hypothetical world consisting of three distinct types of
scalar particles A, B and C, with masses m A , m B , m C . Each is described by
a real scalar field which, if free, would obey the appropriate KG equation; the
ˆ ˆ ˆ
interaction term is gφ A φ B φ C . We shall from now on omit the IP subscript ‘I’,
since all operators are taken to be in the IP. Thus the Hamiltonian is
ˆ
ˆ
′
H = H 0 + H ˆ
(6.43)
where
∫
∑
ˆ
1
2
2 φ ˆ 2
H 0 =
[ ˆ
π i + (∇φ ˆ i )
2 + m i ] d
3
x
(6.44)
2
i
i=A,B,C