180
6. Quantum Field Theory II: Interacting Scalar Fields
where
T (f ˆ (t 1 )f ˆ (t 2 )) = f ˆ (t 1 )f ˆ (t 2 )
for t 1 > t 2
= f ˆ (t 2 )f ˆ (t 1 )
for t 2 > t 1 .
6.2 Verify equation (6.65).
6.3 Let φ ˆ (x, t) be a real scalar KG field in one space dimension, satisfying
(
)
∂
2
∂
2
2
(❗ x + m
2 )φ ˆ (x, t) ≡
−
+ m φ ˆ (x, t) = 0.
∂t
2
∂x
2
(a) Explain why
T (φ ˆ (x 1 , t 1 )φ ˆ (x 2 , t 2 )) = θ(t 1 − t 2 )φ ˆ (x 1 , t 1 )φ ˆ (x 2 , t 2 )
+ θ(t 2 − t 1 )φ ˆ (x 2 , t 2 )φ ˆ (x 1 , t 1 )
(see equation (E.47) for a definition of the θ-function).
(b) Using equation (E.46), show that

d

θ(x − a) = δ(x − a).
dx
(c) Using the result of (b) with appropriate changes of variable, and
equation (5.118), show that
∂ {T (φ ˆ (x 1 , t 1 )φ ˆ (x 2 , t 2 ))}
∂t 1
= θ(t 1 − t 2 )φ ˆ ˙ (x 1 , t 1 )φ ˆ (x 2 , t 2 ) + θ(t 2 − t 1 )φ ˆ (x 2 , t 2 )φ ˆ ˙ (x 1 , t 1 ).
(d) Using (5.117) and (5.122) show that
∂
2
¨ ˆ
{T (φ ˆ (x 1 , t 1 )φ ˆ (x 2 , t 2 ))} = −iδ(x 1 −x 2 )δ(t 1 −t 2 )+T (φ(x 1 , t 1 )φ ˆ (x 2 , t 2 ))
∂t 1
2
and hence show that
(
)
∂
2
∂
2
2
−
+ m T (φ ˆ (x 1 , t 1 )φ ˆ (x 2 , t 2 )) = −iδ(x 1 −x 2 )δ(t 1 −t 2 ).
∂t 1
2
∂x 1
2
This shows that T (φ ˆ (x 1 , t 1 )φ ˆ (x 2 , t 2 )) is a Green function (see appendix G, equation (G.25) – the i is included here conventionally)
for the KG operator
∂
2
∂
2
2
−
+ m .
∂t 1
2
∂x 1
2
The four-dimensional generalization is immediate.
6.4 Verify (6.90).