5.1 Wick’s Theorem
65
In a somewhat symbolic notation, one may write
ˆ
T T T
ˆ
a i ˆ
a j ˆ
a k ˆ
a l . . . ˆ
a r ˆ
a s ˆ
a t
= ˆ
N N N
ˆ
a i ˆ
a j ˆ
a k ˆ
a l . . . ˆ
a r ˆ
a s ˆ
a t
+ ˆ
N N N
sum over all possible pairs of contractions
Wick’s reformulation of a time-ordered product is obtained by moving physical
operators (PO) in the time-ordered product successively to the left. This generates
additional terms whenever a PO does not anticommute with an operator on its left.
A proof of the theorem is given in Appendix A.3.
Wick’s theorem establishes an operator identity. While the right-hand side of this
identity looks rather complicated, the actual benefit of Wick’s theorem becomes
apparent when ground-state expectation values of time-ordered products are to be
evaluated. According to the property of the ˆ
N N N products, only the fully contracted
terms contribute to the expectation value:
0 | ˆ
T T T
ˆ
a i ˆ
a j ˆ
a k ˆ
a l . . . ˆ
a r ˆ
a s ˆ
a t
| 0 = ˆ
N N N
ˆ
a i ˆ
a j ˆ
a k . . . ˆ
a r ˆ
a s ˆ
a t
+ . . .
(5.17)
As indicated on the right-hand side, there is a contribution for each full contraction
scheme. In the ensuing section, we will learn how the distinct contraction schemes
can be expressed in the form of diagrams.
5.2 Zeroth- and First-Order Feynman Diagrams
The perturbation expansion (4.58) of the electron propagator is of the form
i G pq (t, t
) = lim
→0
i ˜
G pq (t, t
)
0 | ˆ
U (∞, −∞)| 0
(5.18)
where the numerator i ˜
G pq (t, t
) is given by
i ˜
G pq (t, t
) =
∞
n=0
(−i)
n
n!
∞
−∞
dt 1 e
−|t 1 |
. . .
∞
−∞
dt n e
−|t n |
0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n )c p (t)c
†
q (t
)
| 0
(5.19)
Using Wick’s result (5.17) for the ground-state expectation values of time-ordered
operator products, the respective nth-order terms in the perturbation expansions, both
for the numerator and denominator, can be determined analytically by generating and
evaluating all possible (full) contraction schemes. As a by far superior procedure,