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Appendix
Using Lemma 1, Wick’s theorem follows readily by induction with regard to the
number n of operators.
Proof of Wick’s theorem: For n = 2, Wick’s theorem (cf. Eq. 5.16) merely reformulates the definition (5.6) of a contraction:
ˆ
T T T
ˆ
a ˆ
b
= ˆ
N N N
ˆ
a ˆ
b
+ ˆ
a
ˆ
b
Suppose that the theorem holds for n factors and consider a product of n + 1 factors
ˆ
a 1 . . . ˆ
a n ˆ
a n+1 . The last operator ˆ
a n+1 may be chosen such that its time argument is
smaller than those of the other factors, which allows us to write
ˆ
T T T
ˆ
a 1 . . . ˆ
a n
ˆ
a n+1 = ˆ
T T T
ˆ
a 1 . . . ˆ
a n ˆ
a n+1
(A.3.9)
Multiplying both sides of Wick’s Eq. (5.16) on the right by ˆ
a n+1 yields
ˆ
T T T
ˆ
a 1 . . . ˆ
a n
ˆ
a n+1 = ˆ
T T T
ˆ
a 1 . . . ˆ
a n ˆ
a n+1
= ˆ
N N N
ˆ
a 1 . . . ˆ
a n
ˆ
a n+1 + ˆ
N N N
ˆ
a 1 ˆ
a 2 ˆ
a 3 . . .
ˆ
a n+1
+ ˆ
N N N
ˆ
a 1 ˆ
a 2 ˆ
a 3 . . .
ˆ
a n+1 + . . . + ˆ
N N N
ˆ
a 1 ˆ
a 2 ˆ
a 3 ˆ
a 4 . . .
ˆ
a n+1 + . . .
(A.3.10)
Applying Lemma 1 individually to all the terms on the right-hand side, one obviously
reproduces the right-hand side of Wick’s operator identity for a product of n + 1
operators:
ˆ
T T T
ˆ
a 1 . . . ˆ
a n ˆ
a n+1
= ˆ
N N N
ˆ
a 1 . . . ˆ
a n ˆ
a n+1
+ ˆ
N N N
sum over all possible pairs of contractions
(A.3.11)
Finally, the initial restriction with regard to the time argument of ˆ
a n+1 can be lifted,
since the factors within the ˆ
T T T and ˆ
N N N products can be changed at will, resulting only
in a common phase on both sides of Eq. (A.3.11).