62
5 Introducing Diagrams
A systematic way to evaluate these quantities is provided by Wick’s theorem [1] to be
addressed below. The treatment based on Wick’s theorem can readily be translated
into the concept of Feynman diagrams, as will be discussed in the next section.
The first observation to be made is that the fermion operators c
†
p , c p can be classified according to their respective action on | 0 :
c
†
p | 0 =
|
N +1
p
for n p = 0
0
f o rn p = 1
(5.2)
c p | 0 =
|
N −1
p
for n p = 1
0
f o rn p = 0
(5.3)
An operator is referred to as physical if the outcome is an (N ±1)-state (first case in
Eqs. (5.2) and (5.3), respectively) and unphysical if the null vector results (second
case in Eqs. (5.2) and (5.3), respectively). This allows us to divide the fermion
operators into two classes comprising exclusively physical and unphysical operators,
respectively:
ˆ
v s
≡
c p , c
†
q ; n p = 1, n q = 0
ˆ
u r
≡
c p , c
†
q ; n p = 0, n q = 1
In the following, we shall use the notations ˆ
v i and ˆ
u i for physical and unphysical
fermion operators, respectively; general fermion operators will be denoted by ˆ
a
or ˆ
b. It should be noted that the physical fermion operators anticommute among
themselves, and so do the unphysical ones:
{ ˆ
v i , ˆ
v j } = 0, { ˆ
u i , ˆ
u j } = 0
( 5 . 4 )
Consider a product ˆ
a i (t i ) ˆ
a j (t j ) ˆ
a k (t k ) . . . of time-dependent fermion operators
in the interaction picture; for brevity, the time arguments will be skipped in the
following. The time-ordering operator, first introduced in Sect. 3.1, generates the
time-ordered product,
ˆ
T T T
ˆ
a i ˆ
a j ˆ
a k . . .
≡ (−1)
P
ˆ
a P(i) ˆ
a P( j) . . .
where P is a permutation of the factors in the product such that operators with larger
time arguments are placed to the left of those with smaller time arguments; (−1)
P is
the sign (or parity) of the permutation. In the case of equal (or absent) time arguments,
the definition can be generalized to the effect that creation operators c
† are placed to
the left of the destruction operators c. Another reordering of the original product is
the normal-ordered product,
ˆ
N N N
ˆ
a i ˆ
a j ˆ
a k . . .
≡ (−1)
P
ˆ
a P (i) ˆ
a P ( j) . . .
5 Introducing Diagrams
A systematic way to evaluate these quantities is provided by Wick’s theorem [1] to be
addressed below. The treatment based on Wick’s theorem can readily be translated
into the concept of Feynman diagrams, as will be discussed in the next section.
The first observation to be made is that the fermion operators c
†
p , c p can be classified according to their respective action on | 0 :
c
†
p | 0 =
|
N +1
p
for n p = 0
0
f o rn p = 1
(5.2)
c p | 0 =
|
N −1
p
for n p = 1
0
f o rn p = 0
(5.3)
An operator is referred to as physical if the outcome is an (N ±1)-state (first case in
Eqs. (5.2) and (5.3), respectively) and unphysical if the null vector results (second
case in Eqs. (5.2) and (5.3), respectively). This allows us to divide the fermion
operators into two classes comprising exclusively physical and unphysical operators,
respectively:
ˆ
v s
≡
c p , c
†
q ; n p = 1, n q = 0
ˆ
u r
≡
c p , c
†
q ; n p = 0, n q = 1
In the following, we shall use the notations ˆ
v i and ˆ
u i for physical and unphysical
fermion operators, respectively; general fermion operators will be denoted by ˆ
a
or ˆ
b. It should be noted that the physical fermion operators anticommute among
themselves, and so do the unphysical ones:
{ ˆ
v i , ˆ
v j } = 0, { ˆ
u i , ˆ
u j } = 0
( 5 . 4 )
Consider a product ˆ
a i (t i ) ˆ
a j (t j ) ˆ
a k (t k ) . . . of time-dependent fermion operators
in the interaction picture; for brevity, the time arguments will be skipped in the
following. The time-ordering operator, first introduced in Sect. 3.1, generates the
time-ordered product,
ˆ
T T T
ˆ
a i ˆ
a j ˆ
a k . . .
≡ (−1)
P
ˆ
a P(i) ˆ
a P( j) . . .
where P is a permutation of the factors in the product such that operators with larger
time arguments are placed to the left of those with smaller time arguments; (−1)
P is
the sign (or parity) of the permutation. In the case of equal (or absent) time arguments,
the definition can be generalized to the effect that creation operators c
† are placed to
the left of the destruction operators c. Another reordering of the original product is
the normal-ordered product,
ˆ
N N N
ˆ
a i ˆ
a j ˆ
a k . . .
≡ (−1)
P
ˆ
a P (i) ˆ
a P ( j) . . .