138
6 Appendix
[A c ] s1,...sn ;r1,...,rm (
p 1 , . . . ,
p n ; ;
q 1 , . . . ,
q m ) := A c ( q 1 , r 1 , . . . ,
q m , r m → →
p 1 , s 1 , . . . ,
p n , s n ) ,
(6.3.4)
[A
†
c ] s1,...sn ;r1,...,rm (
p 1 , . . . ,
p n ; ;
q 1 , . . . ,
q m ) := A
∗
c (
p 1 , s 1 , . . . ,
p n , s n → →
q 1 , r 1 , . . . ,
q m , r m ) .
(6.3.5)
The latter is natural, because the “in” states appear as kets in A c , and therefore on
its “right side”, whereas the “out” states appear as bras on the left. The situation is
reversed for A
∗
c because complex conjugation interchanges kets and bras.
Considering the sets of indices on each side of the semi-column as a generalized
index, we have that f and A c appear as matrices in these generalized indices. We
can therefore define the matrix product notation
[A c (
p 1 , . . . ,
p n ; ;
q 1 , . . . ,
q m ) f (x,
q 1 , . . . ,
q m )] s 1 ,...s n ;r
1 ,...r
m
:= [A c ] s 1 ,...,s n ;r 1 ,...r m (
p 1 , . . . ,
p n ; ;
q 1 , . . . ,
q m ) f r 1 ,...,r m ;r
1 ,...,r
m
(x,
q 1 , . . . ,
q m ) .
(6.3.6)
With these conventions the creation and annihilation terms in Eqs. (5.5.18) and
(5.5.19) now simply read
C
+ (x,
p) ≡
1
2
∞
n,m=0
1
n!m!
(6.3.7)
×
n
k=1
d 3 p k
(2π) 3 2E p k ,s k
m
l=1
d 3 q l
(2π) 3 2E q l ,r l
(2π)
4 δ
(4)
p +
n
k=1
p k −
m
l=1
q l
× Tr +
A c (
p 1 , . . . ,
p n ,
p; ;
q 1 , . . . ,
q m ) f (x,
q 1 , . . . ,
q m )
× A
†
c ( q 1 , . . . ,
q m ; ;
p 1 , . . . ,
p n ,
p) f ◦ (x,
p 1 , . . . ,
p n ,
p)
+ F
+ (x,
p) ,
and
C
− (x,
p) ≡
1
2
∞
n,m=0
1
n!m!
(6.3.8)
×
n
k=1
d 3 p k
(2π) 3 2E p k ,s k
m
l=1
d 3 q l
(2π) 3 2E q l ,r l
(2π)
4 δ
(4)
p +
n
k=1
p k −
m
l=1
q l
× Tr −
f (x,
p,
p 1 , . . . ,
p n ) A
†
c (
p,
p 1 , . . . ,
p n ; ;
q 1 , . . . ,
q m )
× f ◦ (x,
q 1 , . . . ,
q m ) A c ( q 1 , . . . ,
q m ; ;
p,
p 1 , . . . ,
p n )
+ F
− (x,
p) .