80
6 Feynman Diagrams
(−i)
definition of G
(−i)
n
n-th order
i
2n+1
G
0 -lines
⎫
⎪ ⎬
⎪ ⎭
i
n
and multiply by a sign (−1)
L , where L is the number of closed loops.
6.2 Diagrams in the Hartree–Fock Representation
So far, we have confined ourselves to the case where the interaction hamiltonian
consists only of the Coulomb part. The diagrammatic formulation can readily be
extended to the general form
ˆ
H I = ˆ
W + ˆ
V
where
ˆ
W =
w rs c
†
r c s
is a (non-diagonal) one-particle operator (see Eq. 4.3). As a graphical symbol associated with the one-particle interaction, we use the one-particle interaction cross
with one entry and one exit,
(6.4)
According to
1
n!
ˆ
H
n
I =
1
n!
( ˆ
V + ˆ
W )
n
=
1
n!
n
ν=0
n
ν
ˆ
V
ν ˆ
W
n−ν
a general nth order diagram may have ν wiggly interaction lines and μ = n − ν
crosses, ν = 0, . . . , n. Remark 4 can be generalized in an obvious way, as the
(ν!μ!) orderings of the wiggly lines and crosses are equivalent.
The diagram rules can easily be adapted to the general case. The first rule, addressing the generation of diagrams, can be restated explicitly as follows:
(F1’) To generate the nth-order contribution G
(n)
pq (t, t
), draw all topologically distinct connected diagrams with ν wiggly interaction lines and μ = n − ν interaction crosses, ν = 0, . . . , n. The number of free Green’s function lines here
is 2ν + μ + 1; the first G
0 -line begins at the outer vertex (q, t
) and the last
one ends at ( p, t). At any wiggly interaction line, two G
0 -lines begin and two
end; at any interaction cross, one G
0 -line begins and one ends.
Rules (F2) and (F3), dealing with the evaluation of diagrams, apply essentially in
their original form; the phase factor of rule (F4) has to be adapted according to the
number of G
0 -lines (2ν + μ + 1).
6 Feynman Diagrams
(−i)
definition of G
(−i)
n
n-th order
i
2n+1
G
0 -lines
⎫
⎪ ⎬
⎪ ⎭
i
n
and multiply by a sign (−1)
L , where L is the number of closed loops.
6.2 Diagrams in the Hartree–Fock Representation
So far, we have confined ourselves to the case where the interaction hamiltonian
consists only of the Coulomb part. The diagrammatic formulation can readily be
extended to the general form
ˆ
H I = ˆ
W + ˆ
V
where
ˆ
W =
w rs c
†
r c s
is a (non-diagonal) one-particle operator (see Eq. 4.3). As a graphical symbol associated with the one-particle interaction, we use the one-particle interaction cross
with one entry and one exit,
(6.4)
According to
1
n!
ˆ
H
n
I =
1
n!
( ˆ
V + ˆ
W )
n
=
1
n!
n
ν=0
n
ν
ˆ
V
ν ˆ
W
n−ν
a general nth order diagram may have ν wiggly interaction lines and μ = n − ν
crosses, ν = 0, . . . , n. Remark 4 can be generalized in an obvious way, as the
(ν!μ!) orderings of the wiggly lines and crosses are equivalent.
The diagram rules can easily be adapted to the general case. The first rule, addressing the generation of diagrams, can be restated explicitly as follows:
(F1’) To generate the nth-order contribution G
(n)
pq (t, t
), draw all topologically distinct connected diagrams with ν wiggly interaction lines and μ = n − ν interaction crosses, ν = 0, . . . , n. The number of free Green’s function lines here
is 2ν + μ + 1; the first G
0 -line begins at the outer vertex (q, t
) and the last
one ends at ( p, t). At any wiggly interaction line, two G
0 -lines begin and two
end; at any interaction cross, one G
0 -line begins and one ends.
Rules (F2) and (F3), dealing with the evaluation of diagrams, apply essentially in
their original form; the phase factor of rule (F4) has to be adapted according to the
number of G
0 -lines (2ν + μ + 1).