5.3 Linked-Cluster Theorem
73
i G pq (t, t
) =
∞
n=0
(−i)
n
n!
∞
−∞
dt 1 . . .
∞
−∞
dt n
0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n )c p (t)c
†
q (t
)
| 0 C
(5.43)
A validation of this result will be deferred to Sect. 7.2 and Appendix A.4, addressing
the internal time integrations in the Feynman diagrams.
Exercises
5.1 Inspect the contraction patterns in the ground-state expectation value associated
with the second-order term, i ˜
G
(2)
pq (t, t
), in Eq. (5.19). Select the contraction
patterns relating to fully connected contributions (or diagrams).
5.2 Evaluate the ground-state expectation value 0 | ˆ
V | 0 of the Coulomb operator
using Wick’s theorem.
References
1. Wick GC (1950) Phys Rev 80:268
2. Feynman RP (1949) Phys Rev 76:749
3. Brueckner KA (1955) Phys Rev 100:36
4. Goldstone J (1957) Proc R Soc A 239:267
73
i G pq (t, t
) =
∞
n=0
(−i)
n
n!
∞
−∞
dt 1 . . .
∞
−∞
dt n
0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n )c p (t)c
†
q (t
)
| 0 C
(5.43)
A validation of this result will be deferred to Sect. 7.2 and Appendix A.4, addressing
the internal time integrations in the Feynman diagrams.
Exercises
5.1 Inspect the contraction patterns in the ground-state expectation value associated
with the second-order term, i ˜
G
(2)
pq (t, t
), in Eq. (5.19). Select the contraction
patterns relating to fully connected contributions (or diagrams).
5.2 Evaluate the ground-state expectation value 0 | ˆ
V | 0 of the Coulomb operator
using Wick’s theorem.
References
1. Wick GC (1950) Phys Rev 80:268
2. Feynman RP (1949) Phys Rev 76:749
3. Brueckner KA (1955) Phys Rev 100:36
4. Goldstone J (1957) Proc R Soc A 239:267