4.3 Expectation Values of Heisenberg Operators
55
i G pq (t, t
) == 0 | ˆ
T T T
c p [t]c
†
q [t
]
| 0
= lim
→0
∞
n=0
(−i)
n
n!
∞
−∞
dt 1 e
−|t 1 |
. . .
∞
−∞
dt n e
−|t n |
0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n )c p (t)c
†
q (t
)
| 0
0 | ˆ
U (∞, −∞)| 0
(4.58)
where the explicit perturbation expansion of the denominator is given by
0 | ˆ
U (∞, −∞)| 0 =
∞
n=0
(−i)
n
n!
∞
−∞
dt 1 e
−|t 1 |
. . .
∞
−∞
dt n e
−|t n |
0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n )
| 0
(4.59)
Like in the Gell–Mann and Low state (4.42), the resulting expression for G pq (t, t
)
is seen to be the ratio of two perturbation expansions, both depending on the switching parameter . Likewise, the adiabatic limit ( → 0) does not exist independently
for the denominator and numerator, but eventually for their ratio. The linked-cluster
theorem, to be addressed in Sect. 5.3, will show that the denominator cancels a corresponding factor in the numerator, thereby eliminating any diverging contributions
in the adiabatic limit.
The essential ingredients in the perturbation expansions for G pq (t, t
) are expectation values of time-ordered products of creation and destruction operators in
the interaction picture, where the expectation value is to be taken with respect to
the non-interacting ground state | 0 . The evaluation of these expectation values is
the subject of Wick’s theorem considered in Chap. 5.
4.4 Comparison with Rayleigh–Schrödinger Perturbation
Theory
The Gell–Mann and Low expressions (4.42), (4.43) for the interacting ground state
and ground-state energy establish a perturbation theoretical approach, which differs
completely from the familiar RSPT procedure. Of course, the resulting perturbation expansions must be identical, and it is instructive to see explicitly how this
equivalence comes to pass at lowest orders.
The RSPT expansions for the ground state and ground-state energy can be written
in the closed-form expressions presented in Appendix A.1:
| 0 = | 0 +
∞
n=1
ˆ
Q 0
E
(0)
0 − ˆ
H 0
E
(0)
0 − E 0 + ˆ
H I
n
| 0
(4.60)