4.2 The Gell-Mann and Low Theorem
53
The time-reversed form (4.45) can be conveniently written as a bra state,
0 | = lim
→0
0 | ˆ
U (∞, 0)
0 | ˆ
U (∞, 0)| 0
(4.47)
to be used in forming ground-state expectation values.
4.3 Expectation Values of Heisenberg Operators
The Gell-Mann and Low formulation of the interacting ground state can be extended
to ground-state expectation values of operators. In particular, we are interested in
expectation values involving time-dependent Heisenberg operators as encountered
in the definition of the electron propagator (3.6).
Let us consider a general Heisenberg operator
ˆ
O H (t) = e
i ˆ
Ht ˆ
O S e
−i ˆ
Ht
(4.48)
associated with a Schrödinger operator ˆ
O S . Using Eq. (4.16), the ˆ
O S may be replaced
by the corresponding operator ˆ
O I (t) of the interaction picture:
ˆ
O H (t) = e
i ˆ
Ht e
−i ˆ
H 0 t ˆ
O I (t)e
i ˆ
H 0 t e
−i ˆ
Ht
(4.49)
For the time arguments (t, 0), the time-evolution operator in the interaction picture
(Eq. 4.20) can be written as
ˆ
U (t, 0) = e
i ˆ
H 0 t e
−i ˆ
Ht
(4.50)
Likewise, this result can be derived directly from the equation of motion (4.25).
Using Eqs. (4.49) and (4.50), ˆ
O H (t) can be written in the form
ˆ
O H (t) = ˆ
U (0, t) ˆ
O I (t) ˆ
U (t, 0) = lim
→0
ˆ
U (0, t) ˆ
O I (t) ˆ
U (t, 0)
(4.51)
Note that for finite time arguments in the time-evolution operator, the limit → 0 is
unproblematic; that is,
ˆ
U (t 1 , t 2 ) = lim
→0
ˆ
U (t 1 , t 2 )
(4.52)
which justifies the second part of Eq. (4.51).
In the latter form, the Heisenberg operator is compatible with the Gell-Mann and
Low representation (4.42) of the ground state. Noting the normalization (4.44) of
|
0 , the ground-state expectation value of ˆ
O H (t) can be written as