282
Appendix
which is of a form in which the limit → 0 can be taken on both sides. According
to the assumption,
| = lim
→0
| 0 ()
0 | 0 ()
exists in all orders of perturbation theory, and this property is not affected by applying
the operation g
∂
∂g
(in each order n). Then, due to the factor , the right-hand side
vanishes for → 0, and we obtain
( ˆ
H − E 0 ) lim
→0
| 0 ()
0 | 0 ()
= 0
which completes the proof.
Combining Operators and the Time-Evolution Operator:
In deriving Eq. (4.56), the following extended transitivity property of the timeevolution operator has been used:
ˆ
U (∞, t) ˆ
O I (t) ˆ
U (t, −∞) =
∞
n=0
(−i)
n
n!
∞
−∞
dt 1 e
−|t 1 |
. . .
∞
−∞
dt n e
−|t n |
ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n ) ˆ
O I (t)
(A.2.13)
To prove this property, let us consider the nth order term on the right-hand side of
Eq. (A.2.13) for a given value of t,
X
(n)
(t) =
(−i)
n
n!
∞
−∞
dt 1 . . .
∞
−∞
dt n ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n ) ˆ
O I (t)
(A.2.14)
omitting here the switching functions for simplicity. The n-fold time integrations
can be broken up into (n + 1)! distinct contributions according to the different timeorderings of the n + 1 time arguments t 1 , . . . , t n , t. These time-orderings can further
be classified according to the number μ = 0, . . . , n, counting the time arguments
larger than t; correspondingly, ν = n − μ is the number of time arguments smaller
than t. For a given μ, there are
n
μ
ways to select μ time arguments from the set
t 1 , . . . , t n , and for each selection, there are μ! ν! time-orderings in which the first μ
and the last ν time arguments are permuted among themselves. Note that the total
number of time-orderings in class μ is n! =
n
μ
μ!ν!. Let
t i > t j · · · > t k > t l > t > t r > · · · > t u
denote a specific time-ordering of class μ; that is, t i > t j · · · > t k > t l are μ time
arguments. The corresponding contribution in the integrand can be written as
Appendix
which is of a form in which the limit → 0 can be taken on both sides. According
to the assumption,
| = lim
→0
| 0 ()
0 | 0 ()
exists in all orders of perturbation theory, and this property is not affected by applying
the operation g
∂
∂g
(in each order n). Then, due to the factor , the right-hand side
vanishes for → 0, and we obtain
( ˆ
H − E 0 ) lim
→0
| 0 ()
0 | 0 ()
= 0
which completes the proof.
Combining Operators and the Time-Evolution Operator:
In deriving Eq. (4.56), the following extended transitivity property of the timeevolution operator has been used:
ˆ
U (∞, t) ˆ
O I (t) ˆ
U (t, −∞) =
∞
n=0
(−i)
n
n!
∞
−∞
dt 1 e
−|t 1 |
. . .
∞
−∞
dt n e
−|t n |
ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n ) ˆ
O I (t)
(A.2.13)
To prove this property, let us consider the nth order term on the right-hand side of
Eq. (A.2.13) for a given value of t,
X
(n)
(t) =
(−i)
n
n!
∞
−∞
dt 1 . . .
∞
−∞
dt n ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n ) ˆ
O I (t)
(A.2.14)
omitting here the switching functions for simplicity. The n-fold time integrations
can be broken up into (n + 1)! distinct contributions according to the different timeorderings of the n + 1 time arguments t 1 , . . . , t n , t. These time-orderings can further
be classified according to the number μ = 0, . . . , n, counting the time arguments
larger than t; correspondingly, ν = n − μ is the number of time arguments smaller
than t. For a given μ, there are
n
μ
ways to select μ time arguments from the set
t 1 , . . . , t n , and for each selection, there are μ! ν! time-orderings in which the first μ
and the last ν time arguments are permuted among themselves. Note that the total
number of time-orderings in class μ is n! =
n
μ
μ!ν!. Let
t i > t j · · · > t k > t l > t > t r > · · · > t u
denote a specific time-ordering of class μ; that is, t i > t j · · · > t k > t l are μ time
arguments. The corresponding contribution in the integrand can be written as