A Short Course on Quantum Mechanics and Methods of Quantization 45
way:
〈φf|e−ı
tH
|φi〉= lim→ 0 〈φf|(e−ı
H
)M|φi〉
= lim→ 0
∫ (M∏− 1
n=1
dφ∗ndφn
N
)
e−
∑M− 1
n=1φ∗nφn〈φf|e−iH |φM− 1 〉
×〈φM− 1 |e−
ıH
|φM− 2 〉...〈φ 1 |e−
ıH
|φi〉
= lim→ 0
∫ (M∏− 1
n=1
dφ∗ndφn
N
)
e−
∑M− 1
n=1φ∗nφn
(M
∏
n=1
〈φn|e−ı
H
|φn− 1 〉
)
= lim→ 0
∫ (M∏− 1
n=1
dφ∗ndφn
N
)
e−
∑M− 1
n=1φ∗nφne
∑M
n=1φ∗nφn−^1 e−ı
∑M
n=1H(φ∗n,φn−^1 ),
(1.218)
having set:〈φf|≡〈φM|,|φi〉≡|φ 0 〉.
To arrive at this expression we have divided the time intervaltinMintervals
of length ,insertedM−1 resolutions of the identity written in terms of coherent
states:
I=
∫
⎛
⎝
∏
j
dφ∗jdφj
N
e−φ
∗jφj
|φj〉〈φj|
⎞
⎠,
with
φ=
{
z∈C
ξ∈G
, N=
{
2 πi for bosons
1forfermions
.
The last line of (1.218) follows after assuming thatH=H(a†,a) is written in its
normal form so that:
〈φn|e−ı
H
|φn− 1 〉=〈φn| 1 −
ı
H(a†,a)+···|φn− 1 〉=eφ
∗nφn− 1
e−ı
H(φ∗n,φn−^1 )
,
whereH(φ∗n,φn− 1 ) has been obtained fromH=H(a†,a) by means of the substi-
tution:a†→φ∗n,a→φn.
As before, we can interpret{φn}as a discretized trajectory, so that:
φ∗n
φn−φn− 1
→φ
∗
n(t
′)∂
∂t′φ(t
′),
H(φ∗n,φn− 1 )→H(φ∗n(t′),φn− 1 (t′)),
while
−
M∑− 1
n=1
φ∗nφn+
∑M
n=1
φ∗nφn− 1 −
ı
∑M
n=1
H(φ∗n,φn− 1 )
→φ∗(t)φ(t)+
ı
∫t
0
dt′
[
(i)φ∗(t′)
∂φ(t′)
∂t′
−H(φ∗(t′),φ(t′))