From Classical Mechanics to Quantum Field Theory

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′))

]

.