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9 Semiclassical Theory: Path Integrals
a means to link the classical and quantum regimes of a system that is classically
chaotic.
We wish to compute the response function for a classically chaotic system. Let
us first write the semiclassical expression for the response function. From Eqs. (9.6)
and (9.79), we obtain
g osc (e) ≈
1
i ¯
h
1
2πi ¯
h
d−1
2
dx 0 lim
x→x 0
β
|Det( ¯
D d,β )|
× exp
i
¯
h
S β (x 0 , x; e) −
π
2
n β
.
(9.112)
where the set of paths β denotes the paths with energy e. The orbits in chaotic
systems do not lie on tori as they do in integrable systems, so we cannot directly
integrate over x 0 . However, we can evaluate Eq. (9.112) by the method of stationary
phase.
9.6.1 Stationary Phase Approximation
Let us perform the stationary phase approximation on the response function. First
note that for fixed energy e,
lim
x→x 0
∂S(x 0 , x, e)
∂x i
= lim
x→x 0
∂S
∂x i
+
∂x 0j
∂x i
∂S
∂x 0j
= p i − p 0i .
(9.113)
Points of stationary phase occur when p i = p 0i . Such points belong to periodic
orbits. We will let γ denote the subset of periodic orbits contained in the set β.
Useful Identities
Since H (p, x) = H (p 0 , x 0 ) = e, we can write
∂H
∂e
x 0
=
d
i=1
∂p 0i
∂e
∂H
∂p 0i
= −
d
i=1
˙
x 0i
∂ 2 S
∂e∂x 0i
= 1,
(9.114)
∂H
∂e
x
=
d
i=1
∂p i
∂e
∂H
∂p i
=
d
i=1
˙
x i
∂ 2 S
∂e∂x i
= 1,
(9.115)
∂H
∂x i
e,x 0
=
d
j =1
∂p 0j
∂x i
∂H
∂p 0j
= −
d
j =1
˙
x 0j
∂ 2 S
∂x i ∂x 0j
= 0,
(9.116)
∂H
∂x 0i
e,x
=
d
j =1
∂p j
∂x 0i
∂H
∂p j
=
d
j =1
˙
x j
∂ 2 S
∂x 0i ∂x j
= 0,
(9.117)