9.6 Gutzwiller Trace Formula
321
where δ ¯
y and δ ¯
p are 1 × (d − 1)-dimensional column matrices with matrix elements
{δy i } and {δp i }, respectively, and ¯
A, ¯
B, ¯
C, and ¯
D are (d − 1) × (d − 1)-dimensional
square matrices whose matrix elements are A ij , B ij , C ij , and D ij , respectively. The
(2d − 2) × (2d − 2)-dimensional square matrix, ¯
M, is the monodromy matrix for
the given periodic orbit. We will now relate the monodromy matrix to the quantity
under the square root in Eq. (9.122).
Let us first relate deviations in the transverse momenta, {δp 0i } and {δp i }, to
deviations in the transverse coordinates, {δy 0i } and {δy i },
δp 0i =
d−1
j =1
∂p 0i
∂y 0j
δy 0j +
d−1
j =1
∂p 0i
∂y j
δy j =
d−1
j =1
[−a ij δy 0j − b ij δy j ]
(9.126)
and
δp i =
d−1
j =1
∂p i
∂y 0j
δy 0j +
d−1
j =1
∂p i
∂y j
δy j =
d−1
j =1
[( ¯
b
T ) ij δy 0j + c ij δy j ],
(9.127)
where
a ij =
∂ 2 S
∂y 0i ∂y 0j
, b ij =
∂ 2 S
∂y 0i ∂y j
, ( ¯
b
T ) ij =
∂ 2 S
∂y i ∂y 0j
, c ij =
∂ 2 S
∂y i ∂y j
.
(9.128)
If we combine Eqs. (9.123), (9.124), (9.126), and (9.127), we obtain the following
relation between matrices ¯
A, ¯
B, ¯
C, and ¯
D, and the matrices ¯
a, ¯
b, and ¯
c (whose
matrix elements are a ij , b ij , and c ij , respectively):
¯
A = − ¯
b
−1
· ¯
a, ¯
B = − ¯
b
−1 , ¯
C = ¯
b
T
− ¯
c · ¯
b
−1
· ¯
a, and ¯
D = −¯ c · ¯
b
−1 .
(9.129)
Let us now introduce the following function:
F (λ) = Det( ¯
M − λ ¯
I ) = Det
¯
A − λ ¯
I
¯
B
¯
C
¯
D − λ ¯
I
= Det
− ¯
b −1 · ¯
a − λ ¯
I
− ¯
b −1
¯
b T − ¯
c · ¯
b −1 · ¯
a − ¯
c · ¯
b −1 − λ ¯
I
.
(9.130)