Revista Brasileira de Flsica, Vol. 14, n? 1, 1984
Semiclassical Matrix Elements
A.M. OZORIO DE ALMEIDA
Instituto de Física, Universidade Estadual de Campinas, Caixa Postal 1170, Campinas, 13100, SP,
Brasil
Recebido em 27 de junho de 1984
&stract
Semiclassical m a t r i x elements o f an o b s e r v a b l e , w i t h r e s p e c t t o
t h e e i g e n s t a t e s o f a c l a s s i c a l l y i n t e g r a b l e Hamiltonian, a r e d e r i v e d
u s i n g t h e Wigner-Weyl r e p r e s e n t a t i o n i n o r d i n a r y phase space and i n act i o n - a n g l e v a r i a b l e s . For p r o p e r t i e s i n v o l v i n g t h e e n t i r e m a t r i x , such
as t h e eigenvalues, t h e r e i s u s u a l l y l i t t l e d i f f e r e n c e between b o t h r e s u l t s , wliich a r e s i m p l e r t o c a l c u l a t e i n t h e a c t í o n - a n g l e formal i s m . The
r e l a t i v e d i f f e r e n c e f o r s i n g l e m a t r i x elernents can be i m p o r t a n t . T h e
t h e o r y r e l i e s on s e m i c l a s s i c a l approximations o f t h e Moyal m a t r i x o r
crsss-Wigner f u n c t i o n , which g e n e r a l i z e s p r e v i o u s work on t h e p u r e s t a t e
Wigner f u n c t i o n , together w i t h i t s i n t e r p r e t a t i o n i n t e r m s o f t h e
geometry o f the i n v a r i a n t t o r i o f i n t e g r a b l e systems. The s e m i c l a s s i c a l
equivalence between the Weyl t r a n s f o r m o f an o p e r a t o r a n d t h e c o r r e sponding c l a s s i c a l f u n c t i o n i s a l s o discussed.
1. INTRODUCTION
The attempt t o understand t h e quantum mechanicsofsystemswhose
c l a s s i c a l motion i s u n i n t e g r a b l e , i . e . w i t h fewer c o n s t a n t s o f t h e motion
than degrees o f freedom, has hecessari 1 y increased
the in t e r e s t
of
a i t e r n a t i v e approaches t o semiclass i c a l (SC) mechanics. O f these one o f
the most p r o m i s i n g uses t h e Wigner-Weyl f o r m u l a t i o n o f
q u a n t u m meth-
a n i c s , s i n c e t h e rnain mathematical r e s u l t s on u n i n t e g r a b l e systems
are
d e r i v e d i n a phase space r a t h e r than a c o o r d i n a t e space formalism.
In-
i t i a l work on the Wigner f u n c t i o n of i n t e g r a b l e ~ ~ s t e m s ' uncovered
~ ~ ' ~ ' ~
a remarkable geometrical r e l a t i o n between t h e SC Wigner f u n c t i o n a n d t h e
c l a s s i c a l i n v a r i a n t m a n i f o l d (a t o r u s ) t o which i t c 0 1 1 a p s e s
c l a s s i c a l l i m i t where P l a n c k ' s constant A
-+
3
Unfortunately
in the
the
SC
Wigner f u n c t i o n f o r u n i n t e g r a b l e systems has proved as e l u s i v e as o t h e r
r e p r e s e n t a t i o n s o f t h e S C s t a t e , b u t t h e Wigner-Weyl f o r m u l a t i o n may
be
used i n o t h e r ways.
Work
partia1l y
supported by FAPESP, CNPq a n d F I N E P
G o v e r n m e n t A g encies).
( B r a z i li a n
'i'he o b j e c t o f t h i s paper i s t o d e r i v e the SC f o r m o f t h e m a t r i x
elements o f any observable w i t h a c l a s s i c a l l i m i t , w i t h respect
I n the p a r t i c u l a r
e i g e n s t a t e s o f an i n t e g r a b l e system.
t o the
case where t h e
observable i s an u n i n t e g r a b l e Hamiltonian, i t s SC m a t r i x w i l l be n e a r l y
diagonal i f t h e b a s i s used f o r t h e expansion a r e t h e e i g e n s t a t e s
of
a
good i n t e g r a b l e approximation o f t h e Hamiltonian. The spectrum o f a n u n i n t e g r a b l e system thus becomes a c c e s s i b l e by merely d i a g o n a l i z i n g theSC
Hamiltonian. Of course t h e m a t r i x elements have o t h e r uses, such as c a l c u l a t i n g t r a n s i t i o n rates, etc..
.
A
The form o f Weyl transform o f an o p e r a t o r
5,t h e
1( f u n c t i o n o f
cf
and
fundamental c o o r d i n a t e o p e r a t o r s o f a Hami 1 t o n i a n system wi t h
N
degrees o f freedom) whlch I s h a l l use i s
A(&$)
=
/d?$ <;+3/2[1(&3/2>
-f
-+
-f
I t i s n o t general l y t r u e t h a t A(q,p)
if
A
doesri ' t depend e x p l i c i t l y on h
e
-+
= Ac(q,p),
5.
-i;. $/fi
(1.1)
the c l a s s i c a l observable,
However, i t wi 1 1 be shown i n sec-
t i o n 2 t h a t t h i s e q u a l i t y does h o l d s e m i c l a s s i c a l - l y .
The m a t r i x r e p r e s e n t a t i o n o f à i n terms o f a complete b a s i s o f
-+
s t a t e s Irnz i s then
where I d e f i n e the MoyaZ ~ a t r i x o~r , t h e c r o s s Wigner function
l'he Moyal m a t r i x elements a r e simply t h e Weyl t r a n s f o r m o f t h e
I&&(.
operators ( 2 1 ~ h ) - ~
A1 t o g e t h e r they form a complete o r t h o g o n a 1
b a s i s i n the s e t o f Wigner functions corresponding t o pure
states
I+>:
arbit r a r y
The expansion
i n eq.(1.4)
leads t o
The i n a t r i x elements s a t i s f y t h e r e l a t i o n s 6
and
as w e l l as the s e l f - o r t h o g o n a l i t y r e l a t i o n s
r 4 ai; <GIZ> ,> ,q>c
-i
+
=
Z
<$i%>
= 6
and
The Ployal M a t r i x i s o b v i o u s l y Hermitian,
even though t h e o p e r a t o r s
-i
~m><zl a r e
From the form o f eq.(1.2)
not.
i t i s evident t h a t the f i r s t step i n
t h e S C t h e o r y o f m a t r i x r e p r e s e n t ã t i o n o f observables i s t h e g e n e r a l i z a t i o n o f t h e SC t h e o r y o f pure s t a t e Wigner f ~ n c t i o n s ' * ~ 't ~o M o y a l
M a t r i x elements. T h i s i s presented i n s e c t i o n 3. F i n a l l y i n
section
4
the t h e o r y i s reworked i n a c t i o n - a n g l e v a r i a b l e s . Thecomparisonof these
wi t h t h e r e s u l t s d e r i v e d i n t h e untransformed phase space r e v e a ' l s t h a t
t h e s i m p l i c i t y o f t h e a c t í o n - a n g l e formalism may i n some cases e n t a i l a
s i g n i f i c a n t l o s s o f accuracy o f t h e m a t r i x elements.
2. THE SEMICLASSICAL WEYL TRANSFORM
I n t h e simple case t h a t 2 depends o n l y on
t h e d e f i n i t i o n (1.1) t h a t );A
,;(
-f
q,
i t follows
from
= A ( $ ) = ~ ~ ( 4 The
) . a l t e r n a t i v e formof
(1.1) u s i n g t h e momentum r e p r e s e n t a t ion,
A(;,;)
leads t o
i3
=
lftlP-3/2> e -&.$/E
f d; <;+$/2
(2.1)
s i m i l a r i d e n t i t y between t h e Weyl t r a n s f o r m and t h e c l a s s i c a l
l i m i t o f iiny f u n c t i o n o f
8. The
l i n e a r i t y o f t h e Weyl t r _ a n s f o r m t h u s
8 + V(;)
have as Weyl
I n general, however, t h e Weyl correspondence may
n o t a 1wa y s
impl l e s t h a t Hamiltonians w i t h t h e standard form
transforms
2 + V(;).
+
h o l d . Indeed, t h e b e s t way t o q u a n t i z e a coniplicated f u n c t i o n o f p
-P
q i s n o t always c l e a r
5
.
I s h a l l now shaw t h a t these problems a r e
++
c l a s s i c a l l y n e g l i g i b l e by expanding A(q,p)
and
semi-
as a s u p e r p o s i t i o n o f Wigner
f u n c t i o n s . I n t h i s way I s a t i s f y a secondary o b j e c t i v e
the s i m p l s s t instance o f t h e Moyal m a t r i x elements t o
of
introducing
be discussed
in
t h e n e x t ç e c t i o n . Only a s i n g l e degree o f freedom w i l l be considered.
I h e expansion o f t h e observable
1 in
the basis
of
it s
own
eigenfunctions
K
=
1 lrn> ~ ~ < r n i
(2.2)
rn
has t h e Weyl t r a n s f o r m
S e m i c l a s s i c a l l y t h e s t a t e m corresponds t o t h e leve1 c u r v e
Ac(q,p)
= Am
(2.4)
whose area i s 2~ Im
where I
,
m
i s t h e c l a s s i c a l a c t i o n s e l e c t e d by
the
quan t i z a t l o n cond i t l o n
Here a i s t h e
M ~ S Z O Vindex,
o s c i l l a t o r s equals
2.
which f o r t h e e i g e n s t a t e s o f
non-l inear
I n t h e case o f more than one degree o f freedom,
t h e e i g e n s t a t e o f a complete s e t o f commuting observables c o r r e s p o n d s
t o a phase space N- torus. The q u a n t i z a t i o n condi t i o n (2.4)
then appl i e s
t o each orie o f i t s N i r r e d u c i b l e c i r c u i t s , d e f i n i n g t h e s e t
torl
of
basis
.
I t was proved by ~ e r r ~t hla t i n t h e c l a s s i c a l 1 i m i t t h e w i g n e r
f u n c t i o n c o l l a p s e s o n t o the t o r u s o r ,
i n o u r case, t h e leve1 curve:
I n t h i s l i m i t the spacing o f the e i g e n - a c t i o n s (2.5)
tends t o zero,
so
t h a t we may approximate t h e sum by an i n t e g r a l . Thus, i n t e r p o l a t i n g t h e
Am
u s i n g eq. (2.41,
we o b t a i n
The c l a s s i c a l l i m i t (2.6)
o f the Wigner f u n c t i o n does n o t h o l d
a s y m p t o t i c a l 1 y f o r smal 1 non-zero E . The u n i form SC approximat ion, v a l i d
i s'
i n s i d e a n d o u t s i d e the t o r u s , d e r i v e d by ~ e r r ~
where
Ai
7
i s the A i r y function .
I f t h e p o i n t (q,p)
i s i n s i d e the t o r u s ,
t h e area bounded on one s i d e by t h e t o r u s and on t h e o t h e r by t h e t o r u s
chord, which has ( 4 , ~ ) as i t s m i d - p o i n t , i s a(q,p).
c o n s t r u c t t h i s B e r r y chord
(q,p).
2
A
useful
i s t o r e f l e c t the t o r u s through
way
the
The area sandwi tched between t h e t o r u s and the r e f l e c t e d
to
point
'anti-
t o r u s ' i s 2 ~ ( 9 , p ) . T h i s i s shown i n F i g . 1. The chord t i p s have c o o r d i nates ( q .,p .) and t h e f u n c t i o n s
3 3
I n t h e denominator o f eq.(2.8)
I. a r e
3
d e f ined by
t h e r e appear-s t h e Poisson Bracket
w i t h t h e d e r i v a t i v e s evaluated a t t h e o r i g i n .
For p o i n t s ( q , p ) o u t s i d e t h e t o r u s , t h e chords and
u(q,p)
the
areas
a r e complex, so t h a t t h e A i r y f u n c t i o n i s e x p o n e n t i a l l y decaying.
Inside, where t h e argument of
t h e ãsymptotic form
7
of
Ai
Ai
i s much g r e a t e r than one,
t o obtain
we
can use
Fig.1
-
The a n t i t o r u s 'c-'
o b t a i n e d by
reflecting
t o r u s T through
the
a t ion p o i n t ( q , p )
.
is
the
evalu-
The i n t e r -
s e c t i o n o f T and T-'
deter-
mines t h e t i p s o f t h e
Berry
chord. The area a(q,p)
has
been shaded i n .
Thus, as discussed i n r e f . 2 , t h e
c l a s s i c a l l i m i t eq.(2.6)
i s not a r e s u l t
o f t h e amplitude i n s i d e t h e t o r u s going t o zero, b u t o f t h e o s c i i l a t i o n s
o f t h e Wigner f u n c t i o n becoming i n f i n i t e l y f a s t . As (q,p)
the t o r u s {1~,1,) O and a
-+
-+
approaches
O. The indeterminacy i n eq. (2.8)
is
re-
solved i n t h e t r a n s i t i o n a l approximation o f t h e Wigner f u n c t i o n
where
measures
i3
k i n d o f ' c u r v a t u r e ' o f t h e t o r u s , as d i s c u s s e d i n r e f . 3 . l n
li m i t ti + O eq. (2.12)
the
reduces t o t h e 6 - f u n c t i o n (2.5).
The q u e s t i o n i s now whether t h e o s c i l l a t i o n s o f the SC
f u n c t i o n break the e q u a l i t y between the Weyl t r a n s f o r m and
c a l observable. To answer t h i s q u e s t i o n we must improve
on
procedure., o f approximating a d i s c r e t e sum by an i n t e g r a l ,
use o f thí: e x a c t Poisson summation formula
8
Wigner
the c l a s s i the
crude
through t h e
m
1
f,
=
j=-cc
m=-m
6
dm f ( m ) e x p ( 2 n i j r n ) ( 2 . 1 4 )
The terms w i t h j#O supply o s c i l l a t o r y c o r r e c t i o n s t o t h e
p r o x i m a t i o n . A p p l i e d t o eq.(2.3)
where <111>(q,p)
s t a n d a r d ap-
t h e Weyl r e p r e s e n t a t i o n o f
A becomes
stands f o r t h e Wigner f u n c t i o n correspond i n g
to
the
.
Using t h e a p p r o x i m a t i o n
( n o t necessar i 1y quant ized) t o r u s wi t h a c t i o nI
(2.11)
i n s i d e t h e t o r u s f o r each i n t e g r a l , we o b t a i n any o s c i l l a t o r y
SC c o n t r i b u t i o n t o A(q,p)
by e v a l u a t i n g t h e integrais through themethod
o f s t a t i o n a r y phase. Since
Ac(I) i s non- oscillatory,
t h e phase
o f the
j ' t h integral i s j u s t
Note t h a t now i t i s the c e n t r e o f t h e chord (q,p) which i s h e l d
whereas t h e t o r u s
fixed,
I i s v a r i a b i e . The s t a t i o n a r y phase c o n d i t i o n i s
But, s i n c e (2a I) i s t h e area o f t h e whole t o r u s , t h e chord
area can
grow w i t h I o n l y as a f r a c t i o n , n o t as a m u l t i p l e o f 271I. The
only
s t a t i o n a r y p o i n t i s thus f o r
So t h e approximation o f t h e sum i n eq.(2.3)
by an i n t e g r a l i s j u s t i f i e d
a f t e r a l l . Moreover t h e dorninant s t a t i o n a r y a c t i o n i s t h a t of
passing e x a c t l y through (q,p)
o f eq.(2.11),
(2.15).
we g e t
68
-
t h e one w i t h
we must use t h e t r a n s i t i o n a l
oI(q,p)
the torus
= O . Thus, i n s t e a d
approximation
(2.12)
Replacing t h e A i r y f u n c t i o n by i t s i n t e g r a l r e p r e s e n t a t i o n
in
The o n l y s t a t i o n a r y p o i n t i s a t
,
t = o
I*I(~,P)
and t h e Hessian m a t r i x o f t h e phase a t t h i s p o i n t has zero s i g n a t u r e a n d
determinarit (2(2~'6'1)'/~)',
so t h a t t h e method o f s t a t i o n a r y
phase
g i v e s again
~ ( q , p ) = Ac(q,p)
(2.22)
7'he SC o s c i l l a t i o n s o f t h e Wigner f u n c t i o n do n o t
a f f e c ~t h e
Weyl correspondente f o r systems w i t h one degree o f freedom.
For
two
degrees o f freedom t h e t r a n s i t i o n a l approximation o f t o r u s Wigner funct i o n s reduces t o a product o f one dimensional Wigner f u n c t i o n s , f o r t h e
simple c l a s s o f t o r i discussed i n r e f
. 3 .This permi t s t h e general i z a t i o n o f
t h e f o r e g o i n g deduction w i t h t h e same simple r e s u l t .
3. THE
SEMICLASSICAL
MOYAL MATRIX
7'0 d e r i v e the SC l i m i t o f t h e Moyal m a t r i x elements, one
uses
t h e SC form o f t h e wavefunction
1++
=
1
JTn
i
det
-& (q,?;)
aqaI
where
i s t h e cZassicaZ a c t i o n along t h e j ' t h sheet o f t h e
-+
l a b e l e d b y t h e independent a c t i o n v a r i a b l e s I,
accord i n g t o eq. (2.5).
tosheet. F o r a f u l l
The index a
j
classical
each o f
them
torus,
quantized
changes by two i n pass i n g f rom sheet
irreduciblecircuiton thetorusCa
3'
= a,
the
Maslov index.
(A more d e t a i l e d e x p o s i t i o n o f t h e SC wave
given i n ref.2.)
I n s e r t i n g eq.(3.1)
i n t o eq.(1 .3) we g e t
funct ion i s
the
integral
r e p r e s e n t a t i o n o f the SC Moyal m a t r i x :
T h i s i n t e g r a l w i l l be dominated by i t s s t a t i o n a r y p o i n t s even i f these
a r e t o o c l o s e together f o r the simple s t a t i o n a r y phase m e t h o d
to
be
a p p l i e d d i r e c t l y . A l i n e a r canonical ( m e t a p l e c t i c ) t r a n s f o r m a t i o n among
the phase space c o o r d i n a t e s w i l l u s u a l l y b r i n g a11 t h e s t a t i o n a r y p o i n t s
o n t o a s i n g l e p a i r o f sheets j,jl, so t h a t henceforth,
s i n c e the
meta-
p l e c t i c i n v a r i a n c e o f t h e Wigner f u n c t i o n proved i n r e f . 2 can beextended
t o t h e Moyal r n a t r i x , t h e s e i n d i c e s and the surn s h a l l be omrnitted.
The modulus o f t h e Moyal m a t r i x elements
mined, but, u n l i k e the Wigner f u n c t i o n ,
i s uniquely d e t e r -
t h e phase i s p a r t l y
arbitrary.
T h i s a r b i t r a r i n e s s does n o t a f f e c t i n t e g r a l 5 over phase space,
separate i t o u t by d e f i n i n g
The Hoyal m a t r i x then takes t h e form
where
I have used the equal i t y
so
we
Outs i de
The two phases i n eq. (3.5)' have geometrical i n t e r p r e t a t i o n s .
t h e i n t e g r a l appears the a c t i o n sandwitched by t h e mf'th
and
+
-+
t o r i between t h e p o i n t s w i t h c o o r d i n a t e s q 0 and q. For one
the f ' t h
degree o f
freedom t h i s i s j u s t t h e shaded area between t h e two leve1 curves shown
i n F i g . 2. To understand t h e phase i n s i d e t h e i n t e g r a l i t i s b e s t t o
use t h e ' a n t i - t o r u s ' c o n s t r u c t i o n i n t r o d u c e d i n t h e p r e v i o u s s e c t i o n .
This c o n s i s t s i n i n t e r p r e t i n g
Fig.2
-
The f i g u r e
of two t o r i T
shows p o r t i o n s
L,
k and
character-
ized by a c t i o n s I k and Im and
-
the
a n t i t o r u s -ck obta ined by r e f l e c t i n g
T~
through t h e
evaluation
point.
The sum o f rhe cross- hatched
and
t h e b l a c k areas i s p r o p o r t i o n a l t o
l i l i
-
I
90
(I2 q
the phase o f the i n t e g r a n d
( 3 . 5 ) , whereas the sum
of
the
simply hatched and t h e b l a c k areas
q
ql
i n eq.
i s proportional
to
the
phase
outside the integral
+
-+
-+
o b t a i n e d by r e f l e c t i n g t h e t o r u s I =
+
Iz
li,
t o r u sI=
+ -+
through t h e p o i n t (q,p),
i .e.
as the a c t i o n from t h e p o i n t w i t h c o o r d i n a t e q a l o n g t h e
we take
-f--+
Iz(ql
-+
$1)
+
-+
-+
= 1~(-gl+2~,-~'+2;)
(3.8)
The phase o f t h e i n t e g r a n d i s thus t h e a c t i o n between t h e t o r u s
-+
+
a n t i t o r u s T a n d the p o i n t s w i t h c o o r d i n a t e q and q
+
3/2, as
I n t h i s case t h e r e a r e
one degree o f freedom i n Fig.2.
two
~ 2 t h, e
shown f o r
stationary
p o i n t s , where the t o r i i n t e r s e c t , as w i t h t h e Wigner f u n c t i o n , b u t
s t a t i o n a r y p o i n t s ql and q,
arise for
different
moduli o f
Q and
the
the
s t a t i o n a r y phases a l s o have i n general d i f f e r e n t moduli. The d i f f e r e n c e
between the s t a t i o n a r y phases
4.
3
o f t h e Moyal m a t r i x i s i n v a r i a n t under
m e t a p l e c t i c t r a n s f o r m a t i o n s as l o n g as t h e i n i t i a l p o i n t s on
tori,
from
the
which t h e a c t i o n s a r e evaluated, a r e k e p t f i x e d . T h i s
two
is
e x e m p l i f i e d f o r a system w i t h one degree o f freedom i n F i g . 3.
-
The t w o t e r m s
the
Moyal m a t r i x
Fig.3
of
element <klm> ( q , p ) have
p h a s e s ' which d i f f e r by
t h e c r o s s hatched
area
i n b o t h the @,q) and t h e
(p',q')
frames.
+ -f
For p o l n t s ( q p p ) such t h a t t h e a n t i t o r u s
Tz i n t e r s e c t s t h e
t h e s t a t i o n a r y phase e v a l u a t i o n o f eq. (3.5)
t o r u s T+,
f o l lows
that
of
the Wigner f u n c t i o n i n r e f . 2 w i t h the r e s u l t
where t h e m a t r i x o f Poisson Brackets
G;,~I
N
=
n=1
7: = +m
eq.(3.9)
As t h e
3%
3%
aqn
(q
?+)I.
For one degree o f f reedom
jy 7 ' m
reduces t o t h e SC Wigner f u n c t i o n (2.11).
a r e evaluated a t t h e s t a t i o n a r y
and
r ar,---4
a i a,
Ln
1 a
evaZuatíon point
(q,p)
i s taken o u t o f t h e i n n e r
torus
(Ik i n Fig.2), f o r a s i n g l e degree o f freedom, t h e s t a t i o n a r y phases o f
t h e i n t e g r a n d i n eq.(3.5)
become smaller arid, i n t h e l i m i t
where
the
t o r u s and a n t i t o r u s meet n o n - t r a n s v e r s e l y , t h e s i n g l e Poisson b r a c k e t
i n eq. (3.10)
goes t o zero.
'catastrophes
',
It i s easy t o see t h a t t h e l o c u s o f
where s t a t i o n a r y p o i n t s coalesce, i s a
i n t e r p o l a t e d between t h e tw t o r i .
l f m = k t h i s Wígner
these
closed
curve
caustic,
di s-
cussedin r e f s . l , 2 c o i n c i d e s w i t h t h e t o r u s i t s e l f . T h e r e i s another Wigner
which
the
a n t i t o r u s o f Tk touches Tm from t h e i n s i d e i n s t e a d o f t h e o u t s i d e ,
re-
c a u s t i c corresponding t o the l o c u s o f e v a l u a t i o n p o i n t s f o r
f e r r e d t o ais t h e L c u r v e i n r e f . 1 ,
where i t i s shown t h a t i t c o n t a i n s cusps
( F i g . 4 ) . The s i t u a t i o n f o r two degrees o f f reedom anal ysed i n ref.2 i s much
more complicated,
s i n c e t h e Wigner c a u s t i c i s then a connected
dimensional s u r f a c e which envelops t h e two-dimensional
torus
threewhere i t
becomes s i n g u l a r .
Fig.4
-
For the case o f one degree o f
freedom t h e M o y a l m a t r i x e l e m e n t
<klm>(q,p)
has
two W i g n e r c a u s t i c s :
with
The r e f l e c t i o n o f t h e t o r u s
Tk
respect t o any p o i n t o f t h e
interp-
olated torus
;( t h e
be tangent t o
d o t t e d 1 i n e ) wi 1 1
from t h e
outside,
whereas r e f l e c t i o n o f T~ w i t h r e s p e c t
t o a p o i n t on L leads
from t h e i n s i d e o f
to
tangency
L.
The u n i f o r m approxima t i o n o f t h e non- diagonal Moyal m a t r i x e l ments, v a l i d through t h e caust i c , i s more complicated than
eq. ( 2 . 8 ) ,
because o f t h e lower symmetry i n t h e i r geornetrical c o n s t r u c t i o n s .
o u r purpose, which i n v o l v e s i n t e g r a t i o n over phase s p a c e ,
For
it i s s u f
-
f i c i e n t t o d e r i v e the c l a s s i c a l d e l t a - f u n c t i o n type o f a p p r o x i m a t i o n
1 i k e eq. (2.6). To do t h i s f o r t h e case o f one degree o f f reedom we must
expand t h e a c t i o n f u n c t i o n s S(~',I~)
and S(ql,Ik)
near t h e p o i n t s q+Q/2,
f o r which
k
aqr
T h i s cond i t i o n ,
tangents,
t h a t a pa i r o f p o i n t s on t h e two t o r i
have p a r a l l e l
Yocates t h e i r m i d p o i n t on t h e Wigner c a u s t i c as can
be
seen
i n F i g . 5. S e t t i n g Q = Q, i n t h e amplitude o f t h e i n t e g r a n d o f eq.
expandlng t h e exponent t o f i r s t o r d e r i n Q - Q a ( o r second o r d e r ,
eq.(3.11) and u s i n g
(3.5),
u si ng
73
Fig.5
-
The Wigner c a u s t i c i s t h e
l o c u s o f m i d p o i n t s o f segments o f
q-Qo4~q+Q04
q
-
j o i n i n g p o i n t s on -rk and
-rm w i t h
p a r a l l e l tangents.
t h e Moyal ma t r i x becomes
For t h e Wigner f u n c t i o n (m=k, Q ~ = O )t h i s s i r n p l i f i e s t o eq.(2.6).
1i m i t I
k
-i I
m
Q, becomes s m a l l ,
so t o second o r d e r i n (I,-I~)
f o r t h e o f f d i a g o n a l e l e m e n t s . General 1 y, however, t h e Wigner
I n the
and
in
caustic
on which t:he 6 - f u n c t i o n o f eq. (3.13)
t h e geometry o f the t o r i Ik and
Im
i s defined, depends e x c l u s i v e l y on
and i n no way on
any
in t e r v e n ing
t o r u s o f t:he f a m i l y .
l'he der i v a t i o n o f eq. (3.13)
has been phrased i n terms o f
& + L b,u t
Wigner c a u s t i c which c o i n c i d e s w i t h t h e t o r u s when I.
e q u a l l y v a l i d f o r t h e i n n e r Wigner c a u s t i c shown i n F i g . 4.
the
i t i s
The
only
d i f f e r e n c e i s t h a t i n t h i s case t h e o s c i l l a t i o n s i n t h e " amplitude"
the
of
6 - f u n c t i o n a r e much f a s t e r and do n o t disappear i n t h e Wigner func-
tion limit.
The p i c t u r e i s more complicated when another degree o f freedom
1s added. The Wigner c a u s t i c i s then d e f i n e d by t h e c o n d i t i o n
which d e f i n e s a three- dimensional s u r f a c e i n phase space. I n o t h e r words,
one o f fhe eigenvalues o f t h e above m a t r i x i s zero, i n d i c a t i n g t h a t t h e
+++
+++
t a n g r n t planes t o t h e t o r i a t p ( p + ~ o / 2 , ~ )and p(q-~o/2,?z)
have
a
d i r e c t i o r ! i n common. The c o n d i t i o n t h a t t h e o t h e r e i g e n v a l u e
zero
-
is
also
the case o f s t r o n g tangency, occurs on a two dimensional surface.
I n the case o f t h e Wigner f u n c t i o n ,
t h o r o u g h l y analysed i n r e f . 2 , t h i s
is
t h e t o r u s i t s e l f which i s an umbil i c (hyperbol i c o r e11 i p t i c ) s i n g u l a r i t y o f the Wigner c a u s t i c . For Ik c l o s e t o
Im t h i s must a l s o have
the
g l o b a l topology o f a t o r u s :
d e f i n e s an u m b i l i c i n t e r p o l a t i n g t o r u s .
There i s no second o r d e r term i n t h e expansion o f t h e phase o f
t h e i n t e g r a n d o f t h e Moyal m a t r i x i n eq.(3.5)
by eq.(3.16),
so t h a t ,
l e d t o eq.(3.14),
i n the 1 i m i t H
we o b t a i n
+
around t h e p o i n t s d e f i n e d
0, r e p e a t i n g t h e
argument t h a t
For simple tori, uniform and transitional approximations for
the Moyal Matrix can be derived which generalize the results for the
Wigner function3. As in the one dimensional case, these are basically
fringes, peaked on the Wigner caustic and especially cn the interpolating torus eq.(3.16) and decaying outside the caustic. Project ion
integrals of the Wigner function, supplying the wave intensi ty, are
correctly given by the 6-function approximation of the Wigner function,
as shown in ref.3. This is because the integral averages ove the oscillations (where they are not stationary), inspite of the cons iderabl e
amplitude inside and especially on the Wigner caustic but outside the
interpolating torus. The i ntegral (1.2) for matrix elements of
observables, is of twice the dimension of a projection integral and
there are no stationary points. The 6-function appr~ximationover a11
the points which satisfy eq. (3.16) should therefore give correct semiclassical matrix elements. Not only is the 6-function approximation
(3.17) much simpler to work with than morei refined approximations, but
the derivation of the former can be easily generalized to any number of
degrees of freedom, whereas the latter depend on a thorough understanding of the geometry of Lagrõngean manifolds9 of more than two d imens ions.
I have not made a clear distinction between the interpolating
torus, which reduces to the quantized torus in the case of the Wigner
function, and other curves or surfaces which satisfy eq. (3.16), 'such
as the L curve in Fig. 4. The reason is that for neighbouring tori the
exponential term in eq.(3.16) oscillates slowly on the interpolating
torus and very quickly elsewhere. The phase 1s nowhere stationary, so
that the observable matrix elements will be semiclassically negligible
+ +
unless a1 l components of I+-I+ are of order H for the interpolat i ng
m r
torus. The contribution of other surfaces is always negligible.
4. ACTION-ANGLE REPRESENTATION
1s is well known that action-angla variables cannot be quantized directlylO. However the simplicity of the classical motion in these vcariables is reflected in the semiclassical wave functions, so that
an approximate quantization-isextremely useful. The operator f have
the discrete set of semiclasslcal eigenvalues given by the BohrA
(2.5)
-Somnerfeld q u a n t i z a t i o n r u l e
for
each component. The
s t i pu
-
l a t i o n t h i i t t h e corresponding e i g e n f u n c t i o n s
a r e p e r l o d l c I n each conponent o f
3 impl i e s
t h a t i n t h e a n g l e represen-
tation
-i
I
=
-
+
$?i
ala;
214
(4.2)
-+
where a a r e t h e M a ~ l o v ~ i n d i c e sI.t i s sometimes p r e f e r a b l e
to
use
-+
-i
simpler d e f i n i t i o n o f I where t h e a a r e ommitted, a t t h e c o s t o f
Bloch (Foucault) wave f u n c t i o n s " .
a
using
representation?
The two a l t e r n a t i v e
-+
a r e r e l a t e d by a gauge t r a n s f o r m a t i o r i . Thus t h e fundamental o p e r a t o r s I,
8 satisfy
t h e r e q u i r e d romnutation r e l a t i o n
The d i f f i c u l t y i n working w i t h a c t i o n - a n g l e v a r i a b l e s i s t h a t a r b i t r a r y
periodic functions o f
eigenfunc1:ions o f
S
-+
e
can be F o u r i e r analysed I n t o
superpositions o f
-+
I w i t h p o s i t i v e and n e g a t i v e components o f m.
The
freedom 01: choice o f r e p r e s e n t a t i o n prevents us from e x c l u d i n g t h e s e
unphysicar t o r i , though i n t h e s e m i c l a s s i c a l l i m i t t h e expansion
ficients
-+
coef-
o f t h e terms w i t h n e g a t i v e components o f m, must tend t o zero.
The Weyl t r a n s f o r m o f an o p e r a t o r i s d e f ined by
I t i s essent i a 1 t h a t t h e Weyl r e p r e s e n t a t ion c o n t a i n as much i n f o r m a t i o n
about t h e o p e r a t o r  3s t h e a n g l e r e p r e s e n t a t i o n . T h i s would c e r t a i n l y
be t h e case o f t h e F o u r i e r c o e f f i c i e n t s o f
4 + $ I A I ~ - ~ taken
>
p e r i o d i c f u n c t i o n o f O. But i n t h e case o f N = 1 we can w r i t e t h e
F o u r i e r c o e f f i c i e n t as
n
as
a
m'th
I t f o l l o w s t h a t t h e Weyl r e p r e s e n t a t i o n i s complete i f we a l l o w d i s c r e t e
a c t i o n s corresponding t o h a l f i n t e g e r s
o f t h e Weyl t r a n s f o r m a t (IM,9)
M = d 2 . The c u r i o u s p a r t n e r s h i p
and (IM,9+d had a l r e a d y
appeared i n
the treatment o f t h e pure s t a t e c o n d i t i o n f o r the W i g n e r
f u n c t i o n 4,
though no
reference was made i n t h a t i n s t a n c e t o h a l f
tization.
I t i s i n t e r e s t i n g t h a t i n t h e q u a n t i z a t i o n o f l i n e a r maps on
integer
quan-
t h e t o r u s , Hannay and 6 e r r y l 2 do use h a l f i n t e g e r a c t i o n s w i t h o u t a n y
e x p l a n a t i o n . For N degrees o f freedom t h e F o u r i e r c o e f f i c i e n t s take t h e
form
-+
where t h e v e c t o r s E have N components equal t o e i t h e r zero o r one and Y
l a b e l s each o f t h e 2
N possibilities.
The Moyal m a t r i x elements a r e p a r t i c u l a r l y simple i n t h e a c t i o n
-angle r e p r e s e n t a t i o n
+ -+
The need f o r h a l f i n t e ge r a c t i o n l a b e l s L i s once a g a i n m a n i f e s t .
++
important t o compare t h i s r e s u l t w i t h t h e Moyal M a t r i x i n ( q , p )
There we found t h a t f o r purposes o f i n t e g r a t i o n t h e
It i s
space.
ma t r i x e Iemen t s
c o u l d be approzimated by a D i r a c 6 - f u n c t i o n on an i n t e r p o l a t e d
c o n s t r u c t e d from t h e two t o r i w i t h a c t i o n s
-+
torus
3
I+ and 1 ~ Here
. each element
m
i s e x a c t l y a Kronecker 6 - f u n c t i o n on t h e member o f t h e N-parameter fami l y o f t o r i w i t h actions
The t n t e r p o l a t e d t o r u s and t h e one g i v e n $ y eq.
-f
t h e case o f t h e Wigner f u n c t i o n , f o r w h i c h m =
(4.8) o n l y c o i n c i d e ' i n
Z.
l'he Wigner f u n c t i o n f o r a s t a t e corresponding t o
a
member o f
another f a m i l y o f t o r i w i t h a c t i o n f u n c t i o n d e f i n e d by
i s o b t a i n e d from the Weyl t r a n s f o r m o f
I J I > ~ ~ /6
where
-+ -+
If the t o r i w i t h actions variables
= constant are close t o . the
-+3+
-+
pert o r i I(q,p) = c o n s t a n t t h e I($,?) surfaces w i l l be open s u r f a c e s
+.
i o d i c i n 9, as shown i n Fig. 6, f o r one degree
o f freedom.
The r e s u l t
i s again g i v e n by t h e a n t i t o r u s o r B e r r y chord c o n s t r u c t i o n ,
whith the
r e s t r i c t i o n o f e v a l u a t i o n p o i n t s t o those w i t h a c t i o n v a r i a b l e s
by eq.(4.8).
Such general Wigner f u n c t i o n s wi I 1 t h e r e f o r e be
by f r i n g e s r i s i n g approximately t o a Di r a c
given
decorated
6 - f u n c t i o n on t h e t o r u s , as
a l r e a d y noted by ~ e r r and
~ ' decaying e x p o n e n t i a l l y beyond a g i v e n I'Ima
and I < ( l m i n>O)
.
The wave i n t e n s i t y i s g i v e n by eq.
1.31t.1'
(4.6)
as t h e sum
=
I c o u l d al!jo p r e s e n t t h e Moyal M a t r i x elements corresponding
to
such
o t h e r f a m i l i e s o f t o r i , b u t as can be seen from t h e Wigner f u n c t i o n , t h e
advantage o f working I n t h e a c t i o n - a n g l e r e p r e s e n t a t i o n i s
only
felt
when d e a l i n g w i t h v a r i a b l e s s p e c l f i c t o t h e p a r t i c u l a r f a m i l y o f t o r i .
Fig.6
-
The o n l y d i f f e r e n c e between
t h e Berry
chord c o n s t r i ~ c t i o ni n a c t i o n - a n g 1 e v a r i a b l e s ,
w i t h respect t o t h e o r i g i n a l phase, i s t h a t the
t o r i may be open p e r i o d i c curves.
Assuming t h e s e m i c l a s s i c a l equivalente o f t h e Weyl t r a n s f o r m o f
-+
an observable A($
I~)
wi t h t h e corresponding c l a s s i c a l f u n c t i o n AJ+
I~
,),
to
f o l l o w l n g t h e arguments o f s e c t i o n 2, we a r e now i n a p o s i t i o n
-+
M de-
r i v e a simple expression f o r t h e m a t r i x elements. As i n t h e case o f eq.
(1.2) we can w r i t e
=
r
-f
dê d8 <Z16$/2>
<8+;
lbl8 -
8 '8-2d
2>
-f
Im>
(4.12)
-v
where the i n t e g r a t i o n i s over one u n l t c e l l i n t h e
s p a c e as show
in
Fig.
7
foronedegreeof
F o u r i e r s e r i e s w i t h c o e f f i c i e n t s g i v e n by eq.
periodic
freedom. So,
(4.6),
(x,,Cf,)
from
the
-
Fig.7
In the change of variable
(4.12) the domain of integration
the shaded rectangle unit cell
in eq.
becomes
instead of
the square.
we obtain
The greater ,intricacy of this formula compared with eq.(l .2) now disappears wheii we use the expl icit action-angle Moyal Matrix elements
(4.7)
T h i s formula has been presented
before
~ l c h a r d s 'as
~ a g e n e r a l i z a t i o n o f t h e i r w r k on
by
Percival
and
the Heisenberg c o r r e -
spondence p r i n c l p l e f o r systems w i t h one-degree o f freedom. The p r e s e n t
d e r i v a t i o n r e l i e s e x c l u s i v e l y on t h e s e m i c l a s s i c a l v a l i d i t y o f workiny
d i r e c t l y w i t h ã c t i o n a n g l e v a r i a b l e s i n quantum mechanics.
5. CONCLUSION
I have deduced two d i f f e r e n t expressions f o r t h e
SC
matr i x
elements o f an observable w i t h r e s p e c t t o t h e e i g e n s t a t e
o f a classi-
(;,;I
phase space
c a l l y i n t e g r a b l e Hami lt o n i a n . Working i n t h e o r i g i n a l
t h e approximate m a t r i x elements
w i t h eq. (1.2),
<ZJAJ~~>
3
a r e g i v e n by eq. (3.17) t o g e t h e r
-+ -+
which reduces t o an i n t e g r a l o f A ( q , p ) o v e r t h e t o r u s
i n t e r p o l a t e d between t h a t w i t h a c t i o n s
-+
3
I-+
m and t h e one w i t h a c t i o n s
I,$ The o t h e r , a simple F o u r i e r i n t e g r a l (4.15),
working d i r e c t l y i n a c t l o n - a n g l e v a r i a b l e r
-
it
is
is
obtained
exact
by
w i t t i i n the
assumpt ion of SC v a l i d i t y o f t h i s r e p r e s e n t a t i o n .
I t i s easy t o see t h a t as
i n c i d e . Moreover,
if
Z-m
w i l l be negl i g i b l e , as A
+;I(Z~,~,)
of
Fz I
;t h e
-+
two a p p r o x i m a t i o n r co-
has any l a r g e component t h e F o u r i e r
integral
3
iis
arsumed t o be a r r m o t l i f u n c t i o n
ij. S i nce s e m i c l a s s i c a l y t h e q u a n t i z a t i o n c o n d i t i o n
tees t h a t t h e a b s o l u t e v a l u e o f each component
( 2.5)
guaran-
t h e dominant m a t r i x elements a r e nor: a f f e c t e d by t h e c h o i c e o f a p p r o x i mation. The b e s t way t o c a l c u l a t e any p r o p e r t y which
m a t r i x as a whole, such as t h e eigenvalues,
depends o n t h e
i s thus t o use eq. (4.15).
l'ha above statement i s v a l i d i n t h e s t r i c t l i m i t 6
O.
When
d e a l i n g w i t h a l o o s e l y d e f i n a b l e ' s m a l l s e m i c l a s s i c a l pãrameter'
other
f a c t o r s may have t o be taken i n t o account. The most
geometry o f t h e b a s i s t o r i .
-i
o bv ious
is
t he
I n t h e s i m p l e s t case where t h e s e a r e t h e
i n v a r i a n t s u r f a c e s o f a c l a s s i c a l harmonic o s c i l l a t o r o f N degrees
freedom,
tor l
t.he i n t e r p o l a t e d t o r u s always c o i n c i d e s w i t h one o f t h e
.
The d i f f e r e n c e s
$-x
of
basis
are therefore not the only r e l e v a n t p a r -
ameters, b u t a l s o t h e l e s s r e a d i l y q u a n t i f i a b l e n o n l i n e a r i t y o r c o n v o l utedness of t h e t o r i . For one degree o f freedom t h e l e s s
c u r v a t u r e o f t h e b a s i s of c l o s e d curves, t h e c l o s e r
variable
the
wi 1 1 t h e closed
c u r v e o f average a c t i o n approximate t h e i n t e r p o l a t e d c 1 o s e d c u r v e o v e r
which t h e i n t e g r a l should be c a r r i e d o " t .
A r b i t r a r y canonical t r a n s f o r m -
a t i o n s , s i i c h a s those producing I w h o r l s a n d t e n d r i l l s l
in
closed
w i l l s e v e r e l y d i s t o r t t h e i n t e r p o l a t e d t o r u s , as
shown i n
F i g . 8, so t h a t t h e d i r e c t a c t i o n - a n g l e t h e o r y w i l l no longer be v a l i d .
ia)
Fig.8
-
A v e r y non-1 i n e a r canonical t r a n s f o r m a t l o n
(b)
mey r n i s p l a c e con-
s i d e r a b l y t h e i n t e r p o l a t e d t o r u s from t h e t o r u s w i t h t h e average action.
a) For tht! harmonic o s c i l l a t o r they have t h e same shape. b) For a
i l y o f "convolutedl nested curves t h e i n t e r p o l a t e d t o r u s may
t e r s e c t one o f t h e b a s i s t o r i .
even
famin-
Even i n t h e absence o f gross d i s t o r t i o n s , we must be much more
c a r e f u l about f e a t u r e s which depend on a s i n g l e m a t r i x element, such as
be
t r a n s i t i o n r a t e s L 3 . A t r a n s i t i o n r a t e between two s t a t e s may
v i d u a l l y measurable, even i f i t i s n e g l i g i b l e w i t h r e s p e c t t o
tween o t h e r s t a t e s .
I n such instances i t i s t h e r e l a t i v e
indi-
t h a t be-
difference
between t h e r e s u l t s i n sect ions 3 and 4 which may r e q u i r e t h e use o f t h e
former more compl i c a t e d formulae. I n extreme cases when S C t h e o r y i s
extended t o areas o f doubful v a l i d i t y , i t may become
c l u d e i n eq. (3.17)
a i 1 t h e surfaces d e f ined by eq.
necessary t o
(3.16).
in-
Beyond t h a t
a11 6 - f u n c t i o n approximations break down and f u l l u n i f o r m approximations
t o t h e Moyal M a t r i x must be used i n t h e fundamental formula (1.2).
The t r a n s f o r m a t i o n t o a c t i o n - a n g l e v a r i a b l e s
has
l o n g been
known t o s i m p l i f y SC approximations t o quantum mechanics, i n analogy t o
i t s e f f e c t i n c l a s s i c a l mechanics.
I n t h e present case
the
parallel
development o f an untransformed t h e o r y has v i t i a t e d t h i s p r a c t i c e , as a
r u l e , w h i l e s u p c l y i n g t h e necessary c o n d i t i o n s o f v a l i d i t y .
I thank D r . T.B.
Smith f o r : o r i g i n a l l y drawing my a t t b n t i o n
to
t h e problems discussed i n t h i s w r k .
REFERENCES
1 . M.V.Berry,
2. A.M.
P h i l . Trans. Roy. Soc. 287,
Ozorio de Almeida and J.H.Hannay,
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237 (1977).
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138,
115(1982).
745, 100 (1983).
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Shewell, Am. J. Phys. 27,
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U.S.
8 . M.J.
45,
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Stegun,
N a t i o n a l Bureau o f Standards
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A.M.
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Resumo
Os elementos de m a t r i z de um operador observãvel, com r e s p e i t o
a uma base de autoestados de uma Hamiltoniana classicamente i n t e g r á v e l ,
são deduzidos na aproximação semicláss i c a , usando a representação de
Wigner-Weyl no espaço de fases comum e em v a r i á v e i s de ângulo e ação.
0s r e s u l t a d o s d i f e r e m pouco para propriedades que dependem da m a t r i z como um todo, t a l como os a u t o v a l o r e s , embora sejam mais simples de c a l c u l a r com as v a r i á v e i s de ângulo e ação. A d i f e r e n ç a r e l a t i v a para e l e mentos de m a t r i z i s o l a d o s pode ser importante. A t e o r i a r e c a i em aproximações semiclássicas da m a t r i z de Moyal ou função de Wigner c r u z a d a ,
que g e n e r a l i z a t r a b a l h o a n t e r i o r sobre a função de Wigner para estados
puros, assim como a sua i n t e r p c e t a ç a o em t e r m s da geometria dos t o r o s
i n v a r i a n t e s de sistemas i n t e g r a v e i s . Discute- Se tambêm a e q u i v a l ê n c i a
semiclássica e n t r e a transformada de Weyl de um operador e a f u n ç ã o
c l á s s i c a correspondente.