Volume 90A, number 3
PHYSICS LETTERS
28 June 1982
ON CLASSICAL AND QUANTUM INTEGRABILITY
H. JUrgen KORSCH
Fachbereich Physik, Universitàt Kaiserslautern, D-6 750 Kaiserslau tern, Fed. Rep. Germany
Received 24 March 1982
The definition of integrabiity in classical and quantum mechanics is discussed. It is shown quite generally that a classically integrable system is also integrable in the quantum sense, and vice versa.
One of the results of the exciting developments of
classical mechanics during the last decade is that
contrary to common folklore comparatively
“simple” classical systems with hamiltonians
—
—
N
H(p, q)
=
n1 ~
2
+ V(q
1,
...,
q~)
time-dependent one-dimensional harmonic oscillator.
An important implication of the integrability of a systern is the restriction of the trajectories to N-dimensional tori in phase space.
All classically integrable systems known up to now
have been proved to be also integrable in quantum
tm’ m
mechanics, i.e., a set of Hilbert space operators
= 1
N can be constructed, which are constants of the
motion [6] satisfying
[I~ F I = 0
m, n = 1,.. .,N,
(3)
.. .
can exhibit an infinitely complicated chaotic (or “ergodic”) behaviour (see, for example, the review artides [1—4]).The same behaviour also occurs for time
dependent systems with N = 1. The most famous of
these systems is the time independent two dimensional Hénon—Heiles system [5J.Only in very rare cases
which fill, however, the majority of the textbooks
of classical mechanics do we find a “smooth” behaviour of classical orbits in phase space. These systems are called “integrable”. An integrable system in
classical mechanics is one, where N independent singlevalued first integrals of the motion Fm ~ q), m = 1,
.N, exist, which are in involution, i.e. they satisfy
aFm aF
3Fm öFn
where [
is the commutator bracket.
Quantum integrability implies, for instance, that
common eigenstates of the F~,n = 1,
N can be
found. These eigenstates correspond semiclassically
to those tori in the classical phase space which satisfy
Bohr—Sommerfeld quantisation conditions [3]. The
consequences of non-integrability in quantum mechanics are much more intricate than in classical mechanics and still under discussion [7—141.
It is the aim of the present note to discuss the relationship between classical and quantum integrability,
{F~,F~}p =a~ ~ 0
0,
that is to answer the following question: Isaclassicalq
p
ly integrable system also integrable in the quantum
m n= 1
N
(2)
sense, and vice versa; does the concept of integrability
lead to the same classification of quantum and classical
where ~ ~ is the Poisson bracket. For time-indesystems? An affirmative answer to this question has
pendent hamiltonians one of the Fm can be the hamilapparently been taken for granted by some authors,
tonian itself. Examples of integrable systems are the
while others express great surprise when they find it
motion in a three-dimensional spherically potential,
is true for some particular systems under investigation.
the N-dimensional harmonic oscillator, the hydrogenic
We are not aware of any proof of the anticipated oneatom in a linear electric field, the Toda chain, and the
to-one correspondence, which is therefore given in the
following.
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
113
—
—
—~-~~“—~----
,
,
. . .,
Volume 90A, number 3
Since
the early
PHYSICS LETTERS
work by Weyl [15]
and Wigner [161
28 June 1982
ly integrable. Then we have F
1(p, q)
it has become widely known, that there is a one-to-one
correspondence between operatorsA in Hilbert space
and functions A~(p,q)on phase space
satisfying (2), so that
AFmF0 = f F~,F~}= 0,
(4)
(A very useful compilation of the properties of this
Wigner—Weyl equivalent representation can be found
in the appendix of de Groot and Suttorp’s book on
the foundations of electrodynamics [17].) The
Wigner—Weyl transformation (4) maps the quantum
commutator [A, B] onto the Moyal bracket [181
and hence
which
finally
A gives
A ~A~(p,
q)
fA.~.,BW}\l
=
.
(2/h) sin(~hA)A~(p,q)B~(p,q) (5)
with
a(A)
a(B)
a(A)
~lq
ap
ap
a(B)
(6)
acts only on the function A~and ~ ~B) on
0 the Moyal bracket
approaches the Poisson bracket
where ~
B,,~,.In the semiclassical limit h
~
BW}M
-.
~
—*
(7)
{A~,B~~p
.
The Wigner—Weyl representation maps the Lie algebra
of 1-filbert space operators with Lie bracket
,
on-
[
]
to the Lie algebra of phase-space function with Lie
bracket {
The mapping is one-to-one.
(i) Let us now first assume, that the system of interest is integrable quantum-mechanically. In the generic case the integrabiity, i.e. the existence of the operators t,~,
satisfying (3), will be independent of h.
Therefore, after transformation to the Wigner—Weyl
representation, we can take the limit of small h. This
shows, that the Wigner—Weyl equivalents F~(p, q) of
the F~provide a complete set of classical constants
of motion, which are in involution, i.e. they satisfy
(2). This does not of course exclude the possibility of
an accidental quantum integrabiity, where we have
,
}~.
[1~rn,~~ni = ann(JOGmn
,
(8)
with some functions csmn and some operators Gmn,
which satisfy ctmn (h) = 0 for the physically realized
value of h. We do not think, however, that such a situation is of much interest.
(ii) Let us now assume that the system is classical-
114
2FmFn
=
F~(p,q)
(9)
0, A3Fp~F~
= 0 and so on,
(U A2°~1FF =0 (10)
(2v + U!
m
so that the Wigner.-.-Weyl equivalents F,~of the F,~,n
= 1
N satisfy (3) and the system is integrable in
the quantum sense.
(sin ~71A)FmF
n
~
=
v=0
We have therefore shown that the concept of integrabiity is generally the same in quantum and classically mechanics.
References
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