Quantum Chaos?
Benjamin Collar
Introduction
The term science is generally used in reference to a specific
field of study, such as chemistry, biology, or physics. What one
often overlooks is that all sciences have a similar purpose: to
accumulate knowledge, systematize that knowledge, and formulate
general truths that allow the scientist to make predictions about
the world. General science is comprised of separate levels of
specific studies, and the "enterprise of science involves
investigating the laws at all levels, while also worki ng, from
the top down and the bottom up, to build staircases between
them."1 Each level of science is built upon the
previous level and some correlation is necessary between each
level in order to fully describe the world.
Because everything in the universe is comprised of atomic and
sub-atomic particles, and quantum mechanics describes their
interactions, quantum mechanics is the base level of science; it
is the framework into which all other sciences must fit.2
The level directly above quantum mechanics is classical
mechanics. The correspondence principle, a set of equations that
translate classical systems to their quantum counterparts,
provides a perfect example of a bridge between two levels of
science. But there exist some problems with the correspondence
principle, one of which is the existence of classically chaotic
systems; chaos cannot be translated into quantum mechanics
easily. Some kind of reconciliation for chaotic systems with
quantum mechanics is imperative for a complete understanding of
the universe.
Classically Chaotic Systems
Chaotic systems have been discovered over the entire spectrum
of science, excluding quantum mechanics. All chaotic systems have
three elements in common: determinism, sensitivity, and order.
The term determinism means that the system has some equations,
often simple, that regulate the system's overall behavior. The
Mandelbrot set, discovered in 1979 by Benoit Mandelbrot,3
offers an exquisite example of this principle (see figure A). The
set is created by one iterative equation graphed on the complex
plane, z --> z2 + c,
where z begins at 0 and c is the complex number
corresponding to the point being tested.4 The point is
an element of the set if the equation converges. It is also
possible to distinguish between degrees of convergence. For
instance, the program graphing the set may be told to color the
point blue if it converges by two or purple if it converges by
three. The amazing complexity of the set, despite its blatant
simplicity, is revealed when one magnifies the set. For example,
if the set magnification point is at the junction between the two
major spheres, the resulting graph would be one called the Julia
set, reminiscent of sea-horse tails. Increasing the magnification
again will reveal more Julia sets--similar in shape but never
repeating. At some point, after repeated magnification, one will
again find the Mandelbrot set, but it will not be identical to
the initial image. It is apparent that infinitely complex syste
ms can be determined by simple, compact equations.
Michael Barnsley developed a technique called the "chaos
game,"5 which further elucidates the principle of
determinism in chaotic systems. A set of rules that will create a
graph is defined, and simple probabilities determine which rule
will apply to any specific point. For example: 1. Move three
inches north. 2. Move thirty percent closer to the center. After
omitting the first fifty points, in order to equalize the chance
distribution, a shape begins to emerge. The complexity of the
resulting graph is directly proportional to the number of rules;
if one increases the number of rules of the system the complexity
rises accordingly. Barnsley created a fern with a set of only 28
equations with randomly selected starting points (see figure B).
Furthermore, Barnsley realized that the creation process was
reversible: any shape that had repetitive qualities, varying only
on scale, could be represented by a set of simple equations.
The second principle of chaotic systems is sensitivity to
initial conditions. Small, seemingly insignificant changes of
initial parameters will lead to an enormously different and
unpredictable final system. Richard Feynman elucidated this
concept in the 1960's:6
"It's true classically that if we know the position and
velocity of every particle in the world, or in a box of gas, we
could predict exactly what would happen. And therefore the
classical world is deterministic. Suppose, however, that we have
a finite accuracy and do not know exactly where just one atom is,
say to one part in a billion. Then as it goes along it hits
another atom, and because we did not know the position better
than one part in a billion, we find an even larger error in the
position after the collision. And that is amplified, of course,
in the next collision, so that if we start with only a tiny error
it rapidly magnified to a very great uncertainty....Speaking more
precisely, given an arbitrary accuracy, no matter how precise,
one c an find a time long enough that we cannot make predictions
valid for that long a time....The time goes, in fact, only
logarithmically with the error, and it turns out that in only a
very, very tiny time we lose all our information."
During the early 1960's, Edward Lorenz inadvertently provided
statistical proof for Feynman's thought experiment.7
Using a Royal McBee computer he created a fairly accurate
simplified weather system from twelve recursive differential
equations. The computer generated two types of data: numerical
and graphical. It calculated the data internally to six decimal
places, but the output was only accurate to three. After running
the program for some time Lorenz noticed a set of data to which
he was particularly inclined and decided to enter the data by
hand in order to see the system again. But the resulting system,
after a short time, astonished him--it was measurably different
than the original system and also completely unpredictable (see
figure C). Lorenz's weather system provided the first concrete
evidence of the importance of initial conditions to the final
outcome of the system.
The third principle of chaos, that despite the amount of
apparent disorder of any system there is some order-determining
element, was realized by Heinz-Otto Peitgen and Peter H. Richter8
in their mathematical study of the magnetic transition of a
system. They graphed the variations in the arrangement of the
atoms in a transitional system in phase space. The area between
the two types of systems--magnetic or non-magnetic--was their
focus. Often areas between system types are chaotic, with order
on both sides. Repeated magnification of the area revealed
increasingly chaotic figures. The determining factor of the
chaotic region, however, is the interesting aspect: at the
greatest magnification, Peitgen and Richter found the Mandelbrot
set.
The phase space graphics Peitgen and Richter used are often
employed to help translate systems from one level of science to
another. Lorenz offered another discovery to the science of
complexity: the strange attractor. The idea of attractors when
Lorenz published his findings was nothing new. Systems graphed in
phase space often fall to attractors, the final state of the
system. For instance, the phase space diagram of a pendulum (see
figure D) spirals towards the center because of the dissipati ve
force of friction. If one releases the pendulum it will rise to
almost the same level on the opposite side. It returns, again not
as high, and it repeats. The final state of the system is
motionless--the system has arrived at the attractor.
The phase space diagrams of chaotic systems, however, do not
terminate at an attractor. The phase space graph of the solution
of Lorenz's weather equations has two attractors, and the lines
never arrive at either, but they do approach each (see figure E).
Logically it would seem that the lines at some point would
intersect. But because of properties of differential equations
the lines never do. If they did, the system would repeat itself
although a chaotic system, such as weather, never does. This
leads to the concept of infinite length within a finite area. A
visual representation of this is provided by fractals. One of the
first fractals was developed by Helge von Koch in 1904 (see
figure F).9 The Koch curve, or Koch snowflake, is
created by dividing each side of an equilateral triangle into
thirds, placing a smaller equilateral triangle on the middle
third of each side, and repeating the process infinitely. By
tracing the perimeter of the snowflake, one will approach
infinity--within a fini te area. This allows for infinite
variation; a chaotic system graphed in phase space varies
infinitely because it never repeats itself, but stays within a
finite area because it is always self-similar.
An important property of fractals like the Koch curve and the
Mandelbrot set is similarity across scale; by magnifying the
system, almost-repeating images are found. Physical realization
of fractals and similarity across scale was discovered by
Kadanoff Libchaber in the 1977.10 Libchaber created a
very small cell of liquid helium that allowed very few degrees of
freedom in order to study convection currents. Two temperature
probes were attached to the top of the cell and a heat source
surrounded it. At only a few degrees above absolute zero, liquid
helium easily changes to gas with small heat differences. The
outer molecules evaporate first, causing rings of gas. Atoms are
heated and rise but, upon energy loss when they reach the top,
they cool and fall. Chaotic motion becomes apparent as the
temperature increases. Uneven walls of the cell cause differences
in the position of each helium atom. The differences in position
lead to small irregularities in the structure of the rings which
become more pronounced as the energy increases. At some
temperature the loops will double. Increased energy will create
more smaller-scaled loops and when the spectrum of the system is
graphed, spikes occur, corresponding to the changes in the number
and size of the rings.
The Transition to Quantum Mechanics
Bridges between classical and quantum mechanics are necessary
to give an accurate description of the world. One conflict
between classical and quantum mechanics that must be reconciled
is that energy is a continuum in classical physics while, in most
cases, energy comes in packets and is represented by spectra in
quantum physics. When translating, the classical system is
graphed in phase space and the length of the periodic orbits of
the system correspond to the quantum spectral lines. The
correspondence principle often suffices in the task of
translating the systems, but problems arise as the systems become
chaotic. Quantum mechanics calculates the probabilities of
precise histories with no possibility of the indeterminacy
present in chaotic systems:
"However, the difficulty, and a very deep one, arises
form the fact that the former [quantum mechanics] is commonly
accepted to be the universal theory, particularly, comprising the
latter [classical mechanics] as the limiting case. Hence, the
correspondence principle which requires the translation from
quantum to classical mechanics in all cases including the
dynamical chaos. Thus, there must exist a sort of quantum
chaos!"11 Recently a new formula, the Selberg
Trace formula, has been used; it "relates the lengths of
periodic orbits as represented in 'phase space'...to combinations
of quantum energy levels. Using the trace formula, physicists can
convert a system's chaotic behavior in the classical world into
predictions about the statistics of its quantum-mechanical
spectrum."12
Martin Gutzwiller provides an illustration13 of the
difficulties of translation between the two mechanics.
Gutzwiller's example of what he terms "mild chaos" is
centered on the sun-moon-earth system. Most moons in the solar
system are easily described by Kepler's equations for elliptical
motion. Our moon is the exception--it is strongly perturbed by
the gravity of the sun and the earth. On the surface the system
seems simple: it has only three degrees of freedom and one
adiabatically slow perturbation.
The general equations for the motion and position of the moon
have been known since ancient times, allowing for the predictions
of full and new moons and eclipses. The equations of the moon in
the XYZ plane is
x = a0 cos alpha0
+ a1 cos alpha1 + ...
y = a0 sin
alpha0 + a1 sin alpha1
+ ...
z = c0 sin gamma0
+ c0 sin gamma1 + ...
The coefficients ai and ci
depend on the 6 initial conditions of the system. The angular
variables alpha and gamma are all of the same
form, alpha = omega t + theta where
both the frequency omega and the phase theta
are linear combinations with integer coefficients of the four
basic frequencies and their associated phases.14
Unfortunately, new terms for each coordinate are being
continuously discovered and it has been found that the coordinate
equations do not converge, so the conditions that determine the
position of the moon neither are nor ever will be fully known.
This seems to suggest chaos, but Gutzwiller goes on to explain a
more subtle form of complexity:
"The requirements of modern observations demand a
precision of 10-10 in the coordinates of the moon.
Occulations of fixed stars by the moon can be timed with the
precision of the apparent star diameter, ~10-5 arc
seconds. Echoes of light pulses from the reflectors on the
surface of the moon give distances correct to several
centimeters. But at this point, one enters the realm of slight
chaos.
"First there is the problem of proliferating small terms,
as discussed by Gutzwiller. The root of the sum of squares for
all coefficients smaller than 10-10 turns out to be
larger than 10-9. It becomes therefore necessary to
compute all terms down to a precision of 10-12. Since
even such small coefficients should be known with 2 decimals
accuracy, one has to go all the way to 10-14 just to
manage the noise from the small terms in the expansion."15
He goes on to say that, regardless of whether the moon's motion
is chaotic or not, the determinism in its equations gets
lost--the system is not as simple as it appeared. Gutzwiller's
"mild chaos" is a prime example of the difficulty of
finding equivalent equations in quantum and classical mechanics.
It points out our inability as scientists to be precise enough in
our measurements to make accurate predictions.
One approach to solving the chaos-quantum mechanics problem is
the use of random matrices. If the system is chaotic, the chaos
must be represented somehow in the Hamiltonian of the system. The
classical Hamiltonian can be regarded as a matrix and the
elements within the matrix will be determined by the physical
models of the system. In a chaotic system the elements are
exceedingly hard to determine, but mathematical methods permit
some elements to be replaced with random entries, allowing one to
stud y the statistical properties of the matrices.16
As a result of this theory two sets of matrices have been
developed, the Gaussian orthogonal ensemble (GOE) and the
Gaussian unitary ensemble (GUE), both of which only apply to
chaotic systems. Non-chaotic systems have statistical properties
determined by the Poisson ensemble. When a non-chaotic system
becomes chaotic, there is a translation formula that relates
Poisson data to GOE data.
Two classically chaotic systems that are often studied are the
spectrum of actions of a Rydberg atom in a uniform field and
compound nuclei spectrum. The first system is soluble only when
the strength of the magnetic field is zero or infinity.17
As the field increases the phase space diagram reveals an
increasing number of chaotic packets. At some field strength the
entire graph becomes chaotic. The spectrum of the system is given
in figure G, and one notices an increased level of repulsion as
the field increases. The spectral lines fluctuate randomly.
Because this in a non-integrable system, there is a mathematical
transition between Poisson and GOE. The spacing and rigidity of
the spectrum is compared to these in figure H.
Fluctuations in the spectrum of compound nuclei are studied by
firing a neutron at the nuclei and measuring the resulting
energy. In a 166Er atom the distribution of the
spectral lines are smooth for about 108 levels, but at 109 the
lines begin to randomly fluctuate. Both Poisson and GOE
statistics are shown in figure I.
Another possible solution to the translation problem is the
Riemann Zeta function:
zeta(s) = 1 + 1/2s
+ 1/3s + 1/4s + ... + 1/ns
+ ...
and equivalently:
zeta(s) = 1/(1-2-s)(1-3-s)(1-5-s)(1-7-s)
... (1-p-s) ...
where p is a prime number. Riemann discovered that this
equation can also be written as a function of zeroes:
zeta(s) = f(s)(1-s/zeta(1))(1-s/zeta(2))(1-s/zeta(3))
... (1-s/zeta(n)) ...
where zeta(n) are the zeroes of the function
and f(s) is a simple fudge factor. Riemann
realized that this function, after finding the zeroes, will lead
to precise results of the location of prime numbers.
Physicists have discovered many potential uses for the Riemann
Zeta function. Certain aspects of the equation seem to be very
similar to values in quantum mechanics. First of all, the
logarithm of the primes given by the function coincide with the
lengths of periodic orbits of some classical systems. Second, the
equivalency of the two infinite products looks suspiciously like
the Selberg trace formula. Finally, the zeroes of the function
correspond to the density of the spectral lines in the quantum
system. With the advent of computers the zeroes of the function
have become increasingly easy to compute--thus making the Riemann
Zeta function an easily calculable translating scheme for chaotic
systems (see figure J).18
Plotting the Zeta zeroes against the true spectrum of complex
nuclei (see figure K) has increased physicist's resolve in
thinking that the Riemann Zeta function provides an accurate
description of the chaotic system. Similarly, there is a
mathematical translation between the Riemann Zeta function and
GUE (figure L), which in some systems, like the Rydberg atom in a
magnetic field, can be translated to GOE (figure M).
Conclusion
The ultimate goal of science is to correctly describe the
world in all of its various forms. Bridges are created with each
new discovery; the search for more bridges continues. Links to
concepts that are not often directly associated with the world,
like mathematics, are being discovered too. The random matrix
theories and the Riemann Zeta function are methods to not only
bridge the gap between mathematics and physical reality, but also
to reconcile the chaotic systems of the natural world with the sy
stem scientists consider to be the true description of the world.
1 Gell-Mann, Murray, The Quark and the Jaguar,
(New York: W.H. Freeman and Co., 1996) 112.
2 Gell-Mann, 6.
3 Gleick, James, Chaos: Making a New Science,
(New York: Penguin Books, 1987) 223.
4 Gleick, 227.
5 Gleick, 237.
6 Richard Feynman, The Feynman Lectures on
Physics, Vol. 3 (Massachusetts: Addison-Wesley Publishing
Company, 1965) 2-9.
7 Gleick, 16.
8 Gleick, 230.
9 Gleick, 98.
10 Gleick, 202.
11 W. Dieter Heiss ed., Chaos and Quantum Chaos
(Berlin: Springer-Verlag 1992) 5.
12 Barry Cipra, "Prime Formula Weds Number
Theory and Quantum Mechanics," Science 274 (1996):
2014.
13 Casati, Giulio, Chaotic Behavior in Quantum
Systems, (New York: Plenum Press 1983) 153.
14 Casati, 155.
15 Casati, 156.
16 Heiss, 229.
17 Seligman, T.H. and Nishioka, H., eds., Quantum
Chaos and Statistical Nuclear Physics, (New York:
Springer-Verlag 1986) 34.
18 Cipra, 2015.
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