Chapter X
Chaos as a Bridge between
Determinism and Probability in
Quantum Mechanics
Wm. C. McHarris
Departments of Chemistry and Physics/Astronomy
Michigan State University
mcharris@cem.msu.edu
X.1 Introduction
Quantum mechanics is fundamentally a probabilistic science, whereas classical mechanics is deterministic. This dichotomy has led to numerous disputes and confusion,
ranging from the Einstein-Bohr debates of the 1930’s [Einstein, Podolsky, and Rosen
1935; Bohr 1935], through attempts to establish determinism in quantum mechanics
by means of introducing “hidden variables” [de Broglie 1960, 1964; Bohm 1952], to
lengthy discussions of epistemological versus ontological interpretations of quantum
mechanics [Bohm and Hiley 1993]. Throughout most of the twentieth century the
Copenhagen interpretation of Bohr and Heisenberg has endured as the orthodox interpretation, replete with contradictions and paradoxes such as duality, the necessity for
an observer before a quantum system can attain physical meaning, and the reduction
of the wave function upon observation. The reductio ad absurdum of such paradoxes
was the example of Schrödinger’s cat [Schrödinger 1936; Gribben 1984,1995], in
which a cat inside a closed chamber remained in limbo as a linear superposition of
dead cat + live cat until an observer determined whether or not a radioactive nucleus had decayed, releasing a deadly poison.
During the last several decades, nonlinear dynamics and chaos theory have become well-enough developed that they can be used to intercede in some of these
seeming paradoxes of quantum mechanics. Since chaotic systems are fundamentally
deterministic, yet have to be treated statistically, it is worth investigating as to whether chaos can form a bridge between the determinism of classical mechanics and the
probabilistic aspects of quantum mechanics. Perhaps both Einstein and Bohr could
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Chaos and Determinism in Quantum Mechanics
have been correct in their interpretations: At heart chaos theory provides the determinism so dear to Einstein; yet for most practical purposes it reduces to the probabilities of the Copenhagen interpretation. Interestingly enough, although authors of the
earlier (not altogether successful) attempts to impose determinism on quantum mechanics did not have access to modern chaos theory, in many ways they toyed with
ideas that we now can see as arising naturally in chaotic systems.
More recently, nonlinear extensions to quantum mechanics have been promulgated by a number of authors [Weinberg 1989; Gisin 1989, 1990; Mielnik 2001; Czachor
and Doebner 2002]. Again, their efforts have been but partially successful, sometimes introducing superluminal (nonphysical) signals into multi-particle correlations.
Mielnik sums up their thinking, “…perhaps, it resists embedding into too narrow a
scheme…the nonlinear theory would be in a peculiar situation of an Orwellian
‘thoughtcrime’ confined to a language in which it cannot even be expressed…A way
out, perhaps, could be a careful revision of all traditional concepts…” Their efforts
centered around corrections and perturbations applied to traditional, linear quantum
mechanics — and in such weakly nonlinear systems chaotic behavior cannot develop!
From a different perspective, viz., a consideration of the possibility that nonlinearities can arise at the Planck length (10–33 m), has come the recent work of t’Hooft
[2003] and Krasnoholovets [2003]. However, this is more concerned with reconciling
quantum mechanics with relativity than in dealing with the (more simplistic?) imponderables encountered with the Copenhagen interpretation.
Perhaps the time is ripe to inquire into the possibility that chaos and nonlinear dynamics could be an inherent component of quantum mechanics — only heretofore not
recognized as such. In a series of papers [McHarris 2001, 2003, 2004] I have cautiously put forth the idea that certain imponderables — or paradoxes — encountered
with the Copenhagen interpretation of quantum mechanics might have parallel explanations in terms of nonlinear dynamics. This is not to be construed as any sort of
proof of the existence of chaos underlying quantum mechanics — indeed, just as
quantum mechanics itself cannot be derived, but only postulated, such a proof cannot
be forthcoming. And these ideas are still at the “from the bottom up,” even “quasiexperimental” stage: Examples must be collected and analyzed before any broad
formulations can even be considered. Nevertheless, the very existence of such parallel examples makes raising the question(s) worthwhile.
Even if eventually it becomes accepted that quantum mechanics can contain deterministic, if nonlinear components, for the most part this will have relatively little
effect on the day-to-day use of quantum mechanics. This results from the fact that
most scientists use it simply as the superb procedure — or computer formulation —
that it is and care little about the quandaries associated with its foundations. And they
would be trading one set of difficult mathematics for an equally messy set of nonlinear numerologies. The primary exception to this that I foresee lies in the field of quantum information and computing, the very field that has helped so much to reinitiate
interest in reexamining the foundations of quantum mechanics. For the possibility of
massively parallel quantum computing rests firmly on the principle of linear superposition of states, which can be manipulated simultaneously but independently. In a
quantum mechanics containing nonlinear elements, this basis might have to be reexamined.
Chaos and Determinism in Quantum Mechanics
3
X.2 Quantum Mechanical Imponderables with Nonlinear Parallels
At least seven so-called imponderables or paradoxes associated with the Copenhagen
interpretation of quantum mechanics have analogous nonlinear interpretations that
logically are quite compelling. Some of these have been investigated much more
thoroughly than others. They are summarized here in decreasing order of effort spent
in trying to fathom them.
X.2.1 The Exponential Decay Law and the Escape Group from Unimodal Maps
Radioactive decay and atomic and molecular transitions — indeed, all first-order
processes, including first-order chemical reactions — follow exponential decay laws,
exhibiting time-independent half-lives. These are often justified by analogy with
actuarial tables, such as those used by the life insurance industry. Although it is difficult, if not impossible, to determine when an individual will die, given a sufficiently
large sample, the statistical predictions become quite precise, allowing insurance
companies assured profits. By analogy, it is impossible to predict when, say, a given
radioactive nucleus will disintegrate; however, given a large enough sample of nuclei,
a statistical exponential decay law is followed very precisely.
Upon further consideration, such an analogy should not hold up. Actuarial tables
are based on complexity: There are large numbers of different people having widely
diverse causes of mortality. On the other hand, one of the fundamental premises of
quantum mechanics is that identical particles are truly identical and interchangeable;
thus, completely different statistics should apply.
A nonlinear parallel can be found in the iteration of unimodel maps in their chaotic regimes, where extreme sensitivity to initial conditions applies. Consistent with
(but not dependent on) the Uncertainty Principle, an initial, say, radioactive nuclear
state, having a finite width, can be likened to a tiny interval in initial input values.
Different individual nuclei in this state can then be represented by random initial
values selected within this interval. The final nuclear state can similarly be represented by a second interval, corresponding to possible final values. And the decay dynamics — a transition probability is at least quadratic in nature — can be considered
analogous to iterating the map. One keeps a record of the number of initial states
remaining, i.e., the number that have not escaped into the final state, after each iteration. A plot of this remaining number against the number of iterations yields an exponential decay curve.
This analog is treated in considerable detail in [McHarris 2003], where both the
quadratic and sine maps are considered. Tens of thousands of randomly-generated
initial states within intervals typically having widths of 10 –11 were followed, and this
procedure consistently generated exponential decay curves. The process of iteration
can be justified physically if one makes a correspondence with a physical process
such as the number of oscillations of, say, a nuclear dipole or the number of attempts
of an  particle at barrier penetration. Unfortunately, because of the “universality” of
chaos, it is difficult to make further mechanistic predictions, and any sort of time
series analysis is unlikely to retain significant correlations because of the enormous
difference between the laboratory and nuclear times scales. (Nevertheless, because of
the relative ease with which time-delay experiments could be performed with radioac-
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Chaos and Determinism in Quantum Mechanics
tive species, such experiments should be performed on the off chance that some faint
remnant of an attractor could be perceived.)
X.2.2. Bell’s Theorem and Nonextensive Entropy
The EPR paradox [Einstein, Podolsky, and Rosen 1935] was designed to demonstrate
that quantum mechanics was incomplete ; it involved correlations between separated
particles (Einstein’s “spooky action at a distance”), but it was cast in the form of an
abstract Gedankenexperiment. Bohm [Bohm 1951] made the EPR more specific and
less astract by considering a pair of spin-1/2 fermions, but it was Bell [Bell 1964] who
reanalyzed it into something that touched upon physical reality. There have been
various refinements and variants on Bell’s inequality, but perhaps the simplest —
designed to be experimentally friendly — is the CHSH inequality [Clauser, Horne,
Shimony, and Holt 1969], which is used as the illustration here. (Also, cf. [McHarris
2004]).
Consider the following correlation experiments on pairs of particles, using the
standard information theory cartoon characters, Alice and Bob, who are stationed at
an effectively infinite (incommunicado) distance apart. Pairs of particles having
binary properties, e.g., spin up vs spin down or perhaps horizontal vs vertical polarization, are prepared, then one particle from each pair is sent to Alice and the other to
Bob. Examples of such pairs could be two electrons or two photons.
Alice can make measurements Q or R on each of her particles, each measurement
having a possible outcome of +1 or –1. For example, Q could be a measurement of
spin with respect to a vertical axis, while R would be with respect to an oblique axis.
Similarly, Bob can make measurement S or T on each of his particles. Alice and Bob
each choose which measurement to make at random, often waiting until the particles
are already on their way in flight, thus assuring no communication between them or
with the originator of the particles. After making many pairs of measurements in
order to attain statistical significance, the two get together to compare notes. The
quantity of interest based on their measurements is
QS  RS  RT  QT  (Q  R)S  (R  Q)T.
(1)
Note the single minus sign — because Q and R independently can have the values +1
or 1, one of the terms on the right side of the equation must be 0. Either way,

QS  RS  RT  QT  2,
(2)
or in terms of probabilities, where E(QS), for example, is the mean value of the measurements for the combination QS, we come up with the CHSH inequality,

E(QS)  E(RS)  E(RT)  E(QT)  2.
(3)
This so-called “classical” derivation of a specific variant of Bell’s inequality places an
upper limit on the statistical correlations for a specific combination of products obtained by presumably
independent (and randomly chosen) measurements.
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Chaos and Determinism in Quantum Mechanics
The quantum mechanical version is obtained by starting off with the pairs of particles in the entangled Bell singlet state,
  ( 01  10 ) / 2.
(4)
The first qubit from each ket is sent to Alice, the second from each ket to Bob. Measurements are carried out as before, but on the following combinations of observables:

(Z 2  X 2 )
(Z  X 2 )
Q  Z1, R  X1, S 
, T 2
2
2
(5)
Here X and Z are the “bit flip” and “phase flip” quantum information matrices, corresponding to the Pauli 1 and 3 spin matrices. It can be shown that the expectation
values of the pairs QS,
 RS, and RT are all +1/2, while that of QT is 1/2. This
leads to the quantum mechanical analogy to the CHSH inequality,
QS  RS  RT  QT  2 2.
(6)
Thus, quantum mechanics, within the framework of entangled states, predicts a possibly larger statistical correlation than was allowed by the so-called classical inequality.
Bell’s theorem predicts
 that classical systems will obey such inequalities, while quantum systems might violate them under the right circumstances.
During the last several decades, several dozen “Bell-type” experiments have been
performed [Bertlmann and Zeilinger 2003], and they have consistently violated the
inequalities. Quantum mechanics wins, classical mechanics loses! As with most
ideas connected with quantum mechanics, interpretations vary — but most interpretations involve the elimination of “local reality.” Two isolated, far-apart but entangled
particles have some sort of influence on each other. A down-to-earth experimental
example of this might be the following: Two electrons are emitted in a spin-singlet
state. Their individual spin directions are unknown (undefined according to the Copenhagen interpretation), but they must be opposite. When Alice arbitrarily measures
the direction of her electron, say, with respect to a z axis and gets , this information
is instantaneously conveyed to Bob’s electron, whose wave function reduces to 
with respect to this same axis. Einstein’s “spooky” — and superluminal — action at a
distance is real!
But wait a minute. Is this really a contest between quantum vs. classical mechnics, or is it between correlated vs. uncorrelated statistics. In the so-called classical
derivation, the particles were presumably prepared in correlated pairs, but these correlations were then tacitly ignored, while the quantum mechanical entangled pairs necessarily retained the highest correlations. And correlated statistics are known to exist
in nonlinear systems:
The codification of correlated statistics was introduced by Tsallis and his coworkers [Tsallis 1988, Curado and Tsallis 1991], when they formulated so-called
“nonextensive” (meaning nonadditive) thermodynamics. Correlations in classical
systems result in a generalized entropy,
Chaos and Determinism in Quantum Mechanics
6
W
Sq  (1

piq ) /(q 1).
(7)
i1
Here the phase space has been divided into W cells of equal measure, with pi the
probability of being in cell i. For the exponent q (termed the “entropic index”) having
a value of 1, the generalized
entropy reduces to the standard Boltzmann entropy,

W

S1  
pi ln pi .
(8)
i1
As q varies from 1, the deviation from standard distributions becomes greater, with
“long-range” correlations becoming greater. When such correlations are present, the
entropy becomes nonextensive,
and the entropy of the total system becomes

Sq (A  B)
k

Sq (A)
k

Sq (B)
k
 (1 q)
Sq (A)Sq (B)
k2
.
(9)
When q < 1, the entropy of the combined system is greater (superextensive) than the
sum of its parts, and when q > 1, it is less (subextensive) than the sum of its parts.
This concept has found
 widespread applications in classical systems, ranging from
winds velocity distributions in tornadoes to the energy distributions of cosmic rays
[i.a., Gell-Mann and Tsallis 2004].
What is pertinent is that systems “at the edge of quantum chaos” have recently
been studied [Weinstein, Tsallis, and Lloyd 2004], and both for the quantum kicked
top and for the logistic map [Baldovin and Robledo 2002; Borges et al. 2002] values
of q > 1 could be applied. [One has to be careful here with quantitative interpretations because these values were derived within the context of standard (linear) quantum mechanics with “quantum chaos”; however, the crossover into classical chaos is
similar, and it seems that both systems exhibit long-range correlations.] For the logistic equation these groups found that a value of q  2 seemed reasonable.
From the above it seems rather clear that nonlinear classical systems can indeed
exhibit correlations in which “long-range” correlations play an important role. (This
does not necessarily mean long-range forces or “action at a distance,” as has been
known for a long time from the behavior of cellular automata and self-evolving systems.) Thus, the “classical” derivation of the CHSH inequality — and most if not all
of the other guises that Bell’s inequality takes on — is suspect. Classical systems can
easily involve correlated statistics, which raises the apparent upper limit of inequalities such as Eqn. (3). With a value of q in the vicinity of 2, one could easily obtain
something closer to an exponential rather than a Gaussian distribution. In other
words, Bell’s inequality is moot in ruling out the existence of local reality in quantum
mechanics.
Chaos and Determinism in Quantum Mechanics
7
X.2.3 Other Possible Nonlinear Parallels with Quantum Mechanics
Other parallels between nonlinear dynamics and quantum mechanics should also be
considered. These have been less extensively investigated, so are only listed here as a
prod to one’s imagination.
•Attractors and innate quantization. Nonlinear systems have preferred modes
of oscillation, independent of boundary conditions or external influence. Many deterministic but nonlinear classical systems obey eigenvalue equations, i.e., they are
quantized innately without having to invoke such artifices as wave interference.
•Spontaneous symmetry breaking — parity nonconservation. Nonlinear systems can spontaneously break both temporal and spatial symmetry. Odd iterators, for
example, exhibit this property. And practical applications include chemical reactions
and the separation of powders in nonlinear tumblers [cf. McHarris 2004]. Might this
have some bearing on the nonconservation of parity in weak interactions?
•Decoherence and the destruction of KAM tori. Decoherence and the reduction
of wave functions are among the more intriguing paradoxes associated with the Copenhagen interpretation. The nonlinear alternative would be for an observer to perturb, ever so slightly, a knife-edge equilibrium in a Hamiltonian system, perhaps
resulting in its orderly breakdown of KAM tori. This would remove the observer
from having to be an integral part of the system. Weinberg [Weinberg 1989] touches
on ideas similar to this.
•Diffraction — the existence of order in chaos. A possible mode of attack on
problems such as the double-slit experiment would be to examine the intricate mixing
of windows of order with regions of chaos in many chaotic regimes. Windows of
order would correspond to “constructive interference,” whereas chaos would correspond to “destructive interference.” It is well known that two first-order differential
equations can be combined to produce a single second-order equation. Thus, two
identical slits could possibly lead to chaotic behavior, but closing one slit (or doing
anything else to remove one of the equations) would do away with the possibility of
chaos.
•Barrier penetration. Again, many nonlinear systems exhibit a type of barrier
penetration, e.g., water waves crossing a barrier. Chap. 7 of [Hey & Walters 2003]
gives an intriguing overview of this field, which might have deeper implications for
quantum mechanics than they realized.
X.3. Conclusion
This paper is intended more to be thought provoking than to proclaim definite conclusions. Nevertheless, I hope to have raised serious questions about our current, orthodox thinking about quantum mechanics. Two of the paradoxes of quantum mechanics
have been shown to have reasonably quantitative alternative explanations in nonlinear, possibly chaotic dynamics, while five others simply raise reasonable speculations.
Chaos applies to almost every other discipline in nature, so why is quantum mechanics exempt from nature’s preferred feedback and nonlinearities? Perhaps, as Mielnik
suggested, we have unconsciously been using a scientific linear “newspeak,” which
has prevented us from expressing any nonlinear “thoughtcrimes.” The consequences
of nonlinearities and determinism underlying quantum mechanics should shock us,
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Chaos and Determinism in Quantum Mechanics
but then it should make us think again — and investigate the situation without wearing (linear) blinders. After all, perhaps Einstein and Bohr were both right — but
couldn’t realize it at the time.
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