Several of the imponderables associated with the Copenhagen interpretation of quantum mechanics have alternative, parallel explanations in terms of nonlinear dynamics and chaos. For example, an exponential law for processes such as radioactive decay can be obtained by iterating unimodal maps operating in chaotic domains, utilizing their extreme sensitivity to initial conditions. Such alternative explanations raise the question of the possibility of underlying nonlinearities in quantum mechanics. This is a ‘from the bottom up’, empirical approach to the question and will be compared with other, more theoretical, ‘from the top down’ investigations into the possibility of nonlinear extensions to quantum mechanics. Nonlinear dynamics, although deterministic, in its chaotic extreme provides only statistical information about the outcomes of state-space orbits. Thus in principle it provides the determinism so dear to Einstein; yet in practice it exhibits the probabilistic behaviour of Bohr. Perhaps Einstein and Bohr could both have been right.
Several of the imponderables associated with the Copenhagen interpretation of quantum mechanics have alternative, parallel explanations in terms of nonlinear dynamics and chaos. For example, an exponential law for processes such as radioactive decay can be obtained by iterating unimodal maps operating in chaotic domains, utilizing their extreme sensitivity to initial conditions. Such alternative explanations raise the question of the possibility of underlying nonlinearities in quantum mechanics. This is a ‘from the bottom up’, empirical approach to the question and will be compared with other, more theoretical, ‘from the top down’ investigations into the possibility of nonlinear extensions to quantum mechanics. Nonlinear dynamics, although deterministic, in its chaotic extreme provides only statistical information about the outcomes of state-space orbits. Thus in principle it provides the determinism so dear to Einstein; yet in practice it exhibits the probabilistic behaviour of Bohr. Perhaps Einstein and Bohr could both have been right.
Many of the paradoxes encountered in the Copenhagen interpretation of quantum mechanics can be shown to have plausible, more logical parallels in terms of nonlinear dynamics and chaos. These include the statistical exponential decay laws, interpretations of Bell's inequalities, spontaneous symmetry breaking, and perhaps diffractive behavior and even quantization itself. Many of the so-called alternative explanations of quantum mechanics have toyed with ideas that approach chaotic behavior, but as they were formulated before the advent of modern chaos theory, they remained within linear systems or at most nonlinear perturbations to linear systems; however, only strongly nonlinear systems can provide the proper parallels to the Copenhagen paradoxes. Several examples of these will be covered qualitatively. Strongly nonlinear behavior related to quantum mechanics does not involve "hidden variables," but chaos provides a bridge between the statistical behavior of quantum mechanics and deterministic behavior of classical mechanics. Perhaps both Einstein and Bohr were correct in their debates-chaos fundamentally provides the determinism so dear to Einstein, but in practice it must be interpreted statistically in the manner of Bohr.
In recent years we have suggested that many of the so-called paradoxes resulting from the Copenhagen interpretation of quantum mechanics could well have more logical parallels based in nonlinear dynamics and chaos theory. Perhaps quantum mechanics might not be strictly linear as has been commonly postulated, and indeed, during the past year experimentalists have discovered signatures of chaos in a definitely quantum system. As an illustration of what can go wrong when quantum effects are forced into a linear interpretation, I examine Bell-type inequalities. In conventional derivations of such inequalities, classical systems are found to impose upper limits on the statistical correlations between, say, the properties of a pair of separated but entangled particles, whereas quantum systems allow greater correlations. Numerous experiments have upheld the quantum predictions (greater statistical correlations than allowed classically), which has led to inferences such as the instantaneous transmission of information between effectively infinitely separated particles — Einstein's "spooky action-at-a-distance," incompatible with relativity. I argue that there is nothing wrong with the quantum mechanical side of such derivations (the usual point of attack by those attempting to debunk Bell-type arguments), but implicit in the derivations on the classical side is the assumption of independent, uncorrelated particles. As a result, one is comparing uncorrelated probabilities versus conditional probabilities rather than comparing classical versus quantum mechanics, making moot the experimental inferences. Further, nonlinear classical systems are known to exhibit correlations that can easily be as great as and overlap with quantum correlations — so-called nonextensive thermodynamics with its nonadditive entropy has verified this with numerous examples. Perhaps quantum mechanics does contain fundamental nonlinear elements. Nonlinear dynamics and chaos theory could well provide a bridge between the determinism so dear to Einstein and the statisical interpretation of the Copenhagen school. Einstein and Bohr both could have been right in their debates.
Many of the so‐called paradoxes of orthodox quantum mechanics can be shown to have parallel, more logical interpretations in the realm of nonlinear dynamics and chaos theory. Among these are violations of Bell‐type inequalities, which in comparing “classical” mechanics with quantum mechanics implicitly compare uncorrelated and correlated statistics. During the past decade research in the field of nonextensive thermodynamics (including Tsallis entropy) has demonstrated the existence of many statistical correlations in classical, nonlinear systems. When such correlations exist, the conventional classical upper limit on statistical correlations in Bell‐type experiments can easily be raised to overlap with quantum mechanical predictions involving correlated states such as the Bell singlet state, a favorite for deriving Bell inequalities. Thus, arguments based on experimental violations of Bell‐type inequalities, which rule out the existence of “local reality,” become moot. Perhaps quantum mechanics does have a deterministic, ontological basis, albeit one based in nonlinear dynamics and chaos theory. If so, deterministic chaos could provide Einstein’s longed‐for fundamental determinism, but because chaotic systems must be interpreted statistically, this also fits in quite well with the ideas of Bohr — Einstein and Bohr both could have been correct! It should be emphasized that the concept of nonlinear dynamics and chaos underpinning quantum mechanics does not involve hidden variables, nor does the fact that chaos is deterministic interlope on the existence of free will.
A number of the paradoxes encountered in the Copenhagen interpretation of quantum mechanics have apparent resolutions through parallels based in nonlinear dynamics and chaos theory. An overview of these is presented, with Bell’s theorem treated in somewhat greater detail. In these conferences (both “Quantum Theory: Reconsiderations of Foundations” and “Foundations of Probability and Physics”) Bell’s theorem has already been attacked by a number of authors in a number of ways, but here I demonstrate that the so‐called “classical” (or hidden variables) side of the derivation of Bell‐type inequalities is flawed because it is usually based on standard, non‐correlated statistics. When correlated statistics (including nonextensive thermodynamics) is considered, the classical upper limit on correlations is raised, overlapping with quantum predictions using entangled states. Thus, many of the Bell‐type experiments are rendered moot in ruling out “local reality.” Nonlinear dynamics/chaos could well provide a bridge between the determinism of Einstein and the probability of the Bohr school. Possible nonlinear underpinnings for quantum mechanics could also have implications lowering the plausibility of quantum computing.
Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.
By now the “chaos revolution” has reached nearly every branch of the natural sciences. In fact, chaos is everywhere. To name but a few examples, we talk about chaotic weather patterns, chaotic chemical reactions and the chaotic evolution of insect populations. Atomic and molecular physics are no exceptions. At first glance this is surprising since atoms and molecules are well described by the linear laws of quantum mechanics, while an essential ingredient of chaos is nonlinearity in the dynamic equations. Thus, chaos and atomic physics seem to have little to do with each other. But recently, atomic and molecular physicists have pushed the limits of their experiments to such high quantum numbers that it starts to make sense, in the spirit of Bohr's correspondence principle, to compare the results of atomic physics experiments with the predictions of classical mechanics, which, for the most part, show complexity and chaos. The most striking observation in recent years has been that quantum systems seem to “know” whether their classical counterparts display regular or chaotic motion. This fact can be understood intuitively on the basis of Feynman's version of quantum mechanics. In 1948 Feynman showed that quantum mechanics can be formulated on the basis of classical mechanics with the help of path integrals. Therefore it is expected that the quantum mechanics of an atom or molecule is profoundly influenced, but of course not completely determined, by the qualitative behaviour of its underlying classical mechanics.