At a minimum it seems that deterministic quantum mechanics recently proposed by G. ’t Hooft and classical mechanical indeterminism i.e. nonlinear dynamics and deterministic chaos are homomorphic conceptually. The present work gives an introduction to a transfinite spacetime theory based on the geometry and topology of a hierarchal infinite dimensional Cantor set in which both notions are exchangeable.
At a minimum it seems that deterministic quantum mechanics recently proposed by G. ’t Hooft and classical mechanical indeterminism i.e. nonlinear dynamics and deterministic chaos are homomorphic conceptually. The present work gives an introduction to a transfinite spacetime theory based on the geometry and topology of a hierarchal infinite dimensional Cantor set in which both notions are exchangeable.
Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos. We include several recent results in this discussion.
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.
Abstract It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.
Classical physics is generally regarded as deterministic, as opposed to quantum mechanics that is considered the first theory to have introduced genuine indeterminism into physics. We challenge this view by arguing that the alleged determinism of classical physics relies on the tacit, metaphysical assumption that there exists an actual value of every physical quantity, with its infinite predetermined digits (which we name principle of infinite precision). Building on recent information-theoretic arguments showing that the principle of infinite precision (which translates into the attribution of a physical meaning to mathematical real numbers) leads to unphysical consequences, we consider possible alternative indeterministic interpretations of classical physics. We also link those to well-known interpretations of quantum mechanics. In particular, we propose a model of classical indeterminism based on finite information quantities (FIQs). Moreover, we discuss the perspectives that an indeterministic physics could open (such as strong emergence), as well as some potential problematic issues. Finally, we make evident that any indeterministic interpretation of physics would have to deal with the problem of explaining how the indeterminate values become determinate, a problem known in the context of quantum mechanics as (part of) the ``quantum measurement problem.'' We discuss some similarities between the classical and the quantum measurement problems, and propose ideas for possible solutions (e.g., ``collapse models'' and ``top-down causation'').
The true dynamical randomness is obtained as a natural fundamental property of deterministic quantum systems. It provides quantum chaos passing to the classical dynamical chaos under the ordinary semiclassical transition, which extends the correspondence principle to chaotic systems. In return one should accept the modified form of quantum formalism (presented by the Schroedinger equation) which, however, does not contradict the ordinary form and the main postulates of quantum mechanics. It introduces the principle of the fundamental dynamic multivaluedness (redundance) extending the quantum paradigm to complex dynamical behaviour. Moreover, a causal solution to the well-known problems of the foundation of quantum mechanics, those of quantum indeterminacy and wave reduction, is also found using the same method. The concept of the fundamental dynamic uncertainty thus established is universal in character and provides a unified scheme of the complete description of arbitrary complex system of any origin (physics/9806002). This scheme incorporates, in particular, universal definitions of complexity, (non)integrability, and general solution, as well as the physically complete notion of probability. One obtains thus a self-consistent hierarchic picture of the world characterised by a (high) non-zero complexity and containing the intrinsic causal randomness, where the causally extended quantum mechanics can be consistently interpreted as several lowest levels of complexity (quant-ph/990215,16).