Abstract In Norton’s dome, a singular solution of the Newton/Hamilton equations represents a particle starting to move without cause. Similar indeterminism occurs for forces that are not the gradient of a potential, i.e. Newtonian but not Hamiltonian. An exactly solvable quantum partial counterpart displays no trace of the Norton phenomenon. A completely different context is chaotic classical motion of gas molecules in a box, whose walls are penetrable because gravity cannot be screened. The gravity of a single electron at the limit of the observable universe renders the motion of the gas molecules unpredictable after a few nanoseconds. But motion is still determined by the initial conditions. However, the much larger perturbation from the gravity of the computers or ‘Laplace’s demon’ calculating the motion of the molecules introduces self-reference, making it impossible to follow the motion and rendering the determinism irrelevant to physics for such chaotic systems. For the quantum gas molecules, perturbation from the distant electron vastly exceeds the energy level separation for the idealised impenetrable box, rendering individual energies physically insignificant. Calculations are not difficult, and the combination of science and philosophy deserves to be introduced to physics students.
Abstract In Norton’s dome, a singular solution of the Newton/Hamilton equations represents a particle starting to move without cause. Similar indeterminism occurs for forces that are not the gradient of a potential, i.e. Newtonian but not Hamiltonian. An exactly solvable quantum partial counterpart displays no trace of the Norton phenomenon. A completely different context is chaotic classical motion of gas molecules in a box, whose walls are penetrable because gravity cannot be screened. The gravity of a single electron at the limit of the observable universe renders the motion of the gas molecules unpredictable after a few nanoseconds. But motion is still determined by the initial conditions. However, the much larger perturbation from the gravity of the computers or ‘Laplace’s demon’ calculating the motion of the molecules introduces self-reference, making it impossible to follow the motion and rendering the determinism irrelevant to physics for such chaotic systems. For the quantum gas molecules, perturbation from the distant electron vastly exceeds the energy level separation for the idealised impenetrable box, rendering individual energies physically insignificant. Calculations are not difficult, and the combination of science and philosophy deserves to be introduced to physics students.
This thesis is a collection of three case studies, investigating various sources of indeterminism and undecidability as they bear upon in principle unpredictability of the behaviour of mechanistic systems in both classical and quantum physics. I begin by examining the sources of indeterminism and acausality in classical physics. Here I discuss the physical significance of an often overlooked and yet important Lipschitz condition, the violation of which underlies the existence of anomalous non-trivial solutions in the Norton-type indeterministic systems. I argue that the singularity arising from the violation of the Lipschitz condition in the systems considered appears to be so fragile as to be easily destroyed by slightly relaxing certain (infinite) idealizations required by these models. In particular, I show that the idealization of an absolutely nondeformable, or infinitely rigid, dome appears to be an essential assumption for anomalous motion to begin; any slightest elastic deformations of the dome due to finite rigidity of the dome destroy the shape of the dome required for indeterminism to obtain. I also consider several modifications of the original Norton's example and show that indeterminism in these cases, too, critically depends on the nature of certain idealizations pertaining to elastic properties of the bodies in these models. As a result, I argue that indeterminism of the Norton-type Lipschitz-indeterministic systems should rather be viewed as an artefact of certain (infinite) idealizations essential for the models, depriving the examples of much of their intended metaphysical import, as, for example, in Norton's antifundamentalist programme. Second, I examine the predictive computational limitations of a classical Laplace's demon. I demonstrate that in situations of self-fulfilling prognoses the class of undecidable propositions about certain future events, in general, is not empty; any Laplace's demon having all the information about the world now will be unable to predict all the future. In order to answer certain questions about the future it needs to resort occasionally to, or to consult with, a demon of a higher order in the computational hierarchy whose computational powers are beyond that of any Turing machine. In computer science such power is attributed to a theoretical device called an Oracle--a device capable of looking through an infinite domain in a finite time. I also discuss the distinction between ontological and epistemological views of determinism, and how adopting Wheeler-Landauer view of physical laws can entangle these aspects on a more fundamental level. Thirdly, I examine a recent proposal from the area of quantum computation purporting to utilize peculiarities of quantum reality to perform hypercomputation. While the current view is that quantum algorithms (such as Shor's) lead to re-description of the complexity space for computational problems, recently it has been argued (by Kieu) that certain novel quantum adiabatic algorithms may even require reconsideration of the whole notion of computability, by being able to break the Turing limit and "compute the non-computable". If implemented, such algorithms could serve as a physical realization of an Oracle needed for a Laplacian demon to accomplish its job. I critically review this latter proposal by exposing the weaknesses of Kieu's quantum adiabatic demon, pointing out its failure to deliver the purported hypercomputation. Regardless of whether the class of hypercomputers is non-empty, Kieu's proposed algorithm is not a member of this distinguished club, and a quantum computer powered Laplace's demon can do no more than its ordinary classical counterpart.
This paper aims to demonstrate how a rigorous definition of determinism can be formulated and applied to a formal axiomatization of a physical theory. Examining an example of Newtonian classical particle mechanics (CPM) formalized in set-theoretical axioms by Patrick Suppes, the paper shows why definitions of determinism based on the temporal development of physical systems—such as those proposed by Suppes (2002) and Montague (1974)—fail to assess determinism correctly. It is argued that, to deliver a meaningful verdict, the definition of determinism cannot be applied to the entire framework of CPM itself, but rather to its particular theories : definitions of specific physical systems that incorporate the force equation. Therefore, the verdict of determinism or indeterminism typically depends on the uniqueness (or non-uniqueness) of solutions to the differential equations governing the system’s behavior. The paper concludes with a demonstration of how a formally correct definition of determinism can be used to show the indeterminism of the theory of Norton’s dome stated within the framework of CPM. Although the example concerns pre-relativistic Newtonian mechanics, the crucial philosophical distinction between frameworks and theories extends beyond this case and suggests that assessing the determinism of any physical theory requires special caution to distinguish it properly from its framework.
Newtonian mechanics focuses on dx and dt units which tend to zero and so presents, it seems, a smoothed over or averaged type of motion. This motion in principle extends to dx and dt values which are almost zero. There is, however, a priori no reason not to expect that such a smooth averaging is at some length scale based on another kind of motion. The question is: What should that “other” kind of motion be? If it is deterministic in x and t, one may reapply the idea of this motion breaking down at yet some smaller scale. Thus it seems one should consider decoupling of x and t because one cannot follow a particle from one x point to another in time within some small length scale as this would be deterministic. The opposite approach seems to be a probabilistic one with t and x decoupled. The external ruler and clock which exist in the Newtonian picture, however, do not disappear simply because a length scale is small. Thus having x positions based on probabilities at some length scale still uses the notion of the probability itself being smoothed over because it contains x which may take on any value. This is not necessarily a problem because the probabilistic behaviour must still appear to be Newtonian on average and so x and t based on the external clock/ruler with infinite resolution need to “appear” again. (For example, an electron has a wavelength and exhibits interference from two slits, but while moving towards the slits, distance = velocity* time with distance and time measured by an external clock. It is a mathematical generalization, however, to expect that d=vt actually fully describes motion (i.e. other than on average) within a wavelength length. If it did there would be no interference pattern. The idea of a smoothed over (or averaged motion) based on probabilistic motion also appears in the classical world, but for many particles. For example, a Maxwell-Boltzmann gas with no potential has uniform density. If one considers a hypothetical box containing many molecules, it is like a Newtonian stationary object with a balance of pressure. In the case of a potential (1-D) one may consider a dx unit (with many molecules) and balance Force(x) density(x) with the difference of pressure to find the density distribution proportional to exp(- V(x)/T). These ideas are completely Newtonian, but they are smoothed over average notions. At a microscopic level molecules undergo probabilistic collisions linked to T (temperature). Thus the microscopic picture is not static - it is only the equilibrium average or big picture which is static. Even in the case of no potential, focusing on a tiny region of space smaller than the interparticle average separation and arguing that density is constant seems to be an average argument which bypasses much of the reality. If x and t may then become probabilistic and -Et+px is a Lorentz invariant, then why should there not be stochasticity linked with E and p? In fact we argue they are == as shown by Maxwell’s solution of a photon from his electromagnetic equations. E and p (energy and momentum) are average values. Their density is given by: .5C1( Electric field dot Electric field + C2 Magnetic field dot Magnetic field) and the Poynting vector for momentum. Both the Electric field and Magnetic field fluctuate as cos(-Et+px) with E and p being average values. At first this was ascribed to wave motion in a medium, but in 1905 Einstein’s photoelectric arguments against energy being spread out seem to suggest a breakdown of deterministic physics describing a wave in a medium. Instead a photon is probabilistic and has a wavefunction of Electric + i Magnetic field. (2)
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.
Mathematically, Newton's laws of motion are cast in the form of differential equations (specifically, equations which relate quantities to their time dependence). One of the simplest examples can be given in the context of that stone-throwing problem which so interested the early Greeks. Newton did fully realize that the heavens could not be separated, or compartmentalized, into isolated pairs of interacting two-body systems. Newton tried to deal with this problem at least for what he deemed to be the simplest circumstance, namely the case in which one two-body force exerts a much larger influence on the primary motion of concern than the others. In order to probe the effect of chaos in the true quantum regime it is necessary to focus upon the motion of electrons. The electronic 'motion' of an electron about a nucleus cannot be cast, in quantum theory, in terms of classical concepts like orbits.
The nature of the problem The quantum world is inexplicable in classical terms. The predictions pertaining to the interaction of matter and light embodied in Newton's laws for the motion of particles and Maxwell's equations governing the propagation of electromagnetic fields are in flat contradiction with the experimental facts at the microscopic scale. A key feature of quantum effects is their apparent indeterminism, that individual atomic events are unpredictable, uncontrollable and literally seem to have no cause. Regularities emerge only when one considers a large ensemble of such events. This indeed is generally considered to constitute the heart of the conceptual problem posed by quantum phenomena, necessitating a fundamental revision of the deterministic classical world view. Some of the principal phenomena, among those discussed in this book, are as follows: ( a ) Self-interference: a beam of electrons sent one at a time through a barrier containing two apertures builds up through a series of localized detection events on a screen an interference pattern characteristic of waves (cf. the frontispiece and §5.1). ( b ) Tunnelling: an α-particle trapped in a nucleus can pass through a potential barrier in a manner forbidden to a classical particle (§5.3). ( c ) The stability of matter: atoms and molecules are found to exist only in certain discrete, or ‘stationary’, energy states. For this reason they do not ‘collapse’, the result predicted by classical electrodynamics. During transitions between stationary states (quantum jumps) an atom exchanges a discrete quantity of energy with the electromagnetic field (§§4.5, 7.6). […]