On a manifestly covariant classical mechanics or ruminations on “The Computable Universe” and the role of mathematical physics in solving the natural resource problems of the future
Abstract This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics, and of the relation of classical mechanics to relativity and quantum theory. A distinguishing feature is its integration of special relativity into the teaching of classical mechanics. After a thorough review of the traditional theory, Part II of the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. The definition of canonical transformation no longer excludes the Lorentz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics since comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac's formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods such as linear vector operators and dyadics that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics require a rethinking of the quantum–classical connection.
The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the rest of reality as we know it, and the implications of the existence of this structure for quantum theory. Understanding how this structure could relate to space and to the rest of reality requires, I believe, that we consider how space itself relates to reality, and how other so-called “spaces” used in physics relate to reality. In the first chapter I compare space and space-time to other spaces used in physics, such as configuration space, phase space and Hilbert space. I support what is known as the “property view” of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that all these spaces are property spaces. After examining the relationships of these spaces to causality, I argue that configuration space has, due to its role in quantum mechanics, a special status in the microscopic world similar to the status of position space in the macroscopic world. In chapter two prespace itself is considered. One way of approaching this structure is through the comparison of the prespace structure with a computational system, in particular to a cellular automaton, in which space or space-time and all other physical quantities are broken down into discrete units. I suggest that one way open for a prespace metaphysics can be found if physics is made fully discrete in this way. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized, adapting a heuristic principle of this type proposed by Feynman (1982). In the final chapter some of the ideas of the previous chapters are applied in an examination of the physics and metaphysics of quantum theory. I first discuss the “measurement problem” of quantum mechanics: this problem and its proposed solution are the primary subjects of the final chapter. It turns out that considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be modified to give rise to a solution to the measurement problem. The computational heuristic principle reinforces the same solution. I call the modified quantum mechanics Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with some of the other proposed solutions to the measurement problem, in particular the spontaneous localization model of Ghirardi, Rimini and Weber (1986). Finally I argue that the measure of complexity of quantum mechanical states I introduce in CCQM also provides a new definition of entropy for quantum mechanics, and suggests a solution to the problem of providing an objective foundation for statistical mechanics, thermodynamics, and the arrow of time.
Quantum mechanics is considered to be one of the most successful as well as revolutionary theories in the history of physics. It gives the conceptual and mathematical basis of the behavior of matter and energy at atomic and subatomic level. In the course of time quantum mechanics has grown out of a theory that was created to cover experimental anomalies to a comprehensive theory that forms the basis of theoretical physics as known today. Its use is much broader to the early atomic models and now it is central to quantum field theory, particle physics and condensed matter physics, cosmology, and new theories of quantum gravity. Quantum mechanics has a significant impact on the philosophical concept of physical reality because of its abstract nature, the probabilistic nature of its interpretations, wave functions and non-local correlations. The classical physics, in contrast to quantum mechanics, presupposes the determinism and exact predictability of the phenomena; quantum mechanics regards the unpredictability of phenomena as the natural law. This change of thought has enabled physicists to develop a set of theoretical frameworks, which better capture experimental data, especially at gross extremes of energy and space. Quantum mechanics is a theory and a method of modern theoretical physics. It allows the development of field theories of the description of fundamental interactions, predicts the occurrence of previously unknown particles, and the explanation of complex collective phenomena in many-body systems. Moreover, additional quantum concepts have already become essential in the study of the early universe, black holes and microphysical geometry of space-time. The purpose of the research paper is to examine the vast number of theoretical physics studies using quantum mechanics as a tool. The paper has emphasised the contribution of quantum mechanics in the current research on the study of unification of physical theories as it is manifested in key subfields to which quantum mechanics has made contributions. There are also some reflections of where quantum mechanics is likely to make a paradigm shift in the future that confirm its relevance as a fundamental pillar of theoretical physics.
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
While physicists pursue the search for the Grand Unified Theory (GUT) and the expedition to find the Higgs that ought to solidify the Standard Model, philosophers keep inquiring as to what the nature of matter and mind might be and propose various theories. At first sight, the high energy physicist or the philosopher, both human, and the elementary particle seem to have nothing in common, however we contend and argue in this paper that we are looking at different scales of what constitutes universal behaviour and use Herman Dooyeweerd's theory of Modal Aspects to bridge the particle to the mind. To other than a physicist the dual nature of the atom or the so-called elementary particles may be difficult to comprehend and certainly appears unfathomable to intuit. This is due to the fact that the dual nature of matter finds its anchoring within the theoretical and mathematical frameworks provided by quantum mechanics, the latter tends to not be easily accessible to most individual's intuition. What would the consequences be if human nature itself had a dual character? We all know that human nature is both egoistic and altruistic, however it is not often that one considers the fact that egoism is an individual behaviour, while altruism is a collective behaviour. Likewise a particle is an individual manifestation of what we call matter, while wave nature can be considered characteristic of collective behaviour of the same matter.
Recent advances in experimental physics—the queen of reductionist science—aimed at solving the mysteries of the quantum world has led to a deepening of the mystery rather than easing it, dispelling the EPR paradox and pointing out a conceptional error in this now famous gedanken experiment, while lending some authority and validation to quantum mechanics. The strange features of the quantum world are founded on what Bohr dubbed complementarity, or the way a sub-particle entity can behave either as a wave or as a particle; and on Werner Heisenberg's uncertainty principle, which says that a quantum entity such as an electron does not avail itself to a deterministic exact measurement of both position and momentum simultaneously. Such postulates seem to contradict deeply held assumptions by the deterministic and materialistically oriented world of experimental physics where everything, including the atom, can be reduced to a single aspect. There are mounting calls for newer ways of understanding reality that are capable of handling these apparently strange incomprehensible phenomena revealed through the classical observation of the microscopic quantum world. There is also fundamental progress made in bouncing cosmology theories that bridge general relativity to quantum mechanics and provide some insight into the nature of the universe and its transition and pre-history through the big bang.
The thrust of this paper evolves around two pillars. One, is that that the apparent logical fissure emergent in reductionist thought, coming from orthodox science and the application of the scientific method that is manifest in what we know and do not understand in quantum mechanics, can be liaised to our intellectual understanding through the perspective offered by Herman Dooyeweerd's theory of Modal Aspects. The other, is the central and special role humans have in relation to the universe. Through this approach, we will argue that the dual nature of the atom or elementary particles is, rather than being a strange phenomenon, in fact an essential characteristic of what constitutes its nature