7,004 research outputs found
Lorenz, G\"{o}del and Penrose: New perspectives on determinism and causality in fundamental physics
Despite being known for his pioneering work on chaotic unpredictability, the
key discovery at the core of meteorologist Ed Lorenz's work is the link between
space-time calculus and state-space fractal geometry. Indeed, properties of
Lorenz's fractal invariant set relate space-time calculus to deep areas of
mathematics such as G\"{o}del's Incompleteness Theorem. These properties,
combined with some recent developments in theoretical and observational
cosmology, motivate what is referred to as the `cosmological invariant set
postulate': that the universe can be considered a deterministic dynamical
system evolving on a causal measure-zero fractal invariant set in its
state space. Symbolic representations of are constructed explicitly based
on permutation representations of quaternions. The resulting `invariant set
theory' provides some new perspectives on determinism and causality in
fundamental physics. For example, whilst the cosmological invariant set appears
to have a rich enough structure to allow a description of quantum probability,
its measure-zero character ensures it is sparse enough to prevent invariant set
theory being constrained by the Bell inequality (consistent with a partial
violation of the so-called measurement independence postulate). The primacy of
geometry as embodied in the proposed theory extends the principles underpinning
general relativity. As a result, the physical basis for contemporary programmes
which apply standard field quantisation to some putative gravitational
lagrangian is questioned. Consistent with Penrose's suggestion of a
deterministic but non-computable theory of fundamental physics, a
`gravitational theory of the quantum' is proposed based on the geometry of
, with potential observational consequences for the dark universe.Comment: This manuscript has been accepted for publication in Contemporary
Physics and is based on the author's 9th Dennis Sciama Lecture, given in
Oxford and Triest
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Risk: a fiction
Uncertainty creates anxiety so attempts have been made to reduce it using mathematical techniques. In the electronics industry the very large quantities of devices processed have provided reliable statistics and the opportunity to employ statistical methods. However, in fields such as decision-making and risk assessment there are strong criticisms of the probability calculus that have been triggered by discrepancies between the analysis of experts and non-experts. A radically different alternative is to view risk assessment and decision-making as exercises in rhetoric centred on storytelling language games. And to see the risk assessors as part of a political network attempting to influence action
Complex aspects of gravity
This paper presents reflections on the validity of a series of mathematical
methods and technical assumptions that are encrusted in macrophysics (related
to gravitational interaction), that seem to have little or no physical
significance. It is interesting to inquire what a change can occur if one
removes some of the traditional assumptions.Comment: 10 page
The statistical origins of quantum mechanics
It is shown that Schroedinger's equation may be derived from three
postulates. The first is a kind of statistical metamorphosis of classical
mechanics, a set of two relations which are obtained from the canonical
equations of particle mechanics by replacing all observables by statistical
averages. The second is a local conservation law of probability with a
probability current which takes the form of a gradient. The third is a
principle of maximal disorder as realized by the requirement of minimal Fisher
information. The rule for calculating expectation values is obtained from a
fourth postulate, the requirement of energy conservation in the mean. The fact
that all these basic relations of quantum theory may be derived from premises
which are statistical in character is interpreted as a strong argument in favor
of the statistical interpretation of quantum mechanics. The structures of
quantum theory and classical statistical theories are compared and some
fundamental differences are identified.Comment: slightly modified version, 24 pages, no figure
Zeno meets modern science
``No one has ever touched Zeno without refuting him''. We will not refute
Zeno in this paper. Instead we review some unexpected encounters of Zeno with
modern science. The paper begins with a brief biography of Zeno of Elea
followed by his famous paradoxes of motion. Reflections on continuity of space
and time lead us to Banach and Tarski and to their celebrated paradox, which is
in fact not a paradox at all but a strict mathematical theorem, although very
counterintuitive. Quantum mechanics brings another flavour in Zeno paradoxes.
Quantum Zeno and anti-Zeno effects are really paradoxical but now experimental
facts. Then we discuss supertasks and bifurcated supertasks. The concept of
localization leads us to Newton and Wigner and to interesting phenomenon of
quantum revivals. At last we note that the paradoxical idea of timeless
universe, defended by Zeno and Parmenides at ancient times, is still alive in
quantum gravity. The list of references that follows is necessarily incomplete
but we hope it will assist interested reader to fill in details.Comment: 40 pages, LaTeX, 10 figure
Quantum Theory and Determinism
Historically, appearance of the quantum theory led to a prevailing view that
Nature is indeterministic. The arguments for the indeterminism and proposals
for indeterministic and deterministic approaches are reviewed. These include
collapse theories, Bohmian Mechanics and the many-worlds interpretation. It is
argued that ontic interpretations of the quantum wave function provide simpler
and clearer physical explanation and that the many-worlds interpretation is the
most attractive since it provides a deterministic and local theory for our
physical Universe explaining the illusion of randomness and nonlocality in the
world we experience.Comment: Some references updated. Published online in Quantum Studies:
Mathematics and Foundation
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