Nature’s Information and Harmonic Proportion

by Michael A. Sherbon

Abstract: The history of science is polarized by debates over Plato and Aristotle’s holism versus the atomism of Democritus and others. This includes the complementarity of continuous and discrete, one and the many, waves and particles, and analog or digital views of reality. The three-fold method of the Pythagorean paradigm of unity, duality, and harmony enables the calculation of fundamental physical constants required by the forces of nature in the formation of matter; thereby demonstrating Plato’s archetypal viewpoint.

papers.ssrn.com/abstract=1766049

SSRN Classics: Journal of Philosophical & Scientific Texts
Nature’s Information and Harmonic Proportion « Quintessentia

 
 "The Philosopher's Stone is the Holy Grail of alchemy, the ancient art of turning base metal into gold. Its magical and elemental power has fixated explorers, occultists and scientists for centuries. For the Philosopher's Stone, it is said, holds the key, not only to making gold but also to deciphering the riddle of existence and unlocking the secret of eternal life." Following such luminaries as Newton, Jung, St. Thomas Aquinas and Zosimus, who devoted most of their lives to searching for it, Peter Marshall set out to unearth the secrets of alchemy in the lands where it was traditionally practised. The result is a piece of historical, scientific and philosophical detection, as well as an exciting physical and spiritual adventure. Exploring the beliefs and practices, the myths and the symbols of the alchemists, Peter Marshall takes us on a journey into this arcane world."


Goodreads | The Philosopher's Stone: A Quest for the Secrets of Alchemy by Peter Marshall - Reviews, Discussion, Bookclubs, Lists

Division Algebras and Quantum Theory

Abstract
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly "complex" representations), those that are self-dual thanks to a symmetric bilinear pairing (which are "real", in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are "quaternionic", in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds - real, complex and quaternionic - can be seen as Hilbert spaces of the other kinds, equipped with extra structure.
CiteULike: Division Algebras and Quantum Theory

Geometric Algebra: A natural representation of three-space

James M. Chappell, Azhar Iqbal, Derek Abbott
 
(19 Jan 2011) 

 Abstract: Historically, there have been many attempts to define the correct algebra for modeling the properties of three dimensional physical space, such as Descartes' system of Cartesian coordinates in 1637, the quaternions of Hamilton representing rotation in three-space that built on the Argand diagram for two-space, and Gibbs' vector calculus employing the dot and cross products. We illustrate however, that Clifford's geometric algebra developed in 1873, but largely overlooked by the science community, provides the simplest and most natural algebra for three-space and hence has general applicability to all fields of science and engineering. To support this thesis, we firstly show how geometric algebra naturally produces all the properties of complex numbers and quaternions and the vector cross product in a single formalism, whilst still maintaining a strictly real field and secondly we show in two specific cases how it simplifies analysis in regards to electromagnetism and Dirac's equation of quantum mechanics. This approach thus has the immediate advantage of removing complex fields from analysis, so that algebraic entities have a geometric expression in real space. As an example, we show how quadratic equations now can be given two real solutions using GA. This viewpoint also has something interesting to say about the concept of number itself, because numbers in GA can encapsulate scalars, lines, areas and volumes, into a single entity, which can be multiplied and divided, just like normal numbers. This simple and elegant formalism would also seem very appropriate for students learning vectors, algebra and geometry, giving a more natural and intuitive understanding of the properties of three-space.

CiteULike: Geometric Algebra: A natural representation of three-space

 
Physics of the Riemann Hypothesis
 
Schumayer, Daniel; Hutchinson, David A. W.
 
eprint  arXiv:1101.3116  01/2011
 
 
Abstract: Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.

Physics of the Riemann Hypothesis

Fundamental constants from topological insulators
 

Topological Quantization in Units of the Fine Structure Constant

Joseph Maciejko, Xiao-Liang Qi, H. Dennis Drew, and Shou-Cheng Zhang

Phys. Rev. Lett. 105, 166803 (Published October 12, 2010)

Semiconductor Physics

Some of the most precise measurements of fundamental constants in nature come from topological phenomena in condensed matter physics.

...

In an article appearing in Physical Review Letters, Joseph Maciejko and collaborators from Stanford University, in collaboration with SLAC, Microsoft Research, and the University of Maryland, all in the US, propose an optical experiment to measure this. The setup consists of a layer of a generic topological insulator deposited on an ordinary insulator, in a perpendicular external magnetic field. They find that measuring the rotation of light polarization reflected off the top surface (Kerr angle) and transmitted through the two layers (Faraday angle) allows one to extract the quantized magnetoelectric response. If this measurement could be realized, topological phenomena in condensed matter physics could be used to nail down the most precise values for three basic physical constants: the fundamental electric charge e, Planck’s constant h, and the speed of light c. – Sarma Kancharla


Physics - Fundamental constants from topological insulators

Mathematical Constants of Natural Philosophy

Michael A. Sherbon    July 21, 2010


Abstract:

Plato's theory of everything is an introduction to a Pythagorean natural philosophy that includes Egyptian sources. The Pythagorean Table and Pythagorean harmonics from the ancient geometry of the Cosmological Circle are related to symbolic associations of basic mathematical constants with the five elements of Plato's allegorical cosmology: Archimedes constant, Euler's number, the polygon circumscribing limit, the golden ratio, and Aristotle's quintessence. Quintessence is representative of the whole, or the one in four, extraneously considered a separate element or fifth force. This relationship with four fundamental interactions or forces also involves the correlation of constants with the five Platonic solids: tetrahedron, hexahedron, octahedron, icosahedron, and dodecahedron. The values of several fundamental physical constants are also calculated, and a basic equation is given for a unified physical theory in the geometric universe of Plato's natural philosophy.


papers.ssrn.com/abstract=1646568
SSRN Classics: Journal of Philosophical & Scientific Texts