Scientific Essays in Honor of H Pierre Noyes on the Occasion of His 90th Birthday

The following sections are included:

• Preface
• About Pierre
• List of Contributors
• Contents

The familiar notion of the linear unit operator on a Hilbert space is extended to its homogeneous polynomial operator analogues of arbitrary degree : ‘unital’ homogeneous polynomial operators, whose multimatrix representations are identified with generalized Kronecker symbols (tensors). Some properties are lost, many other novel structural properties are established.

The set of unital operators of all degrees on a Hilbert space constitutes a unique object, a ‘Unital System’. It is unit-normed, closed and commutative under functional composition, and closed under functional polar composition. It is a multiplicatively and additively graded set, graded by the positive integer subset of the integer ring. It is the disjoint union of its homogeneous component ‘grade’ subsets each comprised of an equivalence class (under unitary similarity) of unital operators of same degree α ≥ 1, the first grade subset being the only one itself closed under composition.

Some operator equations of Fredholm and Characteristic type involving unital homogeneous polynomial operators are illustrated, their solubility described, and connections with an emerging theory of the spectrum of tensors and homogeneous polynomial operators are noted.

Possible applications to the theory of nonlinear phenomena in the presence of very high field strengths are indicated.

The Combinatorial Hierarchy is a four level bit string hierarchy discovered by Frederick Parker-Rhodes in 1963. Associated with the four levels is the number sequence 3; 10; 137; 1.7 × 10³⁸, the last two closely resembling two of the Structure Constants of Physics. Much has been made of this phenomenon; however this paper concentrates on furthering the understanding of the mathematical structure of the Hierarchy by generalising it to Objects and Morphisms in an arbitrary Category. The author has long argued that if the Combinatorial Hierarchy has any meaning then the important part of it is not the structure of the levels of the Hierarchy, but that of the Level Change Operator, that maps Objects from one level of the Hierarchy to the next. We conclude that if a connection to Quantum Nonlocality is to be made it is more likely to be through the structure of the Level Change Operator on a Topos than through traditional means.

We here strive to increase understanding of Milne quantum cosmology (MQU) by attention to the ordered set of ‘special’ Mersenne primes, 2; 3; 7; 127; 2¹²⁷–1. Each special prime (after the first) is related to its predecessor by Mersenne's prescription. MQU is a T.O.E. founded on an ‘extended-Lorentz’ Lie-group symmetry that defines, for individual universe ‘constituents’, conserved momentum, angular momentum, ‘spirality’, electric charge and energy. Any particle, such as a photon, is a composite of several different MQU constituents.

MQU's 9-parameter symmetry group, with separate ‘left’ complex-number displacement and ‘right’ Lorentz subgroups, immediately invokes the primes 2 and 3. Because the particle-theoretic Standard Model (SM) attens MQU's curved 3-space by contracting the Lorentz subgroup to the Euclidean group — suppressing redshift and replacing MQU symmetry by that of the gauge-augmented Poincaré group — MQU capabilities associating to the remaining special Mersenne primes are susceptible to ‘SM obscuration’. We here speculate about capacity of two ‘electromagnetic’ special Mersenne primes, the pair 7 and 127, (that follow 2 and 3) to reduce SM arbitrariness while underpinning its perturbative recipe for comparison with data. The prime 7 may provide raison d'ètre for quarks as well as leptons and three generations of ‘elementary fermions’, while assigning ‘Majorana’ status to neutrinos.

MQU unification of electromagnetism with gravity allows the huge remaining member of Mersenne's special prime quintet to be called ‘gravitational’ — quantifying the meaning of ‘ination’ by specifying MQU ‘age at big bang’. Particle-energy scale then (in accord with SM) lies in the Gev neighborhood — below the Planck energy (˜10⁻⁵ gm) by a factor of order 10¹⁹. The huge special prime further underpins a ‘macroscopic’ spacetime scale — the scale of life and consciousness — larger than particle scale by another factor of order 10¹⁹.

We introduce the BiEntropy algorithm which measures the order and disorder within a finite binary string of arbitrary length. The BiEntropy algorithm is a weighted average of the Shannon Entropies of the string and all but one of the binary derivatives of the string. The BiEntropy algorithm has been successfully tested empirically in a number of diverse domains. We summarise the history behind and the development of the algorithm and provide some simple algebras to demonstrate its future use in bit-string physics.

We find that the Schrödinger quantization of a rudimentary classical archetype produces operator equations that express most of the rules of classical mechanics. These rules include the Riemannian connectivity of physical space; but this seems to emerge only at the macrophysical scale. The field equations are fourth order PDEs. But they are satisfied by the conventional second order solutions both as regards gravity and EM; extra physics is therefore to be expected. The Maxwell equations, which are deducible from the Feynman-Dyson quantum mechanical argument, are hardly effected. Gravitation is governed by the fourth order Kilmister equation; in conjunction with the Einstein equation, this dictates the allowed distributions of matter-energy-momentum and, perhaps, offers an explanation of the Dark Energy effect. Preliminary machine calculations are encouraging.

It is easy to imagine that in the near future someone will find a theory free from apparent contradiction which will account for all known physical phenomena; but which will soon be replaced by a succession of simpler, and hence more likely, alternatives. Inspired by Feynman, I propose a method for attempting to locate the simplest first:

• 1. Begin with the very simplest structure;
• 2. Find the most conspicuous way in which it is different from the universe;
• 3. In the most natural way, fix that problem, obtaining next candidate;
• 4. Find the most conspicuous way in which it is different from the universe;
• 5. Iterate the process.

We begin by examining a few mathematical seven day creation stories, each beginning with a day of rest. In the first story, each succeeding day, the Powerset operator is applied to the previous days results. Each succeeding story uses a more enriched version of the day to day operator. Our main focus is on the later stories which take us up the tree of double fields of Catalan-Mersenne irreducible trinomials, to the 39 digit Mersenne prime L = 2¹²⁷ − 1.

We reach a tensor product of Cartesian products of disjoint unions of double fields each with L+1 elements, providing steps toward an elegant rooted pedigree for noncommutative, discrete, complex, projective, 3-spaces with coordinates in 2×2 matrices over the field F_L2 or over the ring Z_L2, which we aspire to soon explore with computer graphics.

This should shed light on finite M-theory, Penrose twistors, and Loop quantum cosmology, or so I dream.

The following paper was the basis for my talk at ANPA 16 (1994). At the time I believed its basic idea to be original, a belief supported by a number of knowledgeable readers. However, I subsequently learned that this idea, which I called Boolean geometry, was actually around as early as the 1930s. Having never quite made it into the mainstream of logic, it was reinvented not only by me but by several others, including Gordon Pask and, I believe, Gian-Carlo Rota – the full history here remains to be uncovered. However, the connection I made to negative quantum amplitudes does appear to be new, so my plan became to revise the paper into a larger work in which this connection is developed in detail. This larger work has indeed become larger! What was to be only an introductory section on link states turned into my 90-page IJGS paper, followed by a series of shorter papers on the same topic, and I'm afraid the grand synthesis of link theory and Boolean geometry is still only a sketch. Thus it seems like a good idea to go ahead and release this paper in its present unfinished state. Like the DOC paper, it's a historical record, and also I believe that it's not a bad introduction to its subject, which could well be of interest to other investigators of “strange” logics.

A speculation which arose within the context of the nature of the Bit-string Combinatorial Heirarchy is offered for discussion. It seems to provide a strong clue to how systems must be constructed if they are to support consciousness. It depends on regarding consciousness as a case of ‘relative existence’. So that:‘A is conscious of x’ becomes: ‘x exists for A’.

Using the ANPA annual proceedings we analyze the authors and titles of ANPA papers and detect some long term trends. The author recalls some early conversations with Pierre Noyes and presents the results after following his advice for 27 years.

When a two-body system is bound by a zero-range interaction, the corresponding three-body system – considered in a non-relativistic framework – collapses, that is its binding energy is unbounded from below. In a paper by J.V. Lindesay and H.P. Noyes [1] it was shown that the relativistic effects result in an effective repulsion in such a way that three-body binding energy remains also finite, thus preventing the three-body system from collapse. Later, this property was confirmed in other works based on different versions of relativistic approaches. However, the three-body system exists only for a limited range of two-body binding energy values. For stronger two-body interaction, the relativistic three-body system still collapses.

A similar phenomenon was found in a two-body systems themselves: a two-fermion system with one-boson exchange interaction in a state with zero angular momentum J = 0 exists if the coupling constant does not exceed some critical value but it also collapses for larger coupling constant. For a J = 1 state, it collapses for any coupling constant value. These properties are called “critical stability”. This contribution aims to be a brief review of this field pioneered by H.P. Noyes.

To Pierre Noyes for his 90-th birthday. This paper generalizes our joint work on discrete electromagetism and the Feynman-Dyson derivation of electromagnetism from commutator calculus.

Clive, as a founder of ANPA, was at its heart from the formation in 1979 until just before his death. He served in several important capacities including President, Proceedings Editor, Member of the Executive Council and member of the Advisory Board. In addition he was the most prolific ANPA author. The latter was surprising since Clive did not use personal computers and after writing out his papers in longhand he would make corrections and then use a mechanical typewriter to produce his final text. As secretary of the Advisory Board I worked with Clive on the 2008 report to the Advisory Board and it is included in this festschrift as it summarizes a large part of ANPA achievements from the very beginning up until 2008, a period of more than 50 years.

Advisory Board (2008): Mike Horner (Chairman), Profs G.F.Chew (Berkeley), C.Isham (Imperial College), M.Redhead (Cambridge and LSE), N. Cartwright (LSE), C.W.Kilmister (London), H.Pierre Noyes (Stanford), Rev. Dr. Philip Luscombe (Wesley House, Cambridge).

Perspectives on three and a half decades of joint research with Pierre Noyes on fundamentals of physics are presented. At the core of the discussion is the no/yes nature of quantum detections, consistent with phase coherence and quantum interference. The incorporation of special relativity into non-perturbative scattering theory provides our initial insights into the subtleties connecting quantum physics to classical observations. We furthermore continue to explore quantum effects in general relativity, and tackle a perspective on dark energy as imparting a quantum imprint upon the fluctuations in the energy density of the universe that ultimately result in galaxy formation and life. We will end by exploring some of the late time implications that dark energy has on whether intelligent life can be perpetual, a favorite topic of Pierre's inspired by an article by his colleague Freeman Dyson.

The Ordering Operator Calculus (1982) provided a discrete, strictly finite foundation for differential geometry and the calculus. This abstract mathematical formulation was subsequently incorporated in Foundations of a Discrete Physics (1988), which applied it to a reformulation of relativistic quantum mechanics, vis-à-vis the combinatorial hierarchy and Prof. Noyes' Bit String Physics. The present paper provides a review of ordering operators, discusses principles for applying the ordering operator calculus to physics, and provides a high-level introduction to an OOC notational calculus – the tensor calculus – so that tensor equations can be interpreted as a special case of ordering operator equations.

We provide exact calculations of the information content (and its transformation) of expressions in the geometric algebras over ℤ₃ = {0, 1,−1}. Being fundamentally combinatorial, the results are theory-neutral.

The overall picture is a “Bit Bang” modelled as the algebraic expansion , which expansion is driven by entropy creation via the conversion of information from space-like (non-Shannon) to time-like (Shannon) form. This generates the Standard Model in . further yields a bridge from quantum mechanics to general relativity in the form of time-like quaternion isomorphs called tauquernions, plus dark matter structure [5]. In the end, driving the entropic expansion of the cosmos, emerging like the Cheshire cat, wed the Combinatorial Hierarchy.

Interleaved bit-strings provide a simple mechanism for the encoding of relativistic periodic processes underlying both spacetime and the Dirac equation. We illustrate this using simple model clocks.

The claim that the large scale structure of the Universe is hierarchical has a very long history going back at least to Charlier's papers of the early 20th century. In recent years, the debate has largely focussed on the works of Sylos Labini, Joyce, Pietronero and others, who have made the quantitative claim that the large scale structure of the Universe is quasi-fractal with fractal dimension D ≈ 2. There is now a concensus that this is the case on medium scales, with the main debate revolving around what happens on the scales of the largest available modern surveys.

Space is the only quantity which can be directly observed. Its main characteristics, in its 3-dimensional Euclidean form, are describable by a Clifford algebra. There have been attempts to construct physics on the basis of this parameter alone, involving extensions to extra dimensions and non-Euclidean forms of geometry. However, it seems that adding a second Clifford algebra, identical in structure to the first, but commutative to it, and dual to it in that it contains no extra physical information, with the duality determined by a zero totality condition, gives us the mathematical construction that we need for both quantum mechanics and other forms of physics. The results can be explored from many angles, for example, Clifford algebra, group theory, geometry and topology, to produce significant results in fundamental physics.

In this short informal paper we give a simple and primitive definition of discrete motion beginning prior to the usual notions of space and time. We show how velocity, the relativistic addition of velocities, and the Lorentz factor naturally emerge from simply counting steps in a sequence of discrete abstract motions.

The following sections are included:

• Précis
  • First Explications
  • Second Explications
  • Explication of explications

Over the last four decades, the pharmaceutical industry has been in a continuous decline in effciency with regard to the cost and frequency of approval of new drugs. The pharmaceutical productivity curve is essentially the reciprocal of Moore's law in computer science describing the increase in the speed and effciency of computing, and has been named Eroom's law to reect this fact. The human genome project was conceived as a way to accelerate drug development by describing all of the possible targets of therapy in a single data set. In 2010 Nature published a special news feature on the 10th anniversary of the human genome project, and asked about the benefits of the genome sequence to human health. The answer by the experts who led the project was that health and drug development have not benefitted much. Robert Weinberg one of the leading cancer researchers said that there is little to show for all the money invested in genomic studies of cancer.

The surprising complexity of biology has led to the emergence of the new discipline of systems biology which attempts to organize the myriad details of biology into a coherent framework. In this paper, I provide a personalized description of my several decades long project to develop a robust approach to systems biology using a dynamical systems approach. This was initiated in my Ph.D research where I analyzed the regulatory processes that controlled the growth rate of an intact bacterial cell in different growth media using a reverse engineering approach. While investigating the synthesis of the ribosome, a complex structure that catalyzes the synthesis of all proteins, I discovered a general principle that can serve as a core concept for systems biology. This principle has stood the test of time and many of the predictions have been veried. I will describe this principle and its development and show how it anticipated many important aspects of current systems biology. I will also describe some collaborative work with Pierre Noyes and James Lindesay which resulted in an equation for the core control principle, and a paper by Pierre Noyes showing that this approach to biology is an emergent science as described by Philip Anderson in his paper in Science entitled More is Different.

The following section is included:

• Epilogue