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| viXra.org > Mathematical Physics > 2010 |
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[12] viXra:2010.0233 [pdf] replaced on 2020-11-11 19:17:14
Authors: J.A.J. van Leunen
Comments: 85 Pages. This is part of the Hilbert Book Model Project [Correction to format made by viXra Admin]
This document concerns the discovery of the grand structure of physical reality in terms of mathematical constructs and mechanisms. The grand structure emerges from simple foundations and leads via spaces that show an increasing complexity to a system of Hilbert spaces that will be called Hilbert repository. This structure acts as a powerful storage medium that physical reality applies to store and retrieve prepared data. The repository can easily capture the dynamic geometric data of all objects that will ever exist in our universe. It can also archive the universe as a dynamic manifold and all other physical fields that play in the lifetime of the universe. This enables physical reality to prepare data in a creation episode in which no running time exists and play the story of all archived objects during a running episode in which a flowing time indicates the progression of that life story as an ongoing embedding process.
The investigation concerns a hierarchy of spaces that show increasing degrees of complexity. Finding more complicated spaces is not a big problem. This turns out to be quite simple. The move to a more complicated platform appears always accompanied by several significant restrictions and understanding where these limitations come from is a much bigger problem. Today's mathematics is only just able to explain the restrictions of the Hilbert space. This no longer applies to the system of Hilbert spaces, which in this document is called the Hilbert repository. Current math cannot explain the restrictions and the extra features of the Hilbert repository. This fact will particularly interest those who are curious about the structure and behavior of physical reality. The approach is quite different from the usual path and provides other insights. The usual way tries to deduce new insights from what we know about classical physics. That path appears to be blocked.
The story makes it clear that only mathematics cannot provide a complete picture of reality, while experiments alone also cannot expose physical reality. The combination of mathematics and experimentation produces the best results.
The behavior of fields plays an important role in most theories. Basic physical fields are dynamic fields like our universe and the fields that are raised by electric charges. These fields are dynamic continuums. Most physical theories treat these fields by applying gravitational theories or by Maxwell equations. Mathematically these fields can be represented by quaternionic fields. Dedicated normal operators in quaternionic non-separable Hilbert spaces can represent these quaternionic fields in their continuum eigenspaces. Quaternionic functions can describe these fields. Quaternionic differential and integral calculus can describe the behavior of these fields and the interaction of these fields with countable sets of quaternions. All quaternionic fields obey the same quaternionic differential equations. The basic fields differ in their start and boundary conditions.
The paper introduces the concept of the Hilbert repository. It is part of a hierarchy of structures that mark increasingly complicated realizations of a purely mathematical model that describes and explains most features of observable physical reality. That model is the Hilbert Book Model.
Many of the subjects that are treated in this document cannot be found in standard textbooks. The paper treats the mathematical and experimental underpinning of the Hilbert Book Model.
Category: Mathematical Physics
[11] viXra:2010.0215 [pdf] submitted on 2020-10-26 20:38:18
Authors: Malik Matwi
Comments: 69 Pages. [Corrections made to conform with the requirements on the Submission Form]
Chern-Simons theory is a gauge theory in $2+1$ dimensional spaceime. This theory does not depend on additional structures, like a metric structure, thus it is a topological quantum theory that measures topological invariants like linking numbers, Jones polynomial, and other quantum invariants for knots and 3-manifolds. The equations of motion of Chern-Simons action is vanishing of the curvature $F = 0$. No metric is used in forming the action principle. One might expect the path integral to be a topological invariant of $3$ manifolds. The difference for the equation of motion with the Maxwell theory is that the Maxwell theory has non-trivial solution of curvature $F\ne0$ in absence of matter, while the Chern-Simons theory has solution only with $F=0$. The Chern-Simons theory has non-trivial solution with $F\ne0$ only when the gauge field couples with matter. Since the action functional of the Chern-Simons theory is first order in space-time derivatives, its Legendre transform gives the trivial Hamiltonian $H=0$. So there is no dynamics, and the only dynamics would be inherited from coupling to dynamical matter fields
Category: Mathematical Physics
[10] viXra:2010.0212 [pdf] submitted on 2020-10-26 20:13:46
Authors: Miroslav Josipović
Comments: 3 Pages.
This text is another in a series of supplements to the book [1].
Category: Mathematical Physics
[9] viXra:2010.0195 [pdf] submitted on 2020-10-23 19:50:49
Authors: M. D. Monsia
Comments: 11 Pages.
We perform in this paper a mathematical analysis of a supposed purely nonlinear isotonic oscillator designed to be a generalization of the Ermakov-Pinney differential equation. We calculate its exact and general solution. This allows the determination of new periodic solutions to the Ermakov-Pinney equation as well as non-periodic solutions as complex-valued function. In this context all motions corresponding to this nonlinear isotonic oscillator are not periodic so it is not consistent to consider such differential equations with real coefficients as conservative oscillators which can only have real and periodic solutions like the harmonic oscillator equation.
Category: Mathematical Physics
[8] viXra:2010.0184 [pdf] replaced on 2020-10-23 01:53:37
Authors: Miroslav Josipović
Comments: 4 Pages. A few typos corrected.
This text is for the young people, as an extension to the book [3].
Category: Mathematical Physics
[7] viXra:2010.0174 [pdf] replaced on 2020-10-24 11:42:12
Authors: Bogdan Szenkaryk
Comments: 3 Pages.
The article presents the errors of German researchers who measured the shortest period of time on the basis of one hydrogen H2 molecule.
Category: Mathematical Physics
[6] viXra:2010.0149 [pdf] replaced on 2020-10-20 08:20:20
Authors: Ervin Goldfain
Comments: 5 Pages.
Fractional-time Schrödinger equation (FTSE) describes the evolution of quantum processes endowed with memory effects. FTSE manifestly breaks all consistency requirements of quantum field theory (unitarity, locality and compliance with the clustering theorem), unless the order of fractional differentiation and integration falls close to one. Working in the context of the minimal fractal manifold, we confirm here that FTSE approximates the attributes of gravitational metric and provides an unforeseen generation mechanism for massive fields.
Category: Mathematical Physics
[5] viXra:2010.0112 [pdf] submitted on 2020-10-16 08:20:18
Authors: Miroslav Josipović
Comments: 3 Pages.
This short text is a continuation of a series on the applications of geometric algebra, related to the book [3]. As the book [3], this text is intended for young (or at least young at heart) and open-minded people.
Category: Mathematical Physics
[4] viXra:2010.0086 [pdf] replaced on 2020-12-05 14:00:19
Authors: Richard D. Lockyer
Comments: 34 Pages. Keywords: Sedenion Algebra, Octonion Algebra, algebraic invariance, algebraic variance, algebraic invariant Sedenion zero divisors, Cayley-Dickson doubling, observables, non-observables
The Cayley-Dickson dimension doubling algorithm nicely maps R → C → H → O → S and beyond, but without consideration of any possible definition variation. Quaternion Algebra H has two orientations, and they drive definition variation in all subsequent algebras, all of which have H as a subalgebra. Requiring Octonion Algebra O to be a normed composition algebra limits the possible orientation combinations of its seven H subalgebras to sixteen proper O orientations, which are itemized. Identification of the O subalgebras for Sedenion Algebra S and orientation limitations on these subalgebras provides a fully algebraic proof that all O subalgebras cannot be oriented as proper Octonion Algebras, verifying Sedenion Algebra is not generally a normed composition division algebra. The 168 standard Cayley-Dickson doubled Sedenion Algebra primitive zero divisors are presented, as well as representative forms that will yield primitive zero divisors for all 2048 possible maximal set proper O subalgebra orientations for Sedenion Algebras; algebraic invariant Sedenion primitive zero divisors. A simple mnemonic form for validating proper O orientations is provided. The method to partition any number of O algebraic element products into product term sets with like responses to all possible proper O orientation changes; either to a single algebraic invariant set or to one of 15 different algebraic variant sets, is provided. Most important for O based mathematical physics, the stated Law of Octonion Algebraic Invariance requires observables to be algebraic invariants. Its converse, The Law of the Unobservable suggests homogeneous equations of algebraic constraint built from the algebraic variant sets. These equations of constraint are important to mathematical physics since they will limit the family of solutions for the differential equations describing reality and do not have their genesis in experimental observation. An alternative to the Cayley-Dickson doubling scheme which builds by variations is provided.
Category: Mathematical Physics
[3] viXra:2010.0080 [pdf] replaced on 2020-12-02 10:51:14
Authors: Miroslav Josipović
Comments: 8 Pages.
This text is motivated by the desire to point out some more applications of geometric
algebra in physics. The presentation is simplified, and the reader is referred to the literature.
Category: Mathematical Physics
[2] viXra:2010.0048 [pdf] submitted on 2020-10-08 03:17:49
Authors: Ramzi Suleiman
Comments: 23 Pages.
Many founding fathers of science have underscored the importance of beauty in mathematical representations of natural phenomena and their connection with the beauty of the objects they represent. Paul Dirac, for example, believed that the beauty of a mathematical equation might be an indication that it describes a fundamental law of nature; that it is a feature of nature in that fundamental physical laws are typically described in terms of great beauty. However, despite the growing body of research on beauty in nature and in mathematical representations of natural phenomena, we are unaware of studies in physics and mathematics devoted to the objective beauty induced by motion, regardless of the aesthetic qualities of the moving body. We undertake this objective by focusing on the Doppler formula, which describes the shifts in wave frequencies caused by the motion of the wave's source relative to a human observer or receiver. We underscore the apparent beauty of the equation and uncover several fascinating golden and silver ratios of the base formula and its mathematical moments. Furthermore, we allude to existing applications of the Doppler Effect and golden ratio aesthetics in computer-generated music, and sonar image-detection technology. We also propose a similar usage in the rapidly developing applications of Wi-Fi and smartphones to sense human motion. We point to appearances of the Doppler formula and its moments in quantum physics and the relativity of information, and contemplate the possibility of a deeper level of physical reality.
Category: Mathematical Physics
[1] viXra:2010.0039 [pdf] replaced on 2020-10-09 08:24:51
Authors: Ervin Goldfain
Comments: 10 Pages.
Complex Ginzburg-Landau equation (CGLE) is a paradigm for the onset of chaos and turbulence in nonlinear dynamics of extended systems. Here we point out that the underlying connection between CGLE and the Navier-Stokes (NS) equation bridges the gap between fluid flows, on the one hand, and the mathematics of General Relativity (GR) and classical gauge theory, on the other. The analogy hints to a possible link between the transition from laminar to turbulent flows and the mass generation mechanism of quantum field theory (QFT).
Category: Mathematical Physics
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