Mathematical Physics

The Most Beautiful Form of the Dirac Equation, and Some Speculations re "123 Mystery"

Authors: Warren D. Smith
Comments: 9 Pages.

Dirac 1928's relativistic electron wave equation had involved 4×4 matrices. We show how to reformulate it using 2×2 matrices only, and also that we can get rid even of them provided we make the wavefunction be biquaternion-valued. The Majorana equation then has quaternion-valued wavefunction. We then speculate that extensions of this idea might be able to solve one or two of the oft-cited "great mysteries" of physics.
Category: Mathematical Physics

The Velocity-Wavelength Right Triangles Unit Circles Standard Clock and the Twin Paradox

Authors: Claude Michael Cassano
Comments: 6 Pages.

Using energy and momentum definitions: E=(m_0)c²/sqrt(1-v²/c²) and p=(m_0)v/sqrt(1-v²/c²)their frequencies and wavelengths may be shown as reciprocal transformations: f=c/λ_{c} , λ=c/f_{c} yielding the generally referred to: (λ_{c)}(f_{c})=v=v_{g} group velocity ; and λf=(c/f_{c})(c/λ_{c))=c²/((λ_{c)}(f_{c}))=c²/v=v_{p} phase velocity and yields the right triangles: ((((m_0)c)/h)λ_{c})²(v/c)²=1 and [((((m_0)v)/h))λ]²(v/c)²=1 represent wavelength-velocity right triangles with unit hypotenuses (radii of unit circles) As unit circles, the time rate of change of the angle of these wavelength-velocity right triangles may be considered a standard clock of a relativistic reference frame. Analysis may be made for any planets in a solar system; and similarly for binary star/planet/moon system(s) and also for bodies within rotating galaxies. Using this standard relativistic clock the effects of acceleration concerning the twin paradox may be calculated.
Category: Mathematical Physics

Collision Entropy Estimation in a One-Line Formula

Authors: Alessandro Gecchele
Comments: 14 Pages.

We address the unsolved question of how best to estimate the collision entropy, also called quadratic or second order Rényi entropy. Integer-order Rényi entropies are synthetic indices useful for the characterization of probability distributions. In recent decades, numerous studies have been conducted to arrive at their valid estimates starting from experimental data, so to derive suitable classification methods for the underlying processes, but optimal solutions have not been reached yet. Limited to the estimation of collision entropy, a one-line formula is presented here. The results of some specific Monte Carlo experiments give evidence of the validity of this estimator even for the very low densities of the data spread in high-dimensional sample spaces. The method strengths are unbiased consistency, generality and minimum computational cost.
Category: Mathematical Physics

A Compact CPW-Fed Triple Band Monopole Antenna with Simple Structure for WLAN/WiMAX Applications

Authors: Baek Ye Song, Kim Nam Chol Kim Un Chol, Kim Ju Il, Song Sung Bom
Comments: 5 Pages.

In this paper, a new compact coplanar waveguide (CPW)-fed triple band monopole antenna with dual circular rings and two L-shaped parasitic strips on the ground plane is presented. The fabricated antenna with new simple structure has a compact size of 26 mm × 34 mm. The proposed antenna has impedance bandwidths of 2.39-2.69 GHz, 3.21- 4.12GHz and 5.08 -5.86GHz with a reflection coefficient of less than -10dB. The measured performances are compared with the simulated results and show good agreement. The proposed triple band monopole antenna also has good omnidirectional radiation patterns and available gain. The simple structure, ease of implementation, compact size and good performances make the proposed antenna an excellent candidate for WLAN and WiMAX applications.
Category: Mathematical Physics

Design and Implementation of a Compact CPW-Fed Planar UWB Antenna with Quadruple Band Notched Characteristics

Authors: Kim Dae Il, Kim Un Chol, Ryu Dong Myong, Choe Jong Yol, Pak Chung Jin
Comments: 9 Pages.

A novel coplanar waveguide ()-fed planar UWB antenna with quadruple band-notched property is presented. The quadruple band rejection is achieved by etching a M-shaped slot on the radiation patch and a U-shaped slot on the feed line, and adding two L-shaped parasitic strips near radiator patch and two split ring resonators (SRRs) on the back surface of the substrate. The voltage standing wave ratio (VSWR) results show that the proposed antenna exhibits good wideband performance over an UWB frequency range from 3 to 11.8 GHz with VSWR less than 2, except for four stop-bands at 3.3~3.75GHz, 5.15~5.85GHz, 7. 2~7.76GHz, 8~8.55GHz respectively. It also demonstrates a nearly omnidirectional radiation pattern and available gain. The fabricated antenna has a tiny size, only 34mm 31mm0.508mm. The simulated results are compared with the measured results and good agreement is obtained. The simple structure, compact size and good characteristics make the antenna an excellent candidate for UWB applications.
Category: Mathematical Physics

Gauge Invariance in Counting and Combinatorics

Authors: Koustubh Kabe
Comments: 7 Pages. This paper explores a new kind of gauge invariance that makes its appearance in counting and combinatorics.

There is perhaps no part of mathematics that is more intimately connected with everyday experiences than probability theory and statistics. The element of chance dominates the physical world. Probability is the heart of physics, in particular — quantum physics. At the probability theory lies combinatorics. We make an observation in the combination of n objects taken r objects at a time. We find a sort of combinatorial gauge invariance hidden there in that the combination of �� objects taken �� at a time or �� − �� at a time is the same. It has been argued that the Pauli Exclusion Principle is not a principle or cause at all; rather it is an effect of the combinatorics which essentially delivers the Fermi Dirac Statistics. We explore the consequent applications in quantum mechanics and field theory with particle statistics.
Category: Mathematical Physics

Grondvesten (Foundations)

Authors: Hans van Leunen
Comments: 360 Pages. In Dutch (Dit is onderdeel van het Hilbert Book Model project)

Het Hilbert Book Model project heeft deze publicatie opgeleverd. De zoektocht naar een betrouwbare basis van de fysieke werkelijkheid heeft veel tegenslagen gekend en verloopt traag. Als gevolg hiervan raakte de reguliere natuurkunde op een zijspoor. Kwantumveldentheorie, kwantumelektrodynamica en kwantumchromodynamica gebruiken het minimale actieprincipe als basis. Het Hilbert Book Model laat zien dat continuüms tot de derde fase van een speciale verzameling behoren en niet kunnen werken als basis van de wiskundige fysica. Dit document laat zien hoe de drie fasen van de speciale verzameling leiden tot een vectorruimte en getalsystemen, die van toepassing zijn op een systeem van Hilbertruimten waarin het lokale universum en een parallel multiversum kunnen optreden. Ook laat het document zien dat de wetenschap het Higgs-deeltje of het Higgs-veld niet als onderdeel van het standaardmodel moet beschouwen. In plaats daarvan zou het standaardmodel van de experimentele deeltjesfysici zich moeten beperken tot de elementaire fermionen. De meeste natuurkundigen interpreteren fotonen als excitaties van het elektrische veld. Daarentegen interpreteert het HBM fotonen als ketens van donkere energieobjecten, en de donkere energieobjecten zijn schokfronten die het veld verstoren en zodoende het lokale universum vertegenwoordigen. Hoplandingen van de toestandsvectoren van de fermionen produceren bolvormige schokfronten die met lichtsnelheid wegsnellen van de locatie van deze landing. Dit is in strijd met de ideeën van behoudswetten die in de reguliere natuurkunde spelen. Volgens het HBM heeft een oerknal nooit plaatsgevonden. Het model beschouwt twee gedeelten en aan het begin van het tweede gedeelte begint de tijd samen te lopen met een voortdurende schepping van fermionen.

The Hilbert Book Model project has produced this publication. The search for a reliable basis of physical reality has had many setbacks and is progressing slowly. As a result, mainstream physics got sidetracked. Quantum field theory, quantum electrodynamics and quantum chromodynamics use the minimal action principle as a basis. The Hilbert Book Model shows that continuums belong to the third stage of a special set and cannot function as the basis of mathematical physics. This paper shows how the three phases of the special set lead to a vector space and number systems, applicable to a system of Hilbert spaces in which the local universe and a parallel multiverse can occur. The paper also shows that science should not consider the Higgs boson or the Higgs field part of the Standard Model. Instead, the experimental particle physicists' standard model should be limited to the elementary fermions.Most physicists interpret photons as excitations of the electric field. In contrast, the HBM interprets photons as chains of dark energy objects, and the dark energy objects are shock fronts that distort the field and thus represent the local universe. Hop landings of the fermions' state vectors produce spherical shock fronts that speed away from the location of this landing at the speed of light. This is contrary to the ideas of conservation laws at play in mainstream physics. According to the HBM, a big bang never happened. The model considers two parts, and at the beginning of the second part, time begins to coincide with a continuous creation of fermions.
Category: Mathematical Physics