Theoretical and Mathematical Physics

The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. For instance the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory, while ignoring singularities whenever the Lagrangian is degenerate. Moreover we show that there exist two different definitions of the observable (n-1)-forms which allows one to construct observable functionals by integration (which correspond to two different points of view: generalizing the law {p,q} = 1 or the law dF\dt = {H,F}), oddly enough we prove that these two definitions coincides only in the LP situation. Finally other contributions concerning this subject and examples are also given.

The first two attempts to derive a unified field theory as an extension to general relativity were made by Weyl and Kaluza. The main idea was to develop a geometrical model whereby both gravity and electromagnetism emerged naturally from the same curved space-time continuum. Over the years, each of these attempts has engendered one of the main groupings of classical (non-quantum) unified field theories. Weyl sought to alter the geometry of the continuum while maintaining the number of dimensions at four. Kaluza sought to maintain the geometry of the continuum, but account for electromagnetism by altering the number of dimensions to five. It is generally believed that Kaluza’s was the first five-dimensional model of material reality and that it was largely ignored by the scientific community except for Klein’s attempt to use the model as a basis for quantizing relativity, but the model does have a separate history independent of Klein’s modification.  The mere assumption of a five-dimensional space-time structure can take two basic forms, both of which are not as independent of the other as one would expect or hope. First of all, the five-dimensional space-time may be used as a mathematical formalism alone, which has no more meaning than to allow for the extra variables needed to incorporate the electromagnetic field into a single space-time structure. In this manner, the fifth dimension is assumed to have no physical meaning. It was this form that Kaluza’s theory assumed so Kaluza’s original model merely duplicated Maxwell’s electromagnetic model without adding any new physics. In the second type of these theories, the fifth dimension is given an actual physical reality. It is a basic necessity of this type of theory to explain why the fifth dimension seems to be beyond the natural experience of human perception.

The golden ring to which most physicists aspire is a unified field theory that incorporates all of modern and classical physics. Some scientists and others call this a TOE or ‘theory of everything’, but it is no more than false hubris to believe that humans could possibly know and explain everything about the universe at this time. Einstein chased this goal for the last three decades of his life, basing his theoretical research on his general theory of relativity. Meanwhile, the vast majority of scientists supporting the other major accomplishment of the Second Scientific Revolution were investing all of their time and efforts to advancing the quantum theory and their quest has been extremely successful. They originally had no interest in a unified field theory. After Einstein died in 1955, his efforts were all but abandoned because of his philosophical stance against the prevalent Copenhagen Interpretation of quantum theory even though he had been one of quantum theory’s founders. During the 1970s the tables started to turn and quantum theorists became interested in unifying physics, although not from the foundational principles of relativity theory. They claimed that quantum theory was more fundamental than relativity so they began the same quest from a totally different direction despite their claims to be continuing Einstein’s quest.  Throughout the development of the ensuing standard quantum model, superstring theory and many other theoretical schemes, quantum theorists have remained resolute in their conviction that the quantum and relativity are mutually incompatible so the quantum must completely do away with and replace relativity once and for all. However, the quantum theory and relativity are not actually incompatible and, in fact, say the some of the same things about the nature of physical reality. When the similarities are fully defined and studied and the basic assumptions behind each of the theories are altered to reflect the similarities instead of the incompatibilities, only then can the point of their compatibility be determined and act as a unifying principle resulting in a completed unified field theory of the type that Einstein once sought. The development of this physical model of reality is not without irony. Not only is the quantum theory incomplete as Einstein argued in EPR, but Einstein’s general relativity is also seriously incomplete and true unification cannot be rendered complete at any level of reality until all the theoretical models being unified are themselves complete.

The Big Bang model relies on a growing number of hypothetical entities, things that we have never observed inflation, dark matter and dark energy. Without them, there would be a contradiction between the observations and the theory. Singularities are merely artifices of our current (flawed) physical theories. Our main mistake is that we accept an unverifiable assumption that the portion of the universe which can be observed is representative of the whole, and that the laws of physics are the same throughout the whole universe. It is an oversimplification to generalize that the universe is globally flat. We prove mathematically that the universe is hyperbolic. We modify the laws of gravity from the perspective of General Theory to fix the Big Bang model shortcomings.The manifold of the hyperbolic universe possesses nonsingularity. Our modified Schwarzschild metric in the hyperbolic spacetime possesses nonsingularity. Hence Black holes possess nonsingularity. Instead of the nonphysical inflation. The hyperbolic universe possesses a hyperbolic inflation covers the current observed large structure. We prove that the distance horizon of the hyperbolic universe is infinite.

Detailed notes composed during my Project 2019 on Group Theory.
This reading project is aimed at learning the applications of Group Theory in Physics. Properties of sets, groupoid, magma, groups were studied; Lorentz transformations are derived using Group axioms and non uniqueness of inertial frames; Using representation theory and selection rules the allowed molecular transitions and allowed vibrational modes of H2O are determined; The accidental degeneracy of Hydrogen atom is explained as a consequence of the hidden SO(4) symmetry.

All required Math is also provided in this report.

Stable, holomorphic vector bundles are constructed on an torus fibered, non-simply connected Calabi-Yau threefold using the method of bundle extensions. Since the manifold is multiply connected, we work with equivariant bundles on the elliptically fibered covering space. The cohomology groups of the vector bundle, which yield the low energy spectrum, are computed using the Leray spectral sequence and fit the requirements of particle phenomenology. The physical properties of these vacua were discussed previously. In this paper, we systematically compute all relevant cohomology groups and explicitly prove the existence of the necessary vector bundle extensions. All mathematical details are explained in a pedagogical way, providing the technical framework for constructing heterotic standard model vacua.

The symmetry factor of Feynman diagrams for real and complex scalar fields is presented. Being analysis of Wick expansion for Green functions, the mentioned factor is derived in a general form. The symmetry factor can be separated into two ones corresponding to that of connected and vacuum diagrams. The determination of symmetry factors for the vacuum diagrams is necessary as they play a role in the effective action and phase transitions in cosmology. In the complex scalar theory the diagrams different in topology may give the same contribution, hence inverse of the symmetry factor (1/S) for total contribution is a summation of each similar ones (1/S_i), i.e., 1/S = \sum_i (1/S_i).

This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identites for these functions. The integrals are often used in nuclear scattering calculations, where an analytical result can provide an insight into the reaction mechanism. A technique is developed whereby an integral over several special functions which cannot be found in any standard integral table can be broken down into integrals that have existing analytical solutions

It would seem to many physicists that the unification of physics within a single paradigmatic theory has been the primary goal in science for only the past few decades, but this would not be true.  Unification was the original goal of Einstein and a few other physicists from the 1920s to the 1960s, before quantum theorists began to think in terms of unification. However, both approaches are basically flawed because they are individually incomplete as they now stand. Had either side of the controversy just simplified their worldview and sought commonality between the two, unification would have been accomplished long ago. The point is, literally, that the discrete quantum, continuous relativity, basic physical geometry and classical physics all share one common characteristic – a paradoxical duality between a dimensionless point and an extended length in any dimension – and if the problem of unification is approached from an understanding of how this problem relates to each paradigm all of physics could be unified under a single new theoretical paradigm.

This is a short review article in which we discuss and summarize the works of various researchers over past four decades on Zeeman topology and Zeeman-like topologies, which occur in special and general theory of relativity. We also discuss various properties and inter-relationship of these topologies.

According to the recently proposed wave-particle non-dualistic interpretation of quantum mechanics, the physical nature of Schrödinger's wave function is an 'instantaneous resonant spatial mode' in which a quantum flies akin to the case of a test particle moving along a geodesic in the curved space-time of general theory of relativity. By making use of this physical nature, a causal explanation is provided for the delayed-choice quantum erasure experiment.

ABSTRACT We use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into the reaction-diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear representation, the soliton solutions, and their mutual interactions.

Nearly three decades ago, a physical anomaly was discovered that
should have shaken physics to its very foundations. That anomaly,
called the galaxy rotation problem, resulted from the difference
between the expected velocities of material objects in the disk
portion of galaxies and their observed velocities. To explain this
anomaly, the concept of Cold Dark Matter (CDM) was invoked.
Alternative explanations such as MOdified Newtonian Dynamics
(MOND) have been proposed, but altering the basis of Newtonian
physics by adding a fudge factor to the laws of motion is
aesthetically displeasing as well as completely unnecessary. The
dark matter halo that surrounds spiral galaxies can be easily
explained if scientists are willing to accept the physical reality of a
fourth dimension of space, which amounts to a fifth dimension of
space-time. This solution may seem radical, but there is ample
evidence in other areas of physics to support the existence of a
fourth spatial dimension. What is commonly called dark matter is no
more nor less than space-time curvature that is not associated with
local matter, but is due instead to the total or global curvature of the
universe as a whole.

We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections.

ABSTRACT
Radiation from waste dumpsite as well as certain antropogenic radionuclides released to the environment are major contribution to terrestrial outdoor exposure. This paper takes a look at the radiation level from twenty major waste dumpsites in Ogbomoso metropolis of Oyo State, Nigeria. Soil samples were taken from 100 locations for laboratory analysis with a very sensitive gamma ray spectroscopy Nal(Tl) scintillation detector. The outdoor gamma dose rates were determined by Alarm Dosimeter Geiger counter portable device and measurements were taken in air for two minutes at 1m from the ground. The outdoor gamma dose rate range from 16.53 to 42.51nGy/h. Annual effective dose rate varied from 0.021 to 0.053mSv/y and the excess lifetime cancer risk within the metropolis range from 0.05x10-3 to 0.12x10-3.  It was observed that the concentration of radionuclides viz.226 R  234Th  40K and 134 Cs in samples ranges from  7.30 to 35.06Bq/kg,  10.44 to 28.44Bq/kg,  56.42 to 1000.42Bq/kg and 7.60Bq/kg, respectively. The soil radionuclides” concentrations of the study and annual effective gamma dose were within the worldwide range. Excess lifetime cancer risk is far below worldwide rang of 0.29x10-3    (UNSCEAR, 2000) 134Cs detected was due to fission of heavy radionuclide that may be present in the dump site.

Space-time is not a fictitious fabric but is creation of a vibrating Higgs Bosons. A vibrating particle only defines length and time simultaneously, making them inseparable. An agglomeration of such vibrating particles constitutes space-time continuum. It is the dark energy and defines the universe. The inverse square law is the property of space-time dynamics