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Thread: regrouping quarks

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    Re: regrouping quarks

    Since each quark is composite of spacetime charges, a nucleon (e.g. either proton or neutron) is further decomposed into spacetime charges. At the level of quark composition, a proton is made of 2 up quarks and 1 down quark, while a neutron is made of 1 up quark and 2 down quarks. Since an up quark is made of 5 H+ and 1 H- and a down quark is made of 1 H+ and 3 H-, at the level of spacetime charge composition, a proton is made of 11 H+ and 5 H-, while likewise a neutron is made of 7 H+ and 7 H-. The physical transformation of a proton into neutron can now be viewed as rearrangement of spacetime charges. However, although 11 H+ provide more than enough to arrange 7 H+, 5 H- would not be enough to form any arrangement of 7 H-. To satisfy this full arrangement, 2 H- must be borrowed from the quantum vacuum reservoir of spacetime charges. This need not be a full loan since the excess of 4 H+ can exchange as collateral with 2 H- of the reservoir. High energy experiments already determined that the up quark’s mass is between 1 to 5 MeV. And the mass of the down quark is between 3 to 9 MeV. From these experiment data the average mass of the down quark is twice the mass of the up quarks. Independent calculation has also determined that the coupling constant between spacetime charge of negative curvature and positive curvature is 2.01259 using proton mass of 1.673x10-27 kg and neutron mass of 1.675x10-27 kg and the facts that the proton is made of 11 H+ and 5 H- and the neutron is made of 7 H+ and 7 H- solving simultaneously two simple linear equations: (11H+)+(5H-)=1.673x10-27 and (7H+)+(7H-)=1.675x10-27 for H+ or H-. The solutions are H+=7.9428571x10-29 and H-=15.9857143x10-29. Dividing 15.9857143x10-29 by 7.9428571x10-29 gives 2.01259 to five decimal place accuracy. Geometrically, if the six spacetime charges composing the up quark are arranged as vertices of a regular polyhedron then the up quark can possibly be shaped into a regular octahedron. Likewise, the four spacetime charges of the down quarks as vertices can possibly be shaped into a regular tetrahedron. Geometrically speaking, transforming an up quark into a down quark is topologically equivalent to transforming an octahedron into a tetrahedron plus exchanging four positive curvature spacetime charges with two negative curvature spacetime charges from the quantum vacuum spacetime charges reservoir. The octahedron has 12 convex edges, while the tetrahedron has 6 convex edges. If the lengths of all these 18 edges are equal then the tetrahedron is considered more compact and hence more massive. At this level of spacetime charges arrangement, compactness of a particular spacetime structure is always inversely proportional to edge length of the corresponding polyhedron. In this sense, a strange quark is also a tetrahedron of shorter edge length than the down quark. A bottom quark is also a tetrahedron of shorter edge length than the strange quark. Likewise, a charm quark is also an octahedron of shorter edge length than the up quark. A top quark is also an octahedron of shorter edge length than the charm quark. Incidentally, wholesale exchange of spacetime charges from the quantum vacuum spacetime charges reservoir is also possible by exchanging a real octahedron with a virtual tetrahedron from the quantum vacuum reservoir. Unfortunately, wholesale exchange requires much bigger energy loan and by the Einstein’s quantum uncertainty of changes in time versus changes in energy whose product is always greater than or equal to a fixed factor of Planck’s constant of action, the large energy loan must be paid back to the quantum vacuum reservoir almost immediately. In other words, imitating the nature of spacetime charges rearrangement must not violate the physical principle of least action.
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    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    Suppose that the two irreducible vacuum spacetime charges are 2 by 2 2nd order square symmetric Hadamard matrices: 2H+=[1,-1;-1,1] and 2H-=[-1,1;1,-1]. Similarly, the spacetime charges representing the nucleon are 6th order, 6 by 6 square symmetric Hadamard matrices: 6H+=[1,-1,1,-1,1,-1;-1,1,-1,1,-1,1; 1,-1,1,-1,1,-1; -1,1,-1,1,-1,1; 1,-1,1,-1,1,-1; -1,1,-1,1,-1,1] and 6H-=[-1,1,-1,1,-1,1; 1,-1,1,-1,1,-1; -1,1,-1,1,-1,1; 1,-1,1,-1,1,-1; -1,1,-1,1,-1,1; 1,-1,1,-1,1,-1]. Topologically speaking, borrowing a 6H- from the vacuum spacetime charge reservoir is the same as grouping nine 2H- together. However, computationally speaking, the 2 by 2 matrix serves as the kernel matrix and to its right four column multiplier matrices are (2x3)(3x4)(4x5)(5x6), which can be multiplied associatively but not commutatively. To the left of the kernel matrix are four row multiplier matrices (6x5)(5x4)(4x3)(3x2), which again can be multiplied associatively but not commutatively. The final product of these matrices is a 6 by 6 Hadamard matrix, preferably a square symmetric Hadamard matrix equal to the integer multiples of 6H-=[-1,1,-1,1,-1,1; 1,-1,1,-1,1,-1; -1,1,-1,1,-1,1; 1,-1,1,-1,1,-1; -1,1,-1,1,-1,1; 1,-1,1,-1,1,-1]. The full transformation ordered left-right matrix multipliers is given by (6x5)(5x4)(4x3)(3x2)(2x2)(2x3)(3x4)(4x5)(5x6) ->-> (6x6). Here, the central kernel matrix is denoted by bold typeset. Alternatively, there are also two irreducible vacuum spacetime charges given by 3H+=[1,-1,1;-1,1,-1;1,-1,1] and 3H-=[-1,1,-1;1,-1,1;-1,1,-1]. These two are 3rd order 3 by 3 square symmetric Hadamard matrices whose spacetime curvature traces can never add to zero. Alternatively, borrowing a 6H- from the vacuum spacetime charge reservoir is the same as grouping two 3H+ and two 3H- together to form equally likely one 6H+ or one 6H-. Analogously, the computational sequence of raising a 3rd order square symmetric Hadamard matrix into a 6th order square symmetric Hadamard matrix is given by the full transformation ordered left-right matrix multipliers: (6x5)(5x4)(4x3)(3x3)(3x4)(4x5)(5x6) ->-> (6x6). Note again that the kernel matrix is in bold type and again these seven matrix factors can be multiplied associatively but not commutatively. On the other hand, to return a 6 by 6 nucleon spacetime charge back to the vacuum reservoir the two full transformations left-right matrix multipliers are (2x3)(3x4)(4x5)(5x6)(6x6)(6x5)(5x4)(4x3)(3x2) ->-> (2x2) and (3x4)(4x5)(5x6)(6x6)(6x5)(5x4)(4x3) ->-> (3x3). The central nucleon kernel matrices are denoted by bold types. Note the column matrix multipliers are now on the left and the row matrix multipliers are now on the right of the kernel matrix. Direct wholesale transformations left-right matrix multipliers also exist. However, these would all violate the physical principle of least action: (6x2)(2x2)(2x6) ->-> (6x6), (6x3)(3x3)(3x6) ->-> (6x6), these two show left-right symmetry (that is to say there are the same number of matrix multipliers on the left and on the right of the kernel matrix), while (6x2)(2x2)(2x4)(4x6) ->-> (6x6) and (6x5)(5x3)(3x3)(3x4)(4x5)(5x6) ->-> (6x6) show left-right broken symmetry. The number of left matrix multipliers is not the same as the number of right matrix multipliers. Note the kernel matrices are in bold types. Again, all intermediate matrix products are associative but not commutative. In general, the transformations using left-right matrix multipliers should also give final products that are nonsymmetric Hadamard matrices or matrices with zero elements. In other words, square symmetric matrices can be transformed into square nonsymmetric matrices, vice versa.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    Hypothetically speaking, the quantum statistical mechanics of regrouping quarks is more complicated that the traditional quantum statistical mechanics provided by the statistical distinction of random particle interactions obeying Bose-Einstein statistics or obeying Fermi-Dirac statistics. Both describe the behavior of many particles systems interactions whether distinguishable particles or indistinguishable particles. The Bose-Einstein statistics describes the many-particle interactions of indistinguishable or distinguishable particles called bosons, while the Fermi-Dirac statistics describes the many-particle interactions of indistinguishable or distinguishable particles called fermions. Bosons interactions violate the physical principle called Pauli Exclusion Principle. This physical violation implies that infinitely many indistinguishable or distinguishable bosons can share the same quantum state. The physical phenomena exhibiting this sharing are grouped as a Bose-Einstein condensation. If this sharing is the internal energy of each particle then the total energy of this many-particle system is simply the sum of each individual energy contribution from each single particle of the many-particle system. The concurrent advances of experimental science gives one physical example of this quantum states sharing of energy called the LASER as discovered by Gordon Gould in 1957. The legal priority battle of discovering LASER between Gould and others was finally resolved in 1987 when a Federal judge directed the U. S. Patent Office to award patents to Gould for the optically pumped and the gas discharge LASERs. Gould died September 16, 2005 at age 85. However, the theoretical foundation of LASER was established in 1917 by Albert Einstein in his German paper Zur Quantentheorie der Strahlung (On the Quantum Theory of Radiation). Fermions, on the other hand, always obey Pauli Exclusion Principle. Einstein’s theory of LASER (acronym for light amplification stimulated emission of radiation) is actually a re-derivation of Max Planck’s law of radiation based on probability coefficients, which are now called Einstein coefficients to distinguish them from other probability coefficients of quantum mechanics proper. Nonetheless, in this proper context of a quantum theory of the spacetime continuum, both fermions and bosons are simply composites of spacetime charges. The new statistical mechanics of spacetime charges asserts that each of these can exist in three possible quantum states: (1) the quantum state of positive spacetime curvature, (2) the quantum state of negative spacetime curvature, and (3) the quantum state of zero spacetime curvature, respectively denoted by H+, H-, and H0. Since all these quantum states can also be represented by square Hadamard matrices, the order of each matrix also counts as a distinguishable quantum state of spacetime charges. This matrix order is conventionally indicated by left justified subscripts exclusively for all square matrices and for the 2nd order and 3rd order square matrices these are respectively denoted by 2H+, 2H-, 2H0 and 3H+, 3H-, 3H0. A 6th order positive curvature spacetime charge is then denoted by 6H+. Here again reminded that only spacetime charges of the same matrix order can interact. Note that row and column matrix multipliers are not representations of spacetime charges but act as transformation matrices for lowering or raising the order of square Hadamard matrices. The mathematical reasoning is that only square matrices of the same order can be multiplied or added together. Moreover, all the rules of matrix mathematics apply to spacetime charges of arbitrary order.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    The natural world is full of coincidences. These appear in various guises either mathematical or physical. Any accurate imitation of nature always takes at the least one of these coincidences into consideration, for example: Why all identified living things that fly (birds, insects, flying fishes, etc.) must have at least a pair of wings? The correct physical deduction from this simple encompassing question is that all naturally imitated flying objects must have a pair of wings in order to fly, making them one of the most important aviation’s design challenges for many inexperienced aerospace and aeronautic engineers. However, in connection to regrouping quarks these mathematical and physical coincidences are the more important if they appear frequently in the numerous mathematical formulations or reformulations of physical laws. One of particular interest is the frequent appearance of the virial theorem. This is the same virial theorem as set forth originally by the German physicist Rudolf Clausius in 1870 during the birth of the physics of statistical thermodynamics. In Ludwig Boltzmann own words (see his lectures on gas theory, especially the introduction to Part I of the lectures): It was Clausius who made the physical as well as the theoretical distinction for the special theory of heat and the general theory of heat. The former is a mechanical theory dependent on Newton’s laws of motion and of gravitation, in other words, a microscopic theory. The latter is a thermal theory dependent on temperature, in other words, a macroscopic theory. Combine them together; they form the kinetic theory of heat or of molecular motion (note that before the turn of the 20th century the existence of atoms and molecules was not widely or completely accepted by the scientific community. In fact then, many still believed in the caloric theory of heat). But in using the term “virial” Clausius had in mind the infinitely many microscopic atomic forces within every particle of a dynamic fluid system and the word “virial” is derived from the Latin word for force, distinctly different from the concept of a force field. A fluid system in which every atom having Cartesian coordinates (x,y,z) is acted upon by an infinitesimal primary force from every neighboring atoms with parallel components (X, Y, Z) such that the virial is defined to be equal to the average value with respect to time of the sum of all expressions of the form: -½(xX+yY+zZ). Note that each addend in the parenthesis has units of energy. On the other hand, from thermal consideration the infinite series expansion for the equation of state is given by pv=RT(1+B/v+C/v2+D/v3+E/v4+…) where the constants B, C, D, and E are the temperature dependent virial coefficients. Note that both pv and RT have units of energy therefore each addend in the parenthesis is now unitless, many other scientists prefer to call them dimensionless. Others find it more convenient to express the virial equation in the infinite series expansion of increasing powers of pressure: pv=RT(1+B’p+C’p2+D’p3+E’p4+…) where the constants B’, C’, D’, and E’ are the equivalent temperature dependent virial coefficients. For ideal fluids all the virial coefficients can be identically zero and the equation of state is simply given by pv=nRT where n is the amount of substance involved or the limit of the infinite series expansion. The latter is true if and only if both series converges to their corresponding limit. Some of the more important thermodynamic applications of this theorem can be proved using either classical mechanics or quantum mechanics. One of its many productive applications has been extended into relativistic mechanics (i.e. anything moving near or at exactly lightspeed). In classical mechanics, the virial theorem becomes a special property of central forces relating the time-average value of the negatively signed kinetic energy to half the time-average value of the positively signed potential energy if and only if the central force between two objects is inversely proportional to the square of the distance between them, otherwise known as the inverse square law of physics. A simple proof of this theorem is found in the book by Herbert Goldstein on Classical Mechanics, Addison-Wesley, 1950. A proof in terms of quantum mechanics can be found in the book by Leonard I. Schiff on Quantum Mechanics, 2nd Edition, McGraw-Hill, 1955. The use of the virial theorem to calculate the mass of a cluster of elliptical galaxies was cited by Steven Weinberg in his book on Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, 1972. Of course there are many more comprehensible citations on the virial theorem. However, in this context only one last citation will be mentioned. This is the one cited by Arnab Rai Choudhuri in his book on The Physics of Fluids and Plasma: an Introduction for Astrophysicists, Cambridge University Press, 1998. Choudhuri devoted the last chapter of his book to discuss the important of the virial approach in describing and understanding all types of fluid and plasma systems. Since his book is suitable for graduate level courses, he uses ample advanced mathematics to get all his points across. These would include solving both types of partial differential equations and integral equations relevant to the physics of plasma dynamics. Choudhuri considers the virial approach as a blanket theorem for solving all difficult problems with incomplete knowledge of a particular dynamic system. For him, the theorem always subsumes some global interconnections among the different forms of energy in the system, for example, between kinetic and potential energies. Nonetheless, for a quantum theory of the spacetime continuum, in addition to the Lagrangian and the Hamiltonian functional forms of energy there is one more functional for a square of energy, which represents a single spacetime charge describes by a square symmetric Hadamard matrix, which operates by the same mathematical rules of real matrices in contrast to imaginary or complex matrices use in quantum mechanics and quantum field theories.

    Reference: An excellent but brief exposition of the virial theorem is found in Volume I with the title: Mechanics of the Landau and Lifshitz Course of Theoretical Physics by the Institute of Physical Problems, USSR Academy of Science and translated from the Russian by J. B. Sykes and J. S. Bell, originally published by Pergamon Press, between 1958 and 1965.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    At the start of the 20th century, Max Planck (1900 theory – 1918 Nobel) established and set in motion the progressive development of the quantum theory. Together with Albert Einstein (1905 theory – 1921 Nobel), Niels Bohr (1913 theory – 1922 Nobel), and Louis de Broglie (1925 theory – 1929 Nobel), these four physicists created within a time span of 25 years what is now known as the Old Quantum Theory (OQT). The mathematical solutions of the OQT that are in good agreement with experiments were developed simultaneously by Werner Heisenberg (1932 Nobel) and Erwin Schrödinger (1933 Nobel shared with Dirac), respectively called matrix mechanics and wave mechanics. Planck’s discovery of the quantum law of blackbody radiation in 1900 was awarded the Nobel Prize after 18 years. In hindsight, the new quantum theory would properly incorporate Einstein’s special theory of relativity, which was successfully done by Paul Dirac in 1929. Although Einstein was never awarded a Nobel Prize for his special theory of relativity, which was instrumental for creating the new quantum theory, his 1905 theory on the photoelectric effect instrumental to OQT took 16 years to be recognized by the Nobel Committee. It took only nine years for the Nobel Committee to recognize Bohr’s theory. It took four years for the Nobel Committee to recognize de Broglie’s theory. The success of the new and the old quantum theories did not completely eliminate the finer discrepancies between theories and experiments during the periods of the 1930s and the 1940s. Then all basic quantum researches stop during WWII. After the war, scientists resumed their research then plausible solutions to old problems start to emerge. The continued check and balance between theories and experiments culminated in the formulation of the Standard Model of elementary particles within the discipline of high energy physics. The final missing piece of the puzzle was the spin zero scalar Higgs boson discovered in the start of the second decade of the 21st century. Before his death in 1947 at age 89, Planck had continued since 1900 his independent research on the physical understanding between reversible and irreversible processes of the universe. For Planck, any physical process that is reversible always obeys the principle of least action. On the other hand, any physical process that is irreversible always obeys the 2nd law of thermodynamics, which is also known as the law of increasing entropy. Entropy provides a real arrow of time. This implies that the flow of time is unidirectional, which also implies that time cannot be quantized. If time cannot be quantized then entropy also cannot be quantized. Therefore, any quantum theory of spacetime that also obeys the 2nd law of thermodynamics is not theoretically possible. Philosophical efforts to resolve spacetime quantization had been done by modern philosophers, notably by Hans Reichenbach. Nonetheless, today, this anticipated resolution remains unsolved. Fortunately, a quantum theory of the spacetime continuum is possible if spacetime is replaced by local infinitesimal curvature of two distinct closed Möbius topologies. This distinction of non-equivalence of topology is mathematically possible if local infinitesimal motion (LIM) is postulated. This LIM is meaningful only in one dimension since in two dimensions there are infinite degrees of freedom. However, guided by a new physical principle of directional invariance, the LIMs can be structured into a spacetime charge with multi-dimensional existence as the square of energy, which is properly described and operated mathematically by the algebra of Hadamard matrices. Although the LIM represents a continuous motion, what can be quantized is its direction. This means that the physical concept of spacetime is equivalent to the physical concept of direction. Although time cannot be independently quantized, space and time combined into a quantized direction on one dimensional Möbius topology. The mathematical advantage of quantized abstract directions is solely the use of real Hadamard matrices instead of complex imaginary matrices. Furthermore, the mathematics of spinors and tensors are replaced by the mathematics of deuce vectors and trivectors, the reason being that the concatenations of deuce vectors and trivectors form Hadamard matrices. Fortunately, there are only four irreducible deuce vectors. Likewise, there are only eight irreducible trivectors. These serve as the building blocks for a quantum theory of the spacetime (directional) continuum.
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    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    Virtual reality is multidimensional. In fact, it is infinitely dimensional. It has infinite degrees of freedom. In this sense, some people would say that anything is possible in virtual reality, while not everything is possible in physical reality. Out of this infinity of possible choices, and guided by the principle of least action, nature chooses eight freedom directions to represent a real object existing in physical reality. Likewise, physical reality is also multidimensional. However, since any multiple of eight does not span infinity, physical reality has finite degrees of freedom. Each complete set of eight directions would then represent a directional charge (spacetime charge). Each spacetime charge has an intrinsic continuous motion defined as the local infinitesimal motion (LIM). LIM exists only in one spatial dimension. In turn, each LIM also defines a unit of temporal change, which in layman language is called time. This change in time is simply a change in one dimensional direction. Continuous change in one dimensional direction forms a distinct Möbius topology. Consequently, two degrees of freedom in one dimension form two distinct Möbius topologies denoted as the H+ and the H- topologies. Both topologies can be mathematically described and operated by square symmetric Hadamard matrices. All operational rules of matrices apply with the exception that all square symmetric Hadamard matrices are not invertible, which means their corresponding determinants are all identically zero. Zero determinants is not a mathematical limitation. On the contrary, it provides physical definition for zero spacetime curvatures, representing zero physical mass of real objects in the universe, for example, quarks and leptons as well as scalar and vector bosons. Moreover, zero determinants justify the uniqueness of the two distinct types of directional charges of both virtual and physical realities. Analogous to Russian dolls or layers of an onion, physical reality finite dimensionality is hierarchical. The irreducible building beams of this hierarchical scaffold are the four fundamental deuce vectors and the eight fundamental trivectors. These serve as skeletons for the building blocks of both virtual and physical realities. These building blocks are real Hadamard matrices whose elements are all real numbers with three possible values: +1, -1, and 0. Hierarchical building blocks are constructed from different orders Hadamard matrices. To raise a 2nd order Hadamard matrix into a 3rd order matrix, simply multiply the former by a 3 by 2 left matrix multiplier and a 2 by 3 right matrix multiplier. The product is a 3rd order matrix, which can interact with any arbitrary 3rd order matrix. The mathematics of ordered Hadamard matrices implies that hierarchical zero matrices can be constructed. In other words, a 3rd order zero matrix can be embedded in a 4th order zero matrix. A 2nd order matrix can be embedded in a 3rd order matrix. However, either nine 2nd order matrices can be embedded in a 6th order matrix or four 3rd order matrices can be embedded in a 6th order matrix. Conversely, to lower a 3rd order matrix into a 2nd order matrix simply multiply it by a 2 by 3 left matrix multiplier and a 3 by 2 right matrix multiplier. Strictly speaking, the elements of Hadamard matrices cannot be a mixture of +1, -1 and 0. However, elements of Hadamard matrices can be all zeros or a mixture of +1 and -1. These rules in no way limit the operations of Hadamard matrices for full implementation into a quantum theory of the directional (spacetime) continuum.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    Physical reality is not infinitely dimensional. However, physical dimensionality is hierarchical. In other words, it can be constructively arranged into different Levels of Existence (LoE). Each LoE has three spatial dimensions with an embedding of single temporal dimension (or enveloping temporal dimension, depending on the perspective view of what is outside and what is inside, which is a more abstract topological consideration but the simplest solutions are two distinct Möbius topologies). Traditionally, this is simply called the spacetime continuum of three spatial dimensions coupled to one temporal dimension. But after Einstein formulated his special theory of relativity, the Newtonian separation of absolute space and absolute time was recombined and merged into the Minkowski’s geometry of a four-dimensional continuum. Alternatively, this new idea of a four-dimensional continuum can be viewed as two nested 3-space LoEs sharing the same time dimension, for example, the microscopic universe coexisting with the macroscopic universe. Minkowskian continuum was used by Dirac to formulate his version of relativistic quantum mechanics (QM). Dirac’s version of QM allows him to postulate the existence of antimatter. Eventually, the lightest form of antimatter was discovered in cosmic ray experiments carried out by Carl Anderson in 1932. This is the first successfully detected observation of positron, the antiworld twin of the electron. For this discovery, Anderson shared the 1936 Nobel Prize for Physics with another pioneer of cosmic ray experiments, Austrian physicist Victor Hess. Since the discovery of the positron, physicists believe that all the other anti-twin of all real elementary particles can be found. Subsequently, many of these antiparticles were eventually found. All physical properties being the same, one way to differentiate an antiparticle from its ordinary particle counterpart is their electric charge polarity. That is an electron is negatively charged, while a positron is positively charged. Unfortunately, electric charge could not be used to differentiate the antiparticles of electrically neutral ordinary particles, for examples, neutrons, neutrinos, photons, and gluons. Detecting the antiparticles of electrically neutral elementary particles would require physics of high energy particle acceleration and also involving another new physical property known as the helicity. There are two types of distinct mathematics of helicity, the local and the global. The global integral form of helicity is usually used to define magnetic helicity. It is the integral of the dot product of the magnetic vector potential with the magnetic field over the entire region of enclosed volume. The local differential form of helicity is the dot product of the intrinsic spin angular momentum with the corresponding linear momentum of the particle. Mathematically speaking, both forms of helicity can be derived from the physics of the square of energy. These will be posted momentarily in succeeding posts.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    In the science of fluid dynamics (i.e. science of fluids in motion with nonzero external forces) in contrast to fluid statics (i.e. science of fluids at rest with balanced or zero external forces. The balanced forces is usually defined as the pressure of the fluid in question for both compressible and incompressible fluids), the former is also known as hydrodynamics. The latter is also known as hydrostatics. The units of global helicity as applied in the physics of meteorology is energy per unit of mass, which is equivalent to the square of the speed of spacetime translation or simply equivalent to the square of the absolute magnitude of velocity: h=k|v|2. In this case, the proportionality constant is taken as unity: k=1. Here, the global helicity is denoted by the lowercase italic ‘h’ in order to differentiate it from the ‘H’ for representing Hadamard matrices. This alternative expression for global helicity is true if and only if energy is expressed explicitly as the kinetic energy and that the contribution of potential energy is zero for all practical purposes. In relativistic mechanics energy is given by Einstein’s mass and energy equivalent: E=mc2 and since h=E/m then h=c2. This implies that in relativistic mechanics, the global helicity is equivalent to the square of the speed of light. However, for all velocities much less than lightspeed the global helicity is simply defined as the incremental sum of the scalar or dot product of velocity and the curl of velocity with respect to the infinitesimal spacetime displacement, or for instantaneous of v, the integral of {v,curl v}ds, where ds is the differential spacetime displacement and {v,curl v} denotes scalar product of v and the curl of v. Consequently, in the science or physics of fluid mechanics, the curl of v is also defined as the vorticity. However, in 3-dimensional motion of a fluid the three components of velocity of a fluid particle at Cartesian coordinates (x,y,z) is given by (u,v,w). There are three generalized motion of each fluid particle: (1) a general translation, (2) a pure strain motion, meaning unequally applied parallel shear forces but not enough to cause coupled rotation, and (3) a rotational motion of a parcel of fluid particles about an instantaneous axis. Note that the orientation of the instantaneous axis changes with respect to time. In the case of fluid parcel rotation, the three components of angular velocity are given ½p, ½q, ½r where p is given by the difference of the partial derivative of w with respect to y minus the partial derivative of v with respect to z, q is given by the difference of the partial derivative of u with respect to z minus the partial derivative of w with respect to x, and r is given by the difference of the partial derivative of v with respect to x minus the partial derivative of u with respect to y. Note that (p,q,r) are the three components of vorticity in the Cartesian coordinate system. Furthermore, each component represents a unit of frequency of cycle per second or hertz. Each cycle represents a complete fluid parcel rotation. The frequency components can either be positive or negative depending on whether the difference of partial derivatives is plus or minus. For vorticity to exist, none of these three frequency differences can be identically zero. In other words, the frequency differences must be multiples of the eight irreducible trivectors: (1,1,1), (-1,-1,-1), (1,1,-1), (-1,-1,1), (1,-1,-1), (-1,1,1), (1,-1,1), and (-1,1,-1). Incidentally, global helicity can be used to predict the occurrence of tornadoes. The complex mathematical details of tornado prediction can be found in textbooks about the physics of weather forecasting. But the important point to remember is that the chaotic motion of atmospheric turbulence will always involve positive or negative net helicity. If the global helicity is identically zero then no tornadoes (cyclones, hurricanes, or typhoons) can occur regardless of any other prevailing weather conditions (hot, warm, cool, or cold rain or snow). Next, the mathematics of local helicity will be discussed borrowing the language of quantum mechanics and quantum field theory.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    Before discussing local infinitesimal motion of microscopic helicity, it is important to describe the nonlinear Navier-Stokes equations of fluid mechanics. These equations are of great interest in pure mathematics. In applied mathematics, these equations can be used to model the weather, ocean currents, fluid flows in pipes, and air flow over and under each wing of an airplane. The simplified forms of these equations can help the design of aircraft and cars, the study of blood flows in arteries and veins, the design of distributed electric power stations, the analysis of atmospheric and water pollutions, etc. The Clay Mathematical Institute has included these equations as one of the seven most important open problems in mathematics and has offered $1000000 for a solution of the 3D full forms or a counter-example.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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    Re: regrouping quarks

    The Navier-Stokes equations (N-S) can be easily stated mathematically, although they are extremely difficult to solve computationally, numerically or analytically (as applied to the study of fluids every possible difficulty arises because of the onset of turbulence and chaos, which are fundamentally random processes and unavoidably unpredictable as well as sensitive to initial conditions, for example, see the ‘butterfly effect’ representing a set of strange attractors as described by the meteorologist Edward N. Lorenz in the 1960s). In general, the three dimensional solutions still do not exist. For this reason, the Clay Mathematics Institute is offering one million of dollars for whoever gives the 3D general solutions or counter examples. Solving N-S equations makes the branch of mathematics of infinitesimal calculus the run for its money. It is a real test of the usefulness or true utility of both differential and integral calculus since Isaac Newton and Gottfried Leibniz respectively invented them in the 17th century. The calculus was invented by these two great mathematicians for the sole purpose of describing the physical laws of spatial and temporal displacements of bulk matters. The directional limiting infinitesimal ratio of the former over the latter is physically and mathematically defined as the instantaneous velocity (v). At this point, a good understanding of the theory of limit is required in order to make the mathematical distinction between infinitesimal and differential quantities or entities. Although Newton was able to apply the calculus successfully to describe planetary and projectile motions; the same cannot be said for the French mathematician Claude Louis Marie Henri Navier when he applied the calculus to describe the displacement of fluids. Navier formulated the first version of these equations between 1821 and 1822. He built his mathematics over the foundation of the works of two Swiss mathematicians Daniel Bernoulli and Leonhard Euler. By 1822, Navier was able to generalize Euler’s equations to include the more realistic frictional component of fluid viscosity. In hindsight, there are still mathematicians who think Navier’s mathematical reasoning was flawed even though his final equations were correct. The corrected derivation was supposedly discovered by the famous Irish mathematician George Gabriel Stokes in 1842. Seven years later, Stokes was appointed the Lucasian Professor of Mathematics at Cambridge University in England, a chair once occupied by Newton and today occupies by Stephen Hawking. However, Hawking would rather talk about black holes than to find the general solution of N-S equations. In its full form, N-S equations are simply a recasting of Newton’s second law of motion whose vectorial representation is given by a single equation F=ma=Fgravity+Fpressure+Fviscosity. But in three degrees of freedom are represented by at least three supposedly solvable simultaneous scalar equations. This is made for the purpose of including the body forces contributed from gravity, pressure, and viscosity. Anticipating further mathematical confusion, each of these external forces must be separately defined unambiguously. Moreover, it is implicitly agreed by both mathematicians and physicists that N-S equations are properly used to describe continuum theories of fluids rather than quantum theories of fluids. This suggests that each infinitesimal control volume of fluid is made of infinitely many indistinguishable point particles, and for all practical purposes can be treated as a parcel of inactive volume continuum. Mathematically speaking, a continuum theory is best described by partial differentiation rather than by exact differentiation. However, each parcel of any fluid can be exactly differentiated. Therefore, N-S equations contain both exact and partial spatial or temporal derivatives. However, the nonlinearity of N-S equations comes from the addend term defined as the convective acceleration. This requires a new definition of acceleration as a spatial derivative of instantaneous change of velocity instead of the usual temporal derivative in order to account for the changes of v with respect to position such as the observed increase of fluid speed in passing through a nozzle. Next is to discuss how these different contributions of body forces are incorporated into the 3D full form of the N-S equations.
    Vincent Wee-Foo likes this.
    Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: a(tr(t)=c²

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