Thursday, December 29, 2011

The Theories and Math's that have led to BIGTOE

An overt statement was made to say: "The boson is a natural outcrop of the Higgs mechanism . Why is it so difficult to corner the cagey Higgs Boson?"  To which I replied:

How is the Boson, a natural outcrop of the Higgs mechanism? You may ask better,

"It appears that, the Boson, is a natural outcrop of the Higgs mechanism, how is it that it is so difficult to corner the cagey Higgs boson?"

Otherwise I am going to have to ask you for your math proof for your statement, or reference to whom has proved the boson to be, ..."a natural outcrop of the Higgs mechanism".

It is simple, Bosons, by definition, are slippery. They carry no effective electrically charged surface differential providers, which makes them slippery. Having an effective electrically charged surface differential provider, that's created dynamically, what we view as a spiny constant mono static field of charged electric potential, that is perpendicular to the surface, with the magnetic field counterpart, that is in concentric spheroidal shells, parallel to the electrically charged surface differential provider, is a fermion, like an electron, or Proton.

Neutral particles are Bosons, and are not easily affected by magnets, except for very, very strong magnets that will align the axis of the magnetic moment, that is bipolar in nature and results from the singular average spin axis, and the triangular spinning and rotating structure of the 3, point shaped quarks, that form the triangle, the spinning and rotation of said triangle structure, forms the larger neutral particle.

Photons are a strange case, because they lack a structure, such as that of a sphere that seems to hold for point particles, and larger, one that gives them a more 'solid feel', instead it is more of a 'solid rod'. Photons are more like a rod of a finite length, that spins and rotates through 4D hyperspace, that gives us up to 3 different ways, so far, to view the Photon, each dependant upon the measuring device used, and the relative position / direction of the rod, (photon), at impact / absorption, that gives us the concept of a particle, wave, or field.

Neutrinos are the strangest, in that they have next to zero interaction with anything the size of an atom and larger. They seem to be subnucleonic, and all matter is transparent to them, except to very slow moving ones, that will hit an electron, kicking out a photon out of the electron cloud, to be detected by very sensitive instruments. More at, IMHO, these neutrinos, with a dense flux streaming in from the Sun and elsewhere, actively work with the neutrons, to produce the electron and proton "pairs" that form the nuclei of atoms with the electron clouds around them, leaving an average of one neutron to one proton in all stable atoms, and a few extra neutrons in the not so stable isotopes.          

From wikipedia:
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking.

The above leads to qualifying questions, "How is a gauge boson different from any other boson?"; "What, exactly, is a gauge boson?"; If mass can vanish, and then reappear, that would explain another 3 experiences I have had, if the mechanism is KNOWN in how the mass vanishes, then comes back, let alone be non-vanishing. Who are Nambu and Goldstone, how did they get their bosons, what makes them different from the standard boson, or the gauge boson, or the one I have discovered, an ⓒ"Intermediate Hypervector Boson"™ = IHB, and don't have it named after me as yet, that might be the affect that created an effect that I witnessed.

Hmm let's see ... Gauge Theory - In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a **continuous group of local transformations**.

The term gauge refers to redundant degrees of freedom in the Lagrangian. The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as Lagrangian mechanics. In classical mechanics, the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V:: L = T - V; 

Under conditions that are given in Lagrangian mechanics, if the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation, a particular family of partial differential equations.

The transformations between possible gauges, called gauge transformations, form a Lie group which is referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding vector field called the gauge field.

In mathematics, a Lie algebra ( /ˈliː/, not /ˈlaɪ/) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

A Lie algebra is a vector space over some field F together with a binary operation [·, ·]
[*,*] : g x g -> g, called the Lie bracket, which satisfies the following axioms:

[ax + by , z] = a[x , z] + b[y , z] , [z , ax + by] = a[z , x] + b[z , y]
for all scalars a, b in F and all elements x, y, z in g.

Alternating on g :
[x , x] = 0 for all x in g.

The Jacobi identity:
[x , [y , z]] + [y , [z , x]] + [z , [x , y]] = 0
for all x, y, z in g.

Note that the bi-linearity and alternating properties imply anticommutativity, i.e., [x , y] = -[y , x] for all elements x, y in g, while anticommutativity only implies the alternating property if the field's characteristic is not 2.[1]

For any associative algebra A with multiplication * , one can construct a Lie algebra L(A). As a vector space, L(A) is the same as A. The Lie bracket of two elements of L(A) is defined to be their commutator in A: [a , b] = a * b - b * a;  

Please see : xxx xxx is a nice 2D spherical representation. Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen programme. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics. Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology, by computing the fundamental group of G (an abelian group: a Lie group is an H-space). This was done by Élie Cartan.            

In mathematics, an H-space is a topological space X (generally assumed to be connected) together with a continuous map μ : X × X → X with an identity element e so that μ(e, x) = μ(x, e) = x for all x in X. Alternatively, the maps μ(e, x) and μ(x, e) are sometimes only required to be homo-topic to the identity (in this case e is called homotopy identity), sometimes through base point preserving maps. These three definitions are in fact equivalent for H-spaces that are CW complexes. Every topological group is an H-space; however, in the general case, as compared to a topological group, and from what I have seen, H-spaces do not lack associativity and inverses.

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. 

Roughly speaking, a CW-complex is made of basic building blocks called cells. The precise definition prescribes how the cells may be topologically glued together. The C stands for "closure-finite", and the W for "weak topology".

An n-dimensional closed cell is a topological space that is homeomorphic to an n-dimensional closed ball. For example, a simplex is a closed cell, and more generally, a convex polytope is a closed cell. An n-dimensional open cell is a topological space that is homeomorphic to the open ball. A 0-dimensional open (and closed) cell is a singleton space.
A CW complex is a Hausdorff space (H-space) X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties:

  • For each n-dimensional open cell C in the partition of X, there exists a continuous map f from the n-dimensional closed ball to X such that
    • the restriction of f to the interior of the closed ball is a homeomorphism onto the cell C, and
    • the image of the boundary of the closed ball is contained in the union of a finite number of elements of the partition, each having cell dimension less than n.
  • A subset of X is closed if and only if it meets the closure of each cell in a closed set.
If the largest dimension of any of the cells is n, then the CW complex is said to have dimension n. If there is no bound to the cell dimensions then it is said to be infinite-dimensional. The n-skeleton of a CW complex is the union of the cells whose dimension is at most n. If the union of a set of cells is closed, then this union is itself a CW complex, called a sub-complex. Thus the n-skeleton is the largest sub-complex of dimension n or less.

A CW complex is often constructed by defining its skeleton inductively. Begin by taking the 0-skeleton to be a discrete space. Next, attach 1-cells to the 0-skeleton. Here, the 1-cells are attached to points of the 0-skeleton via some continuous map from unit 0-sphere, that is, S0. Define the 1-skeleton to be the identification space obtained from the union of the 0-skeleton, 1-cells, and the identification of points of boundary of 1-cells by assigning an identification mapping from the boundary of the 1-cells into the 1-cells. In general, given the n-1-skeleton and a collection of (abstract) closed n-cells, as above, the n-cells are attached to the n-1-skeleton by some continuous mapping from Sn − 1, and making an identification (equivalence relation) by specifying maps from the boundary of each n-cell into the n-1-skeleton. The n-skeleton is the identification space obtained from the union of the n-1-skeleton and the closed n-cells by identifying each point in the boundary of an n-cell with its image.
Up to isomorphism every n-dimensional complex can be obtained from its n-1 skeleton in this sense, and thus every finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensional complexes, with the understanding that the result of the infinite process is the direct limit of the skeleta: a set is closed in X if and only if it meets each skeleton in a closed set.

Singular homology and cohomology of CW-complexes is readily computable via cellular homology. Moreover, in the category of CW-complexes and cellular maps, cellular homology can be interpreted as a homology theory. To compute an extraordinary (co)homology theory for a CW-complex, the Atiyah-Hirzebruch spectral sequence is the analogue of cellular homology.  There is a technique, developed by Whitehead, for replacing a CW-complex with a homotopy-equivalent CW-complex which has a simpler CW-decomposition.  Consider, for example, an arbitrary CW-complex. Its 1-skeleton can be fairly complicated, being an arbitrary graph. Now consider a maximal forest F in this graph. Since it is a collection of trees, and trees are contractible, consider the space X / ∼ where the equivalence relation is generated by xy if they are contained in a common tree in the maximal forest F.
The quotient map X \to X/\sim is a homotopy equivalence. Moreover, X / ∼ naturally inherits a CW-structure, with cells corresponding to the cells of X which are not contained in F. In particular, the 1-skeleton of X / ∼ is a disjoint union of wedges of circles.  Another way of stating the above is that a connected CW-complex can be replaced by a homotopy-equivalent CW-complex whose 0-skeleton consists of a single Point. 

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Hn(X). Intuitively spoken, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In brief, singular homology is constructed by taking maps of the standard n-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation on a simplex induces a singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all homo-topically equivalent spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where the homology group becomes a functor from the category of topological spaces to the category of graded abelian groups.  By dualizing the homology chain complex (i.e. applying the functor Hom(-, R), R being any ring) we obtain a cochain complex with co-boundary map δ. The cohomology groups of X are defined as the cohomology groups of this complex; in a quip, "cohomology is the homology of the co- (dual complex)".  The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows:
There are additional cohomology operations, and the cohomology algebra has addition structure mod p (as before, the mod p cohomology is the cohomology of the mod p cochain complex, not the mod p reduction of the cohomology), notably the Steenrod algebra structure. 

The multiplicative structure of an H-space adds structure to its homology and cohomology groups. For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra. Also, one can define the Pontryagin product on the homology groups of an H-space.

The fundamental group of an H-space is abelian. To see this, let X be an H-space with identity e and let f and g be loops at e. Define a map F: [0,1]×[0,1] → X by F(a,b) = f(a)g(b). Then F(a,0) = F(a,1) = f(a)e is homotopic to f, and F(0,b) = F(1,b) = eg(b) is homotopic to g. It is clear how to define a homotopy from [f][g] to [g][f]. 

Adams theorem: S0, S1, S3, S7 are the only spheres that are H-spaces (e.g., using multiplication restricted from the reals, complexes, quaternions, and octonions, respectively). In fact, S0, S1, and S3 are groups (Lie groups) with these multiplications. But S7 is not a group in this way because octonion multiplication is not associative, nor can it be given any other continuous multiplication for which it is a group. However S7 is associative and has other continuous multiplication for which it is a group, when an actual spherical coordinate system, in conjunction with my One equation, is used in place of the octonions with their linearity of cubism restrictions.

Then with the continuous group of local transformations in mind, I came up with: xxx  xxx

So, I would say that: 'The Higgs mechanism is the natural outcrop of the Gauge Field Boson Theory, so why is the Higgs Boson so cagey?' if I thought the Higgs field Boson was even real or remotely possible, let alone findable, because it is like trying to create a particle of space. Space is an empty concept, while particle is a fully solid concept. In my theory, and the One Equation, they are related, but different structures. The primary vectors form dynamic relationship hypervectors that spin on 4 axis', which form into a surface element triangle that rotates with 4 degrees of freedom, that depending on the scalar values of the dynamic relationship hypervectors, with each hypervector perpendicular to the others, and the ratios of any 2 hypervectors that form dynamically a third hypervector, form S7, as there are 7 geometric functions in the group, a group that exhibits continuous local transformations between the 7 geometries within the group. 

Please note that in the above concepts: gauge theory, Lagrangian, Lie group and Lie algebra, H-space, CW complexes, were composited from information found on wikipedia, (don't forget to donate, no ads) to which I added what I know, and hopefully made a coherent contribution.  And on YouTube : Higgs                              

Wednesday, September 7, 2011

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