Archive for November, 2014

The Physics of Superconductivity and Related Phenomena

Note: A PDF version is provided at the end of the article.

This article assumes a basic understanding of quantum-geometry dynamics. The reader should minimally have read An Axiomatic Approach to Physics. For detailed explanation of the concepts of QGD, please refer to the relevant sections of Introduction to Quantum-Geometry Dynamics. You can also browse through the articles on the website.

If each of the axioms of QGD corresponds to a fundamental aspect of reality and if the set of all axioms of QGD is complete, then all physical phenomena can be described and the descriptions follow naturally from the axiom set. We have shown that QGD describes and explains a number of physical phenomena as well as make unique and readily testable predictions.

Now we will show how QGD describes and explains superconductivity and will conclude with testable predictions.

We have seen how the electromagnetic effect results from the interaction between charged polarizing particles (what is usually call charged particles) and free $preon{{s}^{\left( + \right)}}$ which move through a region of quantum-geometrical space neighbouring them.

According to QGD, electrons are composite particles made from $preon{{s}^{\left( + \right)}}$ moving in close helical paths. This regular motion of $preon{{s}^{\left( + \right)}}$ within the electron’s structure allows for sustained directional gravitational interactions with free $preon{{s}^{\left( + \right)}}$ along the direction of motion of the electron’s component $preon{{s}^{\left( + \right)}}$ and as a result affect the direction of free $preon{{s}^{\left( + \right)}}$neighbouring region; thus polarizing it (figure 1). Since the electric charge we associate with electrons and other charged particles results from their interaction with neighbouring free $preon{{s}^{\left( + \right)}}$, thus are not intrinsic to the particles, QGD uses the expression polarizing particle rather than charged particle.

It is important to keep in mind that when QGD refers to gravity, it is not only referring to the weak attraction between massive structures at large scales, but also gravity to at the microscopic scale.

We recall that gravity emerges naturally from the axiom set of QGD which it mathematically describes by the formula $G\left( a;b \right)={{m}_{a}}{{m}_{b}}\left( k-\frac{{{d}^{2}}+d}{2} \right)$ where ${{m}_{a}}$ and${{m}_{b}}$ are the masses of two objects $a$ and $b$ in $preon{{s}^{\left( + \right)}}$ and $d$ is the quantum-geometrical distance measured $preon{{s}^{\left( - \right)}}$ (or preonic leaps) and constant $k=\left| \frac{{{g}^{+}}}{{{g}^{-}}} \right|$ where ${{g}^{+}}$ and${{g}^{-}}$ are the fundamental units of p-gravity and n-gravity; the fundamental forces associated with and acting respectively between $preon{{s}^{\left( + \right)}}$ and $preon{{s}^{\left( - \right)}}$.

Note: The actual value of $k$ has yet to be determined exactly but by resolving the equation for gravity for the distance at which it is equal to zero, that is the minimum distance at which galaxies move appear to move away from each other, we find $k\approx {{10}^{116}}$.

QGD’s gravitational interaction formula implies that when $d$ is large, the magnitude of the gravitational interaction is as observed at the macroscopic scales, which is closely approximated by Newton’s law of gravity. Thus classical gravity corresponds to solutions of the gravitational interaction formula for sufficiently large values of$d$. But when distances are short, even though the masses of the interacting particles are relatively small, the magnitude of the gravitational interaction is orders of magnitude greater than which is observed at large scales.

In fact, in close proximity, gravitational interaction is the interaction that binds $preon{{s}^{\left( + \right)}}$into particles such as electrons, positrons, neutrinos and photons. At slightly larger distances, it allows polarizing particles to interact with neighbouring free$preon{{s}^{\left( + \right)}}$changing their directions and forming what we call a magnetic field.

The changes in the direction of the free $preon{{s}^{\left( + \right)}}$ interacting with a polarizing particle is described is given by $\displaystyle {{{\vec{P}}'}_{{{p}^{\left( + \right)}}}}={{\vec{P}}_{p\left( + \right)}}\vec{+}\Delta \vec{G}\left( {{e}^{-}};{{p}^{\left( + \right)}} \right)$ where $\displaystyle {{\vec{P}}_{p\left( + \right)}}$ is the momentum vector of a $preo{{n}^{\left( + \right)}}$ ${{p}^{\left( + \right)}}$ interacting gravitationally with an electron${{e}^{-}}$ resulting in a momentum vector $\displaystyle {{{\vec{P}}'}_{{{p}^{\left( + \right)}}}}$. The operation $\vec{+}$ is the direction sum which reflects the invariability of the $preon{{s}^{\left( + \right)}}$ fundamental momentum. That is $\displaystyle \left\| {{{{\vec{P}}'}}_{{{p}^{\left( + \right)}}}} \right\|=\left\| {{{\vec{P}}}_{p\left( + \right)}} \right\|=c$. Thus the deflection of a free $preo{{n}^{\left( + \right)}}$ depends on the gravitational interaction between it and electron, itself dependant the mass of the electron and $preo{{n}^{\left( + \right)}}$ and the distance between them.

Figure 1 illustrates the polarization of a neighbouring region by a single electron.

When the electrons of large material structures are similarly oriented the polarizing effect is multiplied creating a proportionally larger magnetic field. Polarized $preon{{s}^{\left( + \right)}}$that form a magnetic field move in both direction as indicated by the double arrows in figure 2. QGD explains that the dynamic structure of the electron causes the magnetic field on one side of the electron to be slightly greater than other.

An implication of the above explanation is that since the polarization of a region is due to the gravitational interaction between electrons (or other polarizing particles) and free$preon{{s}^{\left( + \right)}}$ , and since QGD implies that gravity is instantaneous, than so must be the formation of a magnetic field. If QGD’s description of this electromagnetic effect is correct, the speed of propagation of the Coulomb field must also be instantaneous. This prediction of QGD has been experimentally tested in 2012 (see article here). The experiment, repeated in 2014, confirmed the 2012 results. The paper written by the team who conducted the experiment is available here on arXiv and awaits publication.

From Introduction to Quantum-Geometry Dynamics, we know that a single polarizing particle generates a magnetic field such that $\displaystyle 0<\left\| {{{\vec{P}}}_{{{R}_{{{e}^{-}}}}}} \right\|<{{m}_{{{e}^{-}}}}$, where $\displaystyle \left\| {{{\vec{P}}}_{{{R}_{{{e}^{-}}}}}} \right\|=\sum\limits_{i=1}^{{{m}_{R}}}{{{{\vec{c}}}_{i}}}$and ${{m}_{R}}$ is the number of $preon{{s}^{\left( + \right)}}$ crossing the neighbouring region $R$ (this can be correctly interpreted as mass of $R$). Knowing that any change in momentum of a particle or structure must be an integer multiple of its mass, that is $\Delta \left\| {{{\vec{P}}}_{{{e}^{-}}}} \right\|=q{{m}_{{{e}^{-}}}}$where the expressions on the left and right represent units of momentum (a unit of momentum is the momentum necessary to overcome one unit of n-gravity), the net momentum of the magnetic field generated by an electron is smaller than the minimum allowable change in momentum. Therefore the electron is unaffected by its own magnetic field.

The situation is different when two electrons are close enough for their magnetic fields overlap.

In figure 3, we see how the magnetic field of the electron in the center can affect an electron coming in close proximity. Depending on the position of the second electron relative to the first, it will be either repelled (position $a$), attracted (position $b$) or deflected (position $c$ or $d$).

Note: That electrons can attract each other the way an electron and a positron do may appear to contradict experimental observations but according to QGD’s axioms, all particles are made of $preon{{s}^{\left( + \right)}}$so that all that distinguishes an electron from a positron is the orientation of the helical trajectories of their components$preon{{s}^{\left( + \right)}}$. An electron can behave like a positron which orientation is reversed and vice versa (this is discussed in detail in Introduction to Quantum-Geometry Dynamics).

So, we know that electrons that come into proximity will interact with each other’s magnetic field. One side of the electron interacts only with part of the magnetic field induced by the other causing an asymmetrical distribution of polarized $preon{{s}^{\left( + \right)}}$, increasing the number of free $preon{{s}^{\left( + \right)}}$ moving in one or the other direction between the electrons. And if the electrons are close enough (how close depends on factors explained below), the net momentum of the magnetic field in the neighbouring region of each of the electrons becomes greater than minimum allowable change in momentum.

Motion of Electrons of Current Moving Through a Conductor

The electrons in an electric current moving through conductor interact with each as described above. They are accelerated and decelerated (mostly decelerated since electrons of a current move towards atomic electrons of the conducting material) by discrete changes in momentum or $\Delta \left\| {{{\vec{P}}}_{{{e}^{-}}}} \right\|=q{{m}_{{{e}^{-}}}}$. We know that when $\displaystyle \left\| {{{\vec{P}}}_{{{R}_{{{e}^{-}}}}}} \right\|<{{m}_{{{e}^{-}}}}$ , the momentum of the electron is unaffected, but what happens when but what happens when$\displaystyle \left\| {{{\vec{P}}}_{{{R}_{{{e}^{-}}}}}} \right\|>{{m}_{{{e}^{-}}}}$ but $\displaystyle \left\| {{{\vec{P}}}_{{{R}_{{{e}^{-}}}}}} \right\|\ne q{{m}_{{{e}^{-}}}}$that is when $\displaystyle \left\| {{{\vec{P}}}_{{{R}_{{{e}^{-}}}}}} \right\|=q{{m}_{{{e}^{-}}}}+r$ where $r$ is the remainder of the Euclidian division or$\frac{\left\| {{{\vec{P}}}_{{{R}_{e_{1}^{-}}}}} \right\|}{{{m}_{e_{1}^{-}}}}=q+r$?

QGD optics explains the system will resolve itself by having $e_{1}^{-}$ emit a photon which momentum is exactly equal to $r$or $r=\left\| {{{\vec{P}}}_{\lambda }} \right\|$. It follows that the amount of energy lost when an electrical current gores through a conductor is$\sum\limits_{i=1}^{n}{{{r}_{i}}}=\left\| \sum\limits_{i=1}^{n}{{{m}_{{{\lambda }_{i}}}}{{{\vec{c}}}_{i}}} \right\|=\sum\limits_{i=1}^{n}{\left\| {{m}_{{{\lambda }_{i}}}}{{{\vec{c}}}_{i}} \right\|}$ where $n$ is the number electron interactions. So the lower $r$is, the less energy loss and the better conductor the material is. Conversely, the higher $r$ is, the more energy is lost and the more resistance to conductivity the material must be.

The photons emitted through the mechanism we described will be absorbed by the material, raising its temperature and/or emitted as light which colours depend on the momentum of the photons (wave optics interprets higher momentum photons as having higher frequency).

Note that when applying the general principles outline above to predict the behaviour of systems, factors such the density and momentum of the electrical current, the number and position of the atomic electrons, the chemical composition and molecular structure of the material will also affect the number of interactions and must be taken into account. Conversely, measures of the momentum of photons emitted can inform about the composition and structure of the material (see Mapping the Universe or the section Emission Spectrum of Atoms in Introduction to Quantum-Geometry Dynamics ).

On the Relationship between Temperature and Superconductivity

We have seen how the momentum of an electron changes when interacting with a magnetic field. We also know that the momentum of the magnetic field of a region of quantum-geometrical space depends on the gravitational interaction between an electron (or other charged particle) and the free $preon{{s}^{\left( + \right)}}$ the region contains. Being gravitational in nature, the interaction that induces a magnetic field depends on the mass of the interacting object. The interacting objects here are not electron-electron interactions, but electron-polarized $preon{{s}^{\left( + \right)}}$ interactions. Thus the momentum induced by a magnetic field is proportional the density of $preon{{s}^{\left( + \right)}}$ in the region$\displaystyle R$ which relation is expressed by$\Delta \left\| {{{\vec{P}}}_{{{e}^{-}}}} \right\|\propto \frac{\sum\limits_{i=1}^{{{m}_{R}}}{\left\| {{{\vec{c}}}_{i}} \right\|}}{Vo{{l}_{R}}}$ .

Those familiar with QGD may have noticed the similarity between the expression on the right of this last formula and QGD’s definition of temperature $tem{{p}_{R}}=\frac{\sum\limits_{i=1}^{n}{\left\| {{{\vec{a}}}_{i}} \right\|}}{Vo{{l}_{R}}}$ where $n$ is the number of free particles ${{a}_{i}}$. Since in close proximity the particles${{a}_{i}}$ contained within$R$ are $preon{{s}^{\left( + \right)}}$ then ${{m}_{R}}=n$ and $\frac{\sum\limits_{i=1}^{{{m}_{R}}}{\left\| {{{\vec{c}}}_{i}} \right\|}}{Vo{{l}_{R}}}=\frac{\sum\limits_{i=1}^{n}{\left\| {{{\vec{a}}}_{i}} \right\|}}{Vo{{l}_{R}}}$.

It follows that the number of $preon{{s}^{\left( + \right)}}$ of $R$ whose trajectories brings them simultaneously in close enough proximity for absorption by the electron is proportional the density of $preon{{s}^{\left( + \right)}}$ in $R$ hence proportional to the temperature of $R$ . That is: $\displaystyle \Delta \left\| {{{\vec{P}}}_{{{e}^{-}}}} \right\|\propto \frac{\sum\limits_{i=1}^{n}{\left\| {{{\vec{a}}}_{i}} \right\|}}{Vo{{l}_{R}}}$ .

Now, if ${R}'$ is a neighbouring region of ${{e}^{-}}$ which is close enough so that all the $preon{{s}^{\left( + \right)}}$ it contains can be simultaneously absorbed, then$\displaystyle \Delta \left\| {{{\vec{P}}}_{{{e}^{-}}}} \right\|=\frac{\sum\limits_{i=1}^{n}{\left\| {{{\vec{a}}}_{i}} \right\|}}{Vo{{l}_{{{R}'}}}}-r$ and, as we have seen earlier, then $\displaystyle tem{{p}_{{{R}'}}}=\frac{\sum\limits_{i=1}^{n}{\left\| {{{\vec{a}}}_{i}} \right\|}}{Vo{{l}_{{{R}'}}}}<{{m}_{{{e}^{-}}}}$then electrons do not interact magnetically, hence the momentum of electrons in a current is encounter no resistance. Therefore QGD predicts that an essential condition for a material to become superconductive is when $\displaystyle tem{{p}_{{{R}'}}}<{{m}_{{{e}^{-}}}}$.

Note: When the temperature is below the threshold above, Coulomb fields become undetectable.

Predictions

As often emphasized in earlier articles on QGD, any number of theories can explain a posteriori the behaviour of dynamic systems. To establish its validity and set apart from other theories, a theory must make unique and testable predictions.

From the discussion above and from the principles of QGD optics, we know that an essential condition for a material to become superconductive is that. $\displaystyle tem{{p}_{{{R}'}}}<{{m}_{{{e}^{-}}}}$ Below this threshold, the formula $\left\| {{{\vec{P}}}_{{{R}'}}} \right\|=q{{m}_{{{e}^{-}}}}+r$ is reduced to$\left\| {{{\vec{P}}}_{{{R}'}}} \right\|=r$ which implies three possibilities.

The first is that electron will simply reflect the$preon{{s}^{\left( + \right)}}$ which simultaneously come into absorption proximity to the electron.

The second and most interesting prediction, is that an electron of other polarizing particle may absorb the $preon{{s}^{\left( + \right)}}$ and restore equilibrium by emitting a photon $\lambda$ having equal momentum, that is such $\left\| {{{\vec{P}}}_{\lambda }} \right\|=r$. This implies that it is theoretically possible to convert the momentum of $preon{{s}^{\left( + \right)}}$into photons.

The effect of such mechanism may already have been observed for large polarizing structures such as heavy atomic nuclei. It is conceivable that since the density magnitude of a magnetic field depends on the mass of the polarizing particle, and since the minimum allowable change in momentum is large for a nucleon, allowing for a large value of $r$ , that it may contribute to gamma photons. If correct, the mass of a nucleus, for example, emitting gamma photons through this mechanism would be conserved.

More importantly, this mechanism would be also contributes to the structuring of $\displaystyle preon{{s}^{\left( + \right)}}$ into photons and neutrinos in stages of the universe following the initial stage.

That the mechanism may play a role in beta radiation, as QGD optics may suggest, is also a possibility that should be explored.

Space-Matter-Light Interactions (QGD optics part 3)

Note: The following article assumes that the reader has read part 1 and part 2 of this series on QGD optics.

The reason this section is given the title Space-Matter-Light Interactions and not the more common Matter-Light Interactions is that, according to QGD, quantum-geometrical space interacts dynamically with matter and light in all optical phenomena. In this third part of the series of articles on optics, we will use the principles we have introduced in part 1 and part 2 to describe and explain the phenomena of light absorption, light reflection and the photoelectric effect.

We will also examine the relation the fundamental forces that govern the motion of bodies which determine how they interact with light. We will therefore reinterpret Newtonian laws of motion and show how they are founded in fundamental reality. But first, we need to look at QGD’s definition of a concept central to all dynamics, that is, the notion of speed.

QGD definition of Speed

The speed of an object is defined classically as a function of distance and time. Those who are familiar with the principles of quantum-geometry dynamics know that it considers time to be a purely relational concept (see article here); one which allows us to associate events to natural periodic systems (e.g. movement of celestial objects, emission of an atom) or artificial periodic systems which, when combined with a counting mechanism, constitute what we call clocks. Thus quantum-geometry dynamics proposes that time is not a physical aspect of reality and for this reason defines speed without this concept.

When we measure the speed of an object, we start counting the number of recurrences of a particular state of a periodic reference system when the object is at a chosen initial point along its trajectory and stop counting when it arrives at a second, distinct point. The length of the trajectory between the points divided by the number of recurrences counted (the ticks of a clock) gives what we understand as the speed of the object. This method of measuring speed which compares two quantities has so far defined speed for us, but what allows us to assume that the number of ticks of a clock is a quantity that corresponds to a physical aspect reality?

The fact is: clocks do not measure time. They never have. Clocks are counting devices. What they count is the recurrences of a particular physical state of its inner mechanism, which itself is causally related to all preceding and successive states. This means that there are identifiable physical mechanisms that causally link a clock’s different states. Unlike spatial measuring devices, which relate to the physical aspect of reality we call space, there is no relation whatsoever between clocks and a physical aspects of reality. Clock count the ticks of the clock, nothing else.

This assumption that time is physical is reinforced by the mathematical models we use which represent space and time as geometrical dimensions but without making distinctions between what exist only as a representation, that is, what is pure concept, from a representation of a truly physical aspect of reality. The assumption that time is physical can be singled out as the most misleading idea in the history physics; one that lead to paradoxes, singularities, infinities, all of which create severe inconsistencies in physical theories if not their complete collapse. These problems alone justify abandoning the time-as-physical assumption and even the concept of time itself. But how can we abandon the concept of time when so much of our physics theories are based on it? How, for instance, does one define the essential notion of speed without time?

Using the ideas introduced here we can define timeless speed as follow:

The speed a body $a$ is the ratio of its momentum over its mass or ${{v}_{a}}=\frac{{{P}_{a}}}{{{m}_{a}}}$ where ${{P}_{a}}=\left\| \left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\| \right\|$
and ${{\vec{c}}_{i}}$ is the kinetic energy vector of a $preo{{n}^{\left( + \right)}}$ component of $a$ whose magnitude is fundamental and equal to$c$ . Therefore, the speed of an object is defined using two physical quantities; the magnitude of the vector sum of the kinetic energy vectors of its component $preon{{s}^{\left( + \right)}}$ and its number of$preon{{s}^{\left( + \right)}}$, its mass. We now have a working definition of speed that is timeless, observer independent and natural.

Special cases:

As we have explained early in this book, the energy of any particle or structure is given by ${{E}_{a}}=\sum\limits_{i=1}^{{{m}_{a}}}{\left\| {{{\vec{c}}}_{i}} \right\|}$ and the momentum by ${{P}_{a}}=\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\|$ . In the special cases when the trajectories of the component $preon{{s}^{\left( + \right)}}$ are parallel, $\sum\limits_{i=1}^{{{m}_{a}}}{\left\| {{{\vec{c}}}_{i}} \right\|}=\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\|$ so that${{E}_{a}}={{P}_{a}}$ . In words, the momentum and energy of such particles or structures are equal quantities (but not equal qualities).

This is evident for a single $preo{{n}^{\left( + \right)}}$
$a$ where, since${{m}_{a}}=1$ , we have ${{P}_{a}}=\left\| \sum\limits_{i=1}^{1}{{{{\vec{c}}}_{i}}} \right\|=\left\| {{{\vec{c}}}_{1}} \right\|=c$.

Similarly, if $a$ is a photon or a neutrino, we have ${{P}_{a}}={{E}_{a}}={{m}_{a}}c$ . Also, according to the QGD definition of speed, ${{v}_{a}}=\frac{{{P}_{a}}}{{{m}_{a}}}$ , we have ${{v}_{a}}=\frac{{{m}_{a}}c}{{{m}_{a}}}=c$ . This explains, amongst other things, the constancy of the speed of light without invoking time or the mechanism of time dilation.

The equality ${{P}_{a}}={{E}_{a}}$ also holds for any structure whose component $preon{{s}^{\left( + \right)}}$ have trajectories are parallel regardless of its mass (see here). Such particles include the photon and the neutrino, but theoretically, any particle or structure can achieve a speed equal to $c$ if is subjected to strong enough interactions. And since the maximum momentum any object can have is equal to ${{m}_{a}}c$ , its energy, the maximum speed is $c$ .

Space-Matter Interactions

One of the fundamental assumptions of quantum-geometry dynamics is that space is not continuous, but discrete, made of fundamental particles we call $preon{{s}^{\left( + \right)}}$ and that spatial dimensions emerge from the repulsive interactions between them (n-gravity). This has already been discussed in detail in previous articles (see here. here and here).

In previous articles, we have seen that the gravitational interaction between two particles or material structures is described by the formula $G\left( a;b \right)={{m}_{a}}{{m}_{b}}\left( k-\frac{{{d}^{2}}+d}{2} \right)$ where ${{m}_{a}}$ and ${{m}_{b}}$ are respectively the masses of a bodies $a$ and $b$ measured in$preon{{s}^{\left( + \right)}}$ , $k$ is the proportionality constant relating between n-gravity and p-gravity and $d$ is the number of $preon{{s}^{\left( - \right)}}$ between $a$ and $b$ ; what we traditionally call distance. This formula accounts for the contribution of the two fundamental forces that make up gravity; that is, p-gravity and n-gravity. This is made more apparent when we know that the formula is the simplification of $G\left( a;b \right)=\left( k{{m}_{a}}{{m}_{b}} \right)+\left( -{{m}_{a}}{{m}_{b}}\frac{{{d}^{2}}+d}{2} \right)$ where the components $k{{m}_{a}}{{m}_{b}}$ and $-{{m}_{a}}{{m}_{b}}\frac{{{d}^{2}}+d}{2}$ respectively represents the magnitudes of the p-gravity force and the n-gravity force between $a$ and$b$ .

But while we have so far been using the formula $G\left( a;b \right)={{m}_{a}}{{m}_{b}}\left( k-\frac{{{d}^{2}}+d}{2} \right)$ to calculate the gravitational interaction between two material particles or structure, it can be used to calculate the interaction between matter and space itself. We know that p-gravity only interacts between $preon{{s}^{\left( + \right)}}$ and that interaction is positive (attractive). But $preon{{s}^{\left( + \right)}}$must exist in quantum-geometrical space, hence they exist as $preo{{n}^{\left( + \right)}}/preo{{n}^{\left( - \right)}}$ pairs. The $preo{{n}^{\left( - \right)}}$ part of the $preo{{n}^{\left( + \right)}}/preo{{n}^{\left( - \right)}}$pair interacts with the $preon{{s}^{\left( - \right)}}$ which form quantum-geometrical space through n-gravity. In effect, because it has an effect opposite of p-gravity, we can think of the mass of $preon{{s}^{\left( - \right)}}$ as being negative mass so that, for the component $-{{m}_{a}}{{m}_{b}}\frac{{{d}^{2}}+d}{2}$, the negative sign can be interpreted as attributing a different meaning to the values ${{m}_{a}}$ and ${{m}_{b}}$ which now represent the number of $preon{{s}^{\left( - \right)}}$ $a$ and $b$ contain respectively. And since the ${{m}_{a}}$is the mass of $a$, we can by analogy interpret $-{{m}_{a}}$ as being its negative masses. It follows that all particles and material structures have both positive and negative masses.

From the above, we understand that in the p-gravity component of the formula we must use the positive masses of, the number of $preon{{s}^{\left( + \right)}}$ they each contain. So, assuming that the region $b$ contains no $preon{{s}^{\left( + \right)}}$, then ${{m}_{b}}=0$ and $k{{m}_{a}}{{m}_{b}}=0$. For the second component of the formula, ${{m}_{a}}$ and ${{m}_{b}}$ represent the negative masses of $a$ and $b$, that is, the number of $preon{{s}^{\left( - \right)}}$ pairing with $preon{{s}^{\left( + \right)}}$ or ${{m}_{a}}$ and the number of $preon{{s}^{\left( - \right)}}$ in the region $b$ or ${{m}_{b}}$. It follows that the force acting between an object $a$ and an empty region of quantum-geometrical space $b$is given by $G\left( a;b \right)=-{{m}_{a}}{{m}_{b}}\frac{{{d}^{2}}+d}{2}$ . This explains that, for an object to change direction, it must overcome the force exerted by space itself. Resolving the formula for a change equal to one unit of distance in one given direction, that is where${{m}_{b}}=1$ and$d=1$ , the n-gravity that must be overcome by $a$ is $G\left( a;b \right)=-{{m}_{a}}$.

Note about the Distinction between Mathematical and Physical Meanings

QGD shows that mathematical symbols also have specific physical meaning. The negative sign in the formula for gravitational formula implies that the masses following it are negative masses. The equal sign in the formula $E=mc$ derived from the axiom set of QGD does not represent the equivalence between mass and energy, but proportionality relation between them.

The same holds true for mathematical operations. QGD allows only for integer values of discrete quantities. That means that when we refer to division we mean the Euclidean division since, as we will be exemplified below, the quotient and remainder of the Euclidean division take different physical meaning.

The distinction between the mathematical and physical meanings is essential, not only for quantum-geometry dynamics, but for any theory of physics. In fact, much if not all of the problems with the interpretations of mathematical models of physical phenomena come from having lacked to make the necessary distinction.

Laws of Motion and Optics

The introduction of a new timeless definition of speed and an understanding of the interaction between material objects and between matter and space are necessary prerequisites to a quantum-geometrical description of the phenomenon of absorption and reflection of light and the photoelectric effect. Also essential is the following discussion about the laws of motion (which are given quantum-geometrical explanations).

First Law of Motion

If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero).

In quantum-geometrical terms the first law becomes:

If the magnitude of the vector sum of all the interactions with object$a$ is null, then its momentum, hence its speed, will be constant. Expressed in mathematical terms the first law of motion express becomes:

$\left\| \sum\limits_{i=1}^{x}{\vec{G}\left( a;x \right)} \right\|=0\Rightarrow \Delta {{P}_{a}}=0$

This is a state in which that all external forces acting on $a$ cancel each other. These forces include the interactions with other particles and material structures, as described by the QGD gravity formula, as well as the effect of quantum-geometrical space on it. Thus an object moving at a constant velocity may be understood as one where the external forces acting on it are in equilibrium.

Second Law of Motion

The acceleration of a body is parallel and directly proportional to the net force acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body.

Expressed in QGD terms, Newton’s second law of motion simply says that if $\left\| \sum\limits_{i=1}^{x}{\vec{G}\left( a;x \right)} \right\|>0$ then $\Delta {{P}_{a}}>0$ .

For gravitational interactions, the second law of motion can expressed as $\Delta {{P}_{a}}=\left\| \sum\limits_{i=1}^{x}{\vec{G}\left( a;x \right)} \right\|$.

To illustrate we’ll examine the simplest case. That is, when all interactions acting on two objects cancel each other except for the interaction between them . In this simple case$\Delta {{P}_{a}}=\left\| \vec{G}\left( a;b \right) \right\|={{m}_{a}}{{m}_{b}}\left( k-\frac{{{d}^{2}}+d}{2} \right)$. Since all QGD units are integers (and natural), we know that changes in momentum from interactions will also be integer; that is $\Delta {{P}_{a}}=x{{m}_{a}}$ where $x\in {{N}^{0}}$ .

This is consistent with QGD when you consider that for a body of mass ${{m}_{a}}$ to change momentum in a particular direction, each of its component $preon{{s}^{\left( + \right)}}$ must overcome an integer part of the n-gravity force exerted on it, as we have seen earlier.

Fundamentally, all motion imply that the component $preon{{s}^{\left( + \right)}}$ of an object leap from $preo{{n}^{\left( - \right)}}$ to $preo{{n}^{\left( - \right)}}$ so a fractional value would mean that $preon{{s}^{\left( + \right)}}$ could leap in between$preon{{s}^{\left( - \right)}}$ . That, as quantum-geometrical space implies, is not possible since nothing except the n-gravity field that keeps them apart and dimensionalize space and p-gravity though which matter interacts can exist between $preon{{s}^{\left( - \right)}}$ .

Second Law and Optical Kinetics

Since the QGD gravitational interaction formula can only have integer value, what is the outcome of an event in which the momentum a particle colliding with a body $a$ would result in a non-integer change in momentum?

As we have seen in part 1 and part 2 of this series on QGD optics, if a photon $b$ is absorbed by an object $a$ then $\Delta {{P}_{a}}=x{{m}_{a}}$. What this law implies is that only that part of the photon which obeys the formula $\Delta {{P}_{a}}=x{{m}_{a}}$is absorbed. The question is then: What happens when $\frac{{{m}_{b}}c}{{{m}_{a}}}\ne x$? We’ll look at two possible cases: the first is where $\frac{{{m}_{b}}c}{{{m}_{a}}}\ne x$and $\frac{{{m}_{b}}c}{{{m}_{a}}}<1$ and the second where $\frac{{{m}_{b}}c}{{{m}_{a}}}\ne x$and $\frac{{{m}_{b}}c}{{{m}_{a}}}>1$.

Reflection and the Photoelectric Effect

In the case where $\frac{{{m}_{b}}c}{{{m}_{a}}}\ne x$and $\frac{{{m}_{b}}c}{{{m}_{a}}}<1$, the momentum of the photon $b$ is less than the minimum possible change in the momentum of $a$ . In this case of the proverbial unstoppable force meeting an immovable object, the principle that comes into play is that of conservation of momentum of the system consisting of $a$ and $b$. If the photon cannot be absorbed then it can do two things. Pass through $a$ if it is transparent (this is discussed here) or, if $a$ is not transparent, and the momentum of the system is to be conserved, then $b$ must be reflected.

As we have seen earlier, QGD predicts that the $preon{{s}^{\left( - \right)}}$ which form quantum-geometrical space act on $a$ through n-gravity interactions which force is in direct opposition to the direction of the momentum vector of the photon $b$ (see figure).

Here we see that we can break down the momentum vector of the incident photon $b$ into two components. The component in blue is in opposition to n-gravity interaction between quantum-geometrical space and the object$a$. This component is reverted. The component in purple is not opposition and is thus conserved. The path of the reflected photon, labelled ${b}'$, is as shown. This explains why the angle of incidence is the same as the angle of reflection.

But this raises the question as to whether the incident and reflected photons are the same or distinct particles. Though it is possible that the incident and reflected photons are the same particle, a mechanism by which $a$ would absorb $b$ and emit a distinct particle ${b}'$ to conserve momentum is more consistent with the principles quantum-geometry dynamics. It is also consistent with the well-known phenomenon we call the photoelectric effect.

Now, in the case where $\frac{{{m}_{b}}c}{{{m}_{a}}}\ne x$and $\frac{{{m}_{b}}c}{{{m}_{a}}}>1$, the momentum of $b$ is greater than the minimum allowable change in momentum of $a$ , but here the ratio$\frac{{{m}_{b}}c}{{{m}_{a}}}$ being a non-integer, $a$ can only absorb $b$ by emitting a particle ${b}'$ such that ${{P}_{{{b}'}}}={{r}_{b}}$ where ${{r}_{b}}$ is the remainder of the Euclidian division $\frac{{{m}_{b}}c}{{{m}_{a}}}$. It is here obvious here that ${{r}_{b}}$, the momentum of ${b}'$, if this particle
is a photon, is also equal to its energy. The reflected photon will have a lower momentum and then the incident photon (which we perceive has having a different color). But ${b}'$ can be a particle other than a photon as long as its momentum is equal to${{r}_{b}}$.

For instance, instead of a photon $a$ can emit an electron. Though the emitted electron from the photoelectric effect may have a much greater mass than that of the incident photon which caused its emission, it possesses the same momentum.

A Note on Possible Applications to Optical Computing

Using the notions we have introduced, we see that the optical properties of a transparent structure can be changed predictably by bombarding it with photons having momentum that is equal to or greater than the structure’s mass. Such photons will be absorbed by the structure and will change its mass which in turn will change the allowed value for the momentums of incident photons, thus allowing photons that pass through it to do so and vice versa. QGD optics may be applied to create logical gates using only optical components making the theoretical optical transistor a possibility.

To Conclude this Series on QGD Optics

We have seen in this series how the light-matter interactions can only be understood when taking into account the effect of quantum-geometrical space on matter and light. And once we accept that light, like all material structures, is made of $preon{{s}^{\left( + \right)}}$, the fundamental particle of matter, light-matter interactions are really matter-space interactions. Thus optical phenomena are special cases of matter-space interactions and as such are governed by the same general laws that govern the motion of all matter.

Contrary to classical space, quantum-geometrical space is not a passive medium in which physical events and interactions occur, but an active component in all events and interactions. The effects of quantum-geometrical space are as tangible as the effects of matter.

Finally, it is important to remember that the examples given in the QGD Optics series of articles illustrate specific mechanisms which in nature do not act in isolation. The photons that compose the beam of light that hits an object can be partly absorbed and/or reflected and/or refracted and/or diffracted or all at once depending the properties of the matter they interact with.

The mechanisms we described also explain how the mass of photons change as they travel from a distant source. Photons emitted by a star, for example, will lose mass which equates into loss of momentum (or, since for photons momentum and energy are equal, less energy). This well-known and observed phenomenon is called the redshift effect (though QGD provides a different interpretation of its cause). The variation in the mass of photons provides some indication of the distance from the source (the larger distance it travels the more likely they are to interact with interstellar matter), but though there is a correlation between the distance and the amplitude of the redshift effect, the correspondence is not necessarily proportional. As we will see in a future article (and the upcoming 2nd edition of Introduction to Quantum-Geometry Dynamics), the light from sources at equal distance from Earth can have greatly different redshifts.

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