# Archive for February, 2014

## Quantum-Geometry Dynamics in a Nutshell (update)

It is not the existence of preons that we question here, but rather the consequences that follow the assumption that they do and whether or not these consequences are in agreement with the experimental and observational data.

As the title suggests, the purpose of this post is to provide a summary of the basic notions of quantum-geometry dynamics. This post will be regularly refined and updated.

Particles: There are only two fundamental particles ; the preon(+) and the preon(-).

Forces: There are only two fundamental forces: p-gravity which acts between preons(+) and n-gravity which acts between preons(-)

Matter: All particles and structures are made of preons(+) bound through p-gravity.

Space: Space is discrete or quantum-geometrical, hence it has structure. The discrete components of space are preons(-). Spatial dimensions emerges from the n-gravity interaction between its component preons(-)

Distance: The distance between any two preons(-) is equal to the number of leaps a free preons(+) must make to move from one to the other.

Preons(+): Preons(+) are singularly kinetic and they move by leap from preon(-) to preon(-)

Mass: the mass of any particle or structure corresponds to the number of preons(+) it contains.

Momentum vector: The momentum vector is a vector which describes the direction and momentum of a particle or structure.

Momentum vector of a preon(+): The momentum vector of a preon(+) is fundamental and written as $\vec{c}$. The magnitude of the momentum vector of a preon(+) is a large integer expressed in units of n-gravity, which as we saw above, is the force that acts between preons(-), which force a preon(+) must overcome to move from preon(-) to preon(-). The momentum of a preon(+) is given by $\left\| {\vec{c}} \right\|=c$.

Momentum Vector of a particle or structure: the momentum of a particle or structure is the resultant of the vector sum of all the momentum vectors of all its component preons(+). It is given by ${{\vec{P}}_{a}}=\sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}}$ where ${{m}_{a}}$ is the mass of the particle or structure, the number of preons(+) it contains, and ${{\vec{c}}_{i}}$ is the momentum vector of its ${{i}^{th}}$ component preon(+).

Momentum of a particle or structure: The momentum of a particle or structure is the magnitude of its momentum vector and given by $\left\| {{{\vec{P}}}_{a}} \right\|=\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\|$

Speed: According to QGD, speed is an intrinsic property of particles or structures and is the ratio of the momentum over its mass. Thus the speed ${{v}_{a}}$ of a particle or structure $a$ is given by ${{v}_{a}}=\frac{\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\|}{{{m}_{a}}}$ .

From this definition of speed (one which does not require the use of the time concept) we can see that the maximum possible speed is achieved when all momentum vector of a particle or structure move in the same direction. In such case $\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\|=\sum\limits_{i=1}^{{{m}_{a}}}{\left\| {{{\vec{c}}}_{i}} \right\|}={{m}_{a}}c$, thus $\displaystyle {{v}_{a}}=\frac{{{m}_{a}}c}{{{m}_{a}}}=c$ . So the maximum speed for any particle or structure is $c$ , which is numerically equal to the momentum of a preon(+). We write “numerically equal” because speed and momentum are two different physical properties which are equal in magnitude in special cases only.

Energy: The energy of a particle or structure is equal to the sum of the momentums (not the momentum vectors) of all its component preons(+). It is given by ${{E}_{a}}=\sum\limits_{i=1}^{{{m}_{a}}}{\left\| {{{\vec{c}}}_{i}} \right\|}={{m}_{a}}c$.

Note that though the equation ${{E}_{a}}={{m}_{a}}c$is similar to the $E=m{{c}^{2}}$ , the QGD equation expresses a proportionality relation between mass and energy and not, as the relativistic equation, an equivalence. QGD also provides a fundamental explanation of energy which is based on a fundamental explanation of mass and momentum.

Effects: Since according to QGD there are only two fundamental forces, it follows that all other forces are effects of n-gravity and p-gravity.

Gravity effect: Gravity is an interaction between two particles or structures and results from the combined effect of n-gravity and p-gravity. It is given by $G\left( a;b \right)={{m}_{a}}{{m}_{b}}\left( k-\frac{{{d}^{2}}+d}{2} \right)$ . Gravity is shown to emerge naturally from QGD’s axioms.

Electromagnetic effect: The electromagnetic effect results from absorption of preons(+), which impart their momentums to the interacting particles.

Weak nuclear interaction: The so-called weak nuclear interactions, which results in nuclear decay, results from the effect of gravity (which because $d$ is very small, is very strong at the nuclear scales) and the electromagnetic effect.

Strong nuclear interaction: The strong nuclear interaction also results from the effect of gravity and the electromagnetic effect.

For detailed explanations of any of the notions summarized here, it is suggested to search for and read previous articles using the search window at top left of read the relevant chapters of Introduction to Quantum-Geometry Dynamics or for a shorter (33 pages) yet good overview, you may also read An Axiomatic Approach to Physics.

## A Physics Theory is Required to do Three Things: describe, explain and predict (part 3)

### QGD Cosmology

Though quantum-geometry dynamics is a physics of fundamental reality, its axioms imply a number of predictions at the cosmological scale.

If, as QGD proposes, space is discrete and emerges from the interactions between preons(-) and if the single fundamental component of matter is the preon(+), then the formation of all material structures, from particles to galaxies requires that the Universe evolved from an isotropic state where all preons(+) were free and uniformly distributed through the entire quantum-geometrical space of Universe.

Before I move on, I would like to warn the reader that much of QGD contradicts the dominant theory of the origin and evolution of the Universe, but none of it, as far as I know, contradicts observations. In fact, not only does QGD cosmology not contradict observations that support the dominant theory, but it also accounts for observations that the dominant theory cannot explain and which constitutes strong counter-evidence against it.

As we will see, QGD cosmology proposes that ours is a locally condensing universe rather than an expanding universe. A locally condensing universe, as defined below, is nearly undistinguishable from an expanding universe. From an observer’s point of view, the galaxies of both types of universe will appear to recede from each other at an accelerated rate. But QGD not only agrees with all observations that support the Big Bang theory, it also agrees with the observations of redshift anomalies, which is strong counter-evidence against the Big Bang theory. QGD not only describes and explains, but predicts the conditions that will produce redshift anomalies.

### The Material and Spatial Dimensions of the Universe

When we think of the Universe, we think of what we observe through telescopes. We think of planets, stars, galaxies, galaxy clusters; the material structures of the Universe. When we study the Universe, we do so by observing the material structures, but not the entire Universe is observable. We can only observe what is made of matter because only matter can interact with the instruments we use for observation. Yet the Universe is made of more than structures of matter; it is also made of space. And if QGD is correct, that space is quantum-geometrical.

Virtually all of physics considers space to be an amorphous expanse in which physical systems exist and interact. As a consequence all physics theories are theories of matter (or matter and energy to be precise). Quantum-geometry dynamics too is a theory of matter, but it is also a theory of space. Also, according to QGD, not only is space quantum-geometrical and emergent, it also determines the very structure of matter.

### Conservation of Space

QGD proposes that it be the repulsive force of n-gravity acting between preons(-) that generates space (n-gravity being the fundamental force intrinsic to preons(-)). Since preons(-) are fundamental particles, they obey the law of conservation which states that nothing fundamental can be created or destroyed. It follows that there must be a finite number of preons(-), which in turn implies that there is a finite number of interactions, thus a finite amount of quantum-geometrical space. Therefore, as large as it appears to be, space must be finite.

### Particle Formation and Strict Causality

QGD follows the principle of strict causality, which is short for saying that the formation of any non-fundamental physical object requires the pre-existence of its constituents. Fundamental objects, being fundamental, pre-exist everything else (post-exist everything else as well).

The strict causality implies that any structure requires the pre-existence of its components may appear trivial, but it is a principle that some theories feel is fine to violate. Other theories, such as string theory, can’t tell which particles may be components of which other particles (see Leonard Susskind’s lectures of reductionism). As a result, theories that violate strict causality may ambiguously indicate that reality can get more complex the closer we approach the fundamental scale.

There is no such ambiguity in quantum-geometry dynamics. Strict causality implies that reality get simpler at the fundamental scale. QGD predicts the existence of only two fundamental particles and two fundamental forces. Reality can’t get any simpler.

QGD shows that all laws of physics can be derived from a simple of set of axioms which is complete and consistent.

This, of course, contradicts Gödel’s incompleteness theorem. But if the Universe is made of a finite set of fundamental particles which combine in accordance to a finite set of fundamental laws to produce physical reality, then it follows that Gödel’s first incompleteness theorem is, at least in its present form, wrong. Also, if you believe that the fundamental components and laws are a consistent and that the Universe is a coherent system, then Gödel’s second incompleteness theorem must also be wrong.

It follows the Universe is found to be complete and consistent system, then Gödel must be revised and Hilbert’s program must be reinstated.

An acquaintance once commented that we should make a distinction between a mathematical demonstration and a physical demonstration. My take on the question is that it makes no difference.

If the Universe is found to be both coherent and complete (that is, fundamental particles and the laws that govern them are consistent and all that they produce remains part of the Universe (completeness), then all physical processes are emergent from the axiomatic set of fundamental particles and laws. Now, that means that not only are the basic physical interactions emergent, but all processes, including environmental, social, cultural and neurological processes emerge from the fundamental axiomatic set.

One can argue that, as abstract mathematics may be from reality, they are the result of mental processes, which are necessarily physical so that they, themselves, can be derived from the fundamental laws of physics. In that context, it doesn’t matter what the construct is (a painting, a film, a poem or a mathematical theory), it must be emergent and can theoretically be derived from the fundamental axiomatic set.

### The Cosmic Microwave Background Radiation

Quantum-geometry dynamics describes the initial state of the Universe as being one in which preons(+) were free and distributed uniformly throughout the quantum-geometrical space. Following this initial state, the simple structures we call photons started to form. The formation of photons happened throughout the Universe uniformly and resulted in the cosmic microwave background. The density of the preonic field being greater than it is now, the photons produced were more massive (see mass/energy equation in Introduction to Quantum-Geometry Dynamics).

Most preons(+) are still free today and still form photons (though at the lower rate). The collective gravitational effect of those free preons(+) have been observed and correspond to what has been called dark matter.

### Small Structure Formation

The strict causality principle, which requires the pre-existence of a structure’s components, implies that photons combined to form the electrons, positrons and neutrinos. In fact, the well-known electron-positron annihilation is simply the reverse of the mechanism of particle formation. This is explained in the book.

### Large Structures

The formation of large structures also follows the principle of strict causality. It implies the formation of larger particles, then nuclides (the components of the atomic nucleus) then light atoms. These eventually formed stars and galaxies. The formation of increasingly massive structures (elements) continued in stars where the gravitational interactions are sufficient fusion of elements.

It has also been observed that particles, nuclides in particular, have certain sizes. The lower and higher boundaries on the size of any particle determine the island of stability. The mechanisms which limit the size of a particle are explained in chapter 14 of the book. This chapter explains the notion of equilibrium and how only particles that are within the range of equilibrium are stable and why particles that are lighter or heavier will decay.

### Locally Condensing Universe

This is one the most distinctive aspect of the QGD cosmology. If follows from the axioms of QGD that the size of the Universe, defined as the space emerging from the interactions between preons(-) is constant, but within that space massive structures will gradually collapse towards their center.

To the observer, a locally condensing universe is nearly indistinguishable from an expanding universe. For instance, the distance between galaxies progressively increases in both locally condensing universe (LCU) and expanding universe (EU). And in both the rates at which the galaxies retract from each other increases, which indicates that galaxies retract at accelerated rate in both the LCU and EU. So if both LCU and EU are nearly indistinguishable to the observer, how do we know which is correct? Is there any evidence which would support LCU?

The observational evidence exists and has been known from some time as redshift anomalies.

The redshift is simply the shift of the frequency of light coming from a moving source (which is understood to be analogous to the Doppler Effect for sound). The faster the relative speed of the source of light away from the Earth, the greater the redshift of the light coming from that source. The magnitude of the redshift is used to calculate the distance between galaxies and the rate at which they recede from each other. Hence it is used to infer the expansion of the Universe.

According to the Big Bang theory, which is the dominant theory of the expanding universe, the further away galaxies are, the faster they will recede from us. This implies that neighboring cosmic structures (galaxies, quasars, etc.) would recede from us at the same rate, thus have the same redshift. This is generally true, but there are an increasing number of observations that show neighboring cosmic structures having significant differences in their redshifts. This would indicate that the rate at which they recede from us differs by many orders of magnitude. Redshift anomalies (and there are now thousands of them) are in direct opposition with the Big Bang and other expanding universe theories.

Yet, redshift anomalies support the idea of a locally condensing universe. Not only do redshift anomalies support QGD cosmology, they are predicted by QCD cosmology. Redshift, according to QGD, is not a measure of the rate at which they galaxies recede, but the rate at which they collapse (which itself is a function of the density of the galaxy or cosmic structure). The acceleration of the rate at which galaxies recede is also consistent with rate at which they would collapse under the gravitational effect described by QGD.

The rate of collapse of cosmic structures obeys the QGD law of gravitation which is described by the equation found in chapter 8 of my book. As such it is affected by their mass and density, but also by the gravitational interactions between them. Using the QGD gravitational interaction equation, the rate of collapse between galaxies will be affected by dark energy or dark matter effects depending on the distance between them. The dark energy and dark matter effect will also determine the shapes of the interacting galaxies. Given certain distance exceeding a certain a value (see part 1 and part 2), n-gravity will be dominant (the dark energy effect) resulting in a flattening of the galaxies along the axis that connects them. While at distances lower than the equilibrium point, p-gravity becomes dominant (the dark matter effect) and the shape galaxies will expand along the axis connecting them.

The same principle explains why the material universe (that part of the universe where matter is concentrated) is nearly flat (something that the Big Bang and EU theories can’t explain). Thus the universe should become flatter as it evolves.

### Universe as a Finite and Closed Structure

We mentioned earlier that the quantum-geometrical space must be finite. If QGD is correct, it must also be closed. That is, a photon going in a straight trajectory would eventually go forth to its point of origin, whichever point we arbitrarily chose as origin.

So though the Universe may be finite, there would be no edge to it.

### Summary of QGD Cosmology Predictions

• The Universe evolved from an isotropic state. This eliminates all problems associated with singularities.
• The Universe is a finite and closed system. This eliminates all problems associated with infinities.
• The Universe in strictly causal

### Consequences for Particle Physics

Particle accelerators, such as CERN’s large hadron collider, are extraordinary tools that attempt to recreate on a microscopic scale the conditions that prevailed at the beginning of the Universe. The hope is that recreating the conditions immediately following the Big Bang will reveal the fundamental particles and states that existed at the very beginning of the Universe. This is a valid approach if the Big Bang theory’s assumption that the Universe evolved from a singularity is correct. But what if, as QGD suggest, the Universe evolved from an isotropic state?

If such is the case, then the conditions recreated in particle colliders are not those that prevailed at the beginning of the Universe, but conditions we could expect to be found much later and only in dense preonic structures such as those existing prior to the formation of stars. Thus, particles colliders do not reveal fundamental reality, but an emergent reality. In other words, trying to discover fundamental reality using particle accelerator is like looking at the wrong end of microscope to reveal the microcosm or at the wrong end of a telescope to observe the macrocosm.

That is not to say that such instruments as the LHC are useless. On the contrary, such instruments are essential to our understanding of reality. It’s only that what they show us is not fundamental reality, but processes that came into existence at states that followed the initial isotropic state of the Universe.

## QGD Locally Realistic Explanation of Quantum Entanglement Experiments (part 1)

Note that the following article assumes that the reader is familiar with the basic notions about quantum-geometry dynamics.

Those who are familiar with quantum-geometry dynamics know that it excludes quantum entanglement. Yet, as we will show in this article, QGD remains consistent with the results from all experiments which are thought to support quantum entanglement. Not only is QGD consistent with the experimental data of so-called quantum entanglement experiments but, unlike quantum mechanics, precisely explains the mechanisms responsible for the apparent quantum entanglement effect without violating the principle of locality. In the setup shown in figure 1, which is called a Mach-Zehnder Interferometer, we have a source of light which beam is split in two by a half-silvered mirror. The classical prediction is that 50% of the light will be reflected to the mirror on the top left (path 1) and 50% will be refracted to the mirror at the bottom right (path 2). The light which arrives at the top left mirror will be reflected towards the back side of the half-silver mirror on the top right where it will be split into two beams towards detector 1 and detector 2, each of which should be receiving 50% of the photons coming through path 1 or with 25% of photons emitted by the source.

The photons that follows path 2 (50% of the photons from the source) are reflected by the mirror at the bottom right towards the half-silvered mirror at the top right where it will be split into two beams each having 50% of the photons following path 2 (or 25% of the photons from the source beam). So classical optics predicts that 50% of the photons from the source will reach D1 and the other 50% will each D2. But, see figure 2, experiments show that 100% of the photons from the source reach D2 and none reach D1. The explanation provided by quantum mechanics, which is similar to that given for the results of double-slit experiments, proposes that the wave function of each individual photon travels both paths and engages in interference at the half-silvered mirror on the top right and that they interfere destructively at D1 and constructively at D2 (a detailed explanation can be found here).

Applying the QGD optics to the setup, we arrive a different and much simpler explanation.

We know from QGD’s description of optics and motion that an electron $\displaystyle e_{0}^{-}$ can absorb a photon ${{\gamma }_{0}}$ only if $\displaystyle {{m}_{{{y}_{0}}}}c=x{{m}_{e_{0}^{-}}}$. After absorption of a photon the mass of the electron will be increased by the mass of the photon, that is ${{m}_{e_{1}^{-}}}={{m}_{e_{0}^{-}}}+{{m}_{{{y}_{0}}}}$. And since electrons $\displaystyle e_{1}^{-}$ can only absorb a photon ${{\gamma }_{1}}$ if $\displaystyle {{m}_{{{y}_{1}}}}c=x\left( {{m}_{e_{0}^{-}}}+{{m}_{{{y}_{0}}}} \right)$ then $e_{1}^{-}$ electrons will reflect ${{\gamma }_{0}}$photons .

So what happens according to QGD is that photons following path 1 are absorbed by the atomic electrons of the transparent part of the half-silvered mirror at the top right (since the distance from the source is shorter, they reach that part earlier). Since ${{m}_{e_{1}^{-}}}>{{m}_{e_{0}^{-}}}$ and $\displaystyle {{m}_{{{y}_{0}}}}c, all photons which momentum is smaller than $\displaystyle {{m}_{{{y}_{1}}}}c$ will be reflected back by these electrons. So, if the transparent part of the mirror exceeds a minimum thickness, the photons coming in from path 2 going through it will meet $e_{1}^{-}$ electrons along their path and, since which mass ${{m}_{e_{1}^{-}}}>{{m}_{{{\gamma }_{0}}}}c$, will be reflected by them. (see figure 3, electrons at phase $e_{1}^{-}$ are in yellow area). As for the photons from path 1 that reach the silvered surface of the top right mirror (figure 4), half will pass through towards D2, half will be reflected back towards their surface of entry. Those last photons will encounter $e_{1}^{-}$electrons along their path back, but since their momentum of $\displaystyle {{m}_{{{\gamma }_{0}}}}c$ is less the momentum required for absorption by the $e_{1}^{-}$ electrons , $\displaystyle {{m}_{{{\gamma }_{0}}}}c, they will be reflected back towards the half-silvered surface of the mirror where half of these photons will be refracted towards D2 and the other half reflected back towards their surface of entry. This will continue until all photons are directed towards D2. Now consider the setup shown in figure 5. Here we have 50% of the photons that will reach D3, 25% of the photons that will reach each of D1 and D2. According to quantum mechanics, the photons moving along path 2 that reach D1 can only do so if the photons moving along path 1 are deflected towards D3. This raises the question: How the photons that reach D1 know that the photons of path 1 were deflected towards D3?

Quantum mechanics’ answer to that question is that the photons from path 1 and path 2 are entangled, a phenomenon known as quantum entanglement, by which a change done to photons on path 1, by a measurement for example, and which instantly affects the photons moving on path 2 even though the photons of path 1 are not in contact with the photons of path 2. And, according to quantum mechanics, it does so instantly and independently of the distance that separate the two groups of photons. If quantum mechanics is correct, that entangled photons be separated by a meter or billions of light-years does make any difference. This explanation of course violates locality, but this violation is essential to quantum mechanics if it is to correctly describe the result of experiments such as the ones we described above and which in turn, as interpreted by quantum mechanics, support the existence of quantum entanglement and non-locality.

That said, QGD optics provides a simpler interpretation of the results. Based on QGD’s explanation of the results from the setup shown figure 1 and figure 2, we understand that if photons in the second setup that move along path 2 will reach D2 simply because, without incoming photons from path 1, the atomic electrons of the transparent material remain in the $e_{0}^{-}$state. Hence the photons from path 2 that are not reflected within the transparent part of the top right half-silvered mirror.

Thus quantum-geometry dynamics describes and explains the results without quantum-entanglement and without violating locality. That in itself does not mean that QGD better describes reality. It does however offer a much simpler and local realistic explanation. As such, it contradicts Bell’s theorem which implies that no local hidden theory can explain the correlations of such experiments as those described in this article. That said, any number of theories can be made to be consistent with data and thus explain physical phenomena a posteriori. The only valid tests of a theory are the predictions that it makes that are original to it and that can be verified experimentally.

## QGD Experimental Predictions of “Non-Locality” Experiments

If QGD’s explanation of the result of the first setup is correct, and since as we explained, the refractive material needs to be of a certain minimum thickness so that any photons from path 2 will be intercepted by an atomic electron with mass ${{m}_{e_{1}^{-}}}$, then reducing the thickness below a certain value (figure 6) will allow photons from path 2 to reach D1. The minimum thickness being a number of electrons that exceeds the number of photons arriving from path 2 plus the photons from path 1 that are reflected back towards D1 by the reflecting surface of the top right mirror.

Also implied by QGD’s explanation is that, if the number of photons from path 2 moving through the top left mirror exceeds the number of $e_{1}^{-}$ electrons, then photons from path 2 will reach D1. This means that if the photon density of the source is increased passed a certain value, past the number of electrons which have absorbed photons from the path 1, then photons from path 2 will reach D1. Similarly, photons from path 1 reflected by the silvered side of the mirror will also reach D1. The actual number of photons that will reach is the difference between the number of photons from path 1 and path 2 moving towards D1 and the number of electrons in the $e_{1}^{-}$ state along their paths. Each articles of this series will examine other experiments understood to support quantum entanglement and non-locality.

If you are interested in knowing more about QGD, I suggested reading earlier articles or the pdf book Introduction to Quantum-Geometry Dynamics.

## The Casimir Effect as Explained by Quantum-Geometry Dynamics Quantum mechanics essentially describes the Casimir effect as being caused by virtual photons which spring into existence out of the nothingness of the vacuum to impart the plates of the apparatus and pushing them towards each other. The introduction of virtual particles, which is required if quantum mechanics is to explain the phenomenon, also creates a number of problems (though not seen as problems within the framework of quantum mechanics) among which is the violation of the principle of conservation of energy and the consequential infinite vacuum energy.

QGD also attributes the Casimir effect to free preons(+); the single elementary particle of matter the theory requires. Thus QGD’s explanation of the effect does not violate the law of conservation or introduce any of the problems that arise from the assumption of virtual particles.

As we know, preons(+) are isotropically distributed throughout quantum-geometrical space. They constantly bombard all material objects. Consider the figure on the left where we have a plate ${{R}_{a}}$. The plate divides quantum-geometrical space into two symmetrical regions ${{R}_{1}}$ and ${{R}_{2}}$ . Since the regions are symmetrical, and since the preonic density is the same in both, the resulting momentum of the preons(+) simultaneously hitting the plate from ${{R}_{1}}$ is equal that that from ${{R}_{2}}$ . That is: $\left\| {{{\vec{P}}}_{{{R}_{1}}}} \right\|=\left\| {{{\vec{P}}}_{{{R}_{2}}}} \right\|$. The momentum of the free preons(+) hitting the plate cancel each other out.

But when we put two plates in close proximity as is the Casimir effect apparatus, we divide quantum-geometrical space into three regions; ${{R}_{1}}$ , ${{R}_{2}}$ and ${{R}_{3}}$.

In this arrangement, the same number of preons(+) hit ${{R}_{a}}$ from ${{R}_{1}}$ as does the number of preons(+) that hit ${{R}_{b}}$ from ${{R}_{3}}$, but the number of preons(+) that hit ${{R}_{a}}$ from ${{R}_{2}}$ is equal the number of preons(+) from ${{R}_{3}}$ minus the preons(+) that are absorbed or reflected by ${{R}_{b}}$ . Hence, if $\left\| {{{\vec{P}}}_{{{R}_{1}}}} \right\|$ and $\left\| {{{\vec{P}}}_{{{R}_{2}}}} \right\|$ are respectively the resultant momentum of the preons(+) hitting ${{R}_{a}}$ from ${{R}_{1}}$ and ${{R}_{2}}$ , then we must have $\left\| {{{\vec{P}}}_{{{R}_{1}}}} \right\|>\left\| {{{\vec{P}}}_{{{R}_{2}}}} \right\|$. And, if $\left\| {{{\vec{P}}}_{{{R}_{1}}}} \right\|-\left\| {{{\vec{P}}}_{{{R}_{2}}}} \right\|\ge x{{m}_{{{R}_{a}}}}$, then net momentum imparted to ${{R}_{a}}$ will be $\Delta \left\| {{{\vec{P}}}_{{{R}_{a}}}} \right\|=x{{m}_{{{R}_{a}}}}$ . In other words, if ${{R}_{a}}$ was free, it would be pushed towards ${{R}_{2}}$ and its speed will increase by $\displaystyle \left\lfloor \frac{\left\| {{{\vec{P}}}_{{{R}_{1}}}} \right\|-\left\| {{{\vec{P}}}_{{{R}_{2}}}} \right\|}{x{{m}_{{{R}_{a}}}}} \right\rfloor$. Similarly, ${{R}_{3}}$ would move towards ${{R}_{2}}$ at $\displaystyle \Delta {{v}_{{{R}_{b}}}}=\left\lfloor \frac{\left\| {{{\vec{P}}}_{{{R}_{3}}}} \right\|-\left\| {{{\vec{P}}}_{{{R}_{2}}}} \right\|}{x{{m}_{{{R}_{a}}}}} \right\rfloor$.

## Cosmological Implication

If QGD is correct then the Casimir effect must affect all objects which divide quantum-geometrical space asymmetrically. It follows that the Casimir effect must affect objects at the cosmic scale as well; pushing cosmic structures towards each other and thus contributing to the dark matter effect.

(The above is an excerpt from Introduction to Quantum-Geometry Dynamics)