We have shown in "The Imagineer’s Chronicles" one can define a universe in terms of the existence of four *spatial* dimensions in a manner that makes predictions identical to those of both the General and Special Theories of Relativity while at the same time describing the theoretical advantages to doing so.

One of these is that it would allow for the development of a theoretical interpretation of the Heisenberg’s Uncertainty Principal in terms of the laws Classical of Physics.

The Heisenberg Uncertainty Principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known. This is not a statement about the limitations of a researcher’s ability to measure particular quantities of a system; it is a statement about the nature of the system itself as described by the equations of quantum mechanics.  According to the uncertainty principle, it is, for instance, impossible to measure simultaneously both position and velocity of a microscopic particle with any degree of accuracy or certainty.

In the article "Why is mass and energy quantized?" Oct 4, 2007 it was shown one can theoretically derive the properties of a particle in terms of a classically resonating system if one assumes it is a result of matter wave moving on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.  Additionally, it was showed why all energy exists in these resonant systems and therefore is quantized.

There are four conditions required for resonance to occur in a classical Newtonian environment, an object, or substance with a natural frequency, a forcing function at the same frequency as its natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial.

(In a latter article "The geometry of quarks" it will be shown how and why quarks join together to form these resonant systems in terms of the geometry of four *spatial* dimensions.)

The existence of four *spatial* dimensions would give three-dimensional space the ability to oscillate spatially on a "surface" between a third and fourth *spatial* dimensions thereby fulfilling one of the requirements for classical resonance to occur.

These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital.  This would force the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.

However, these oscillations in a "surface" of a three-dimensional space manifold, according to classical mechanics would generate a resonant system or "structure" in space.  These resonant systems are known as particles.

The only way to dampen the frequency of a classically resonating system is to add or remove energy from it, which results in changing the characteristics of that system. 

Additionally the energy in a classically resonating system is discontinuous and can only take on the discrete values associated with its fundamental or harmonic of its fundamental frequency.

However, these properties of a classically resonating system are the same as those found in a particle in that they are made up of discreet or discontinuous packets of mass/energy and when energy is either added or removed from it, its characteristics change. 

The reason why we do not observe energy in its continuous wave form is that, as mentioned earlier all energy is propagated through space in the discrete components associated with its resonant structure. Therefore, when observing the energy of a wave function it would appear as though its energy was concentrated in a single point in space associated with the center of its resonant structure.

This is analogous to how the potential energy of water in a sink is release by allowing it to go down the drain.  If all we could observe is the water coming out of the drain we would have to assume that it was concentrated in the region of space defined by the diameter of the drain.  However, in reality the water occupies a much larger region. 

However, this also defines a Classical reason for the validity of Heisenberg’s Uncertainty Principle, which states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision.

Classical wave mechanics tells us a wave’s energy is instantaneously constant at its peaks and valleys or the 90 and 270-degree points as its slope changes from positive to negative while it changes most rapidly at the 180 and 360-degree points.

Therefore, the precise position of a particle could be only be defined at the peaks and valleys of the matter wave responsible for its resonant structure because those points are the only place where its energy or "position" is stationary with respect to a fourth *spatial* dimension.  Whereas its precise momentum would only be definable with respect to where the energy change or velocity is maximum at the 180 and 360-degree points of that wave.  All points in between would only be definable in terms of a combination of its momentum and position.

However, to measure the exact position of a particle one would have to divert or "drain" all of the energy at the 90 or 270-degree points to the observing instrument leaving no energy associated with its momentum left to be observed by another instrument.  Therefore, if one was able to determine precise position of a particle he could not determine anything about its momentum. Similarly, to measure its precise momentum one would have to divert all of the energy at the 180 or 360 point of the wave to the observing instrument leaving none of its position energy left to for an instrument trying to measure it position.  Therefore, if one was able to determine a panicles exact momentum one could not say anything about its position.

The reason we observe a particle as a point mass instead of an extended object is because, as mentioned earlier the article "Why is mass and energy quantized?" showed its energy must be packaged in terms of its resonant frequency.  Therefore, when we  observe or "drain" the potential energy continued in its wave function it will appear to come form a specific point in space similar how the potential energy of water flowing down a sink drain appears to be coming from a "point" source with respect the extended volume of water in the sink.

As mentioned earlier, all points in-between are a dynamic combination of both position and momentum.  Therefore, the degree of accuracy one chooses to measure one will affect the other.  For example, if one wants to measure the position of a particle to within a certain predefined distance "m" its wave energy or momentum within that distance will "move" or change by an amount defined by its wave function or the energy change within that distance or ("m"kg / s).  Summarily if one wants to measure the momentum ("m"kg / s) of a particle in a predefined volume or distance ”m" its position will change within that distance.

However, because of the dynamic interaction between the position and moment component of the matter wave responsible for generating the resonant system associated with a particle defined in the article  "Why is mass and energy quantized?" the change or uncertainty of one with respect would be defined by the product of those factors or m^2 kg / s

This defines the reason in terms of Classical Wave Mechanics for Heisenberg uncertainty principal and why it has the units of m^2 kg / s

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2010

(In a PDF format)

As Brian Greene pointed out in his book "The Elegant Universe", one of the unsolved mysteries of modern particle physics is why every fundamental particle encountered to date can be group into three families.

"Physicists have recognized a pattern among these particles displayed in the following table.  The matter particles neatly fall into three groups, which are often called families.  Each family contains two of the quarks an electron or one of its cousins and one of their neutrino species.  The corresponding particle types across the three families have identical properties except for their mass, which grows larger in each successive family."

The answer to Brian Greene’s question regarding why the particles in the above table can be group into three families is related to the resonant "structures" that defined their properties in the article “Why is mass and energy quantized?” Oct. 4, 2007

That article showed one can derive a particle’s properties in terms of a classically resonating system formed on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.  Additionally it was shown mass can only take on the energy values associated with its resonant or a harmonic of its resonant frequency.

If true, one should be able to derive the relative masses of each family member by extrapolating laws governing classical resonance in three-dimensional space to a fourth *spatial* dimension.

The article "Defining gravity" Dec. 15, 2007 showed one can derive the mass of objects and particles in terms of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.  Additionally, it was shown the magnitude of their mass is directly related to a *separation* in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by that curvature.

(This curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension is analogous to the space-time curvature the Einstein postulated is responsible for an objects mass.)

Finally the article "Defining potential and kinetic energy?" Nov. 26, 2007 showed one can derived all forms of energy, including gravitational, electrical and thermal, in terms of a spatial "separation" between different "surfaces" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

As mentioned earlier the article "Defining gravity" showed the mass of all objects including particles are a result of a separation in a "surface" of three-dimensional space manifold with respect to a fourth "spatial" dimension.  However, as was shown in article "Defining potential and kinetic energy?" the separation between "surfaces" of a three-dimensional space manifold is dependent on the temperature or energy content of the environment associated with them.  Therefore, the masses of all objects including particles would also be dependent on the energy content or temperature of the environment they are occupying.

Therefore, as was shown in the articles "Why is mass and energy quantized?" and "Defining gravity" the individual masses of each family member would be dependent on the energy of their associated resonant systems and energy of the environment they are occupying.

However, because there are only specific points in space where environments can resonant, each of the corresponding particle types across the three families would have a specific mass related to the energy of the environment in which they were created

The particles in the first family are found in relativity low energy environments, are relatively stable, and for the most part can be observed in nature.  However, the particles in the second and third families are for the most part unstable and can be observed only in high-energy environments of particle accelerators.  The exception is the Muon in the second family, which is only observed in the high-energy environment of cosmic radiation.

The relative masses of the fundamental particles increases in each successive family because as shown in the article "Defining potential and kinetic energy?" the higher-energy environments where they are generated in result in the corresponding particles in each successive family to be formed with a greater relative "separation" in the “surfaces” of a three-dimensional space manifold with respect to a fourth *spatial* dimension..

Therefore, the corresponding particles in the second family will have a greater mass than the particles in the first family because the "separation", with respect to a fourth *spatial* dimension of the three-dimensional space manifold associated with them is greater than the "separation", associated with the first family.

Similarly, the corresponding particles in the third family will have a greater mass than those in the second family because the "separation", with respect to a fourth *spatial* dimension, of the three-dimensional space manifold associated with them is greater than the spatial "separation", associated with the second family.

Additionally the corresponding particle types across the three families have "identical properties" because as shown in the article "The geometry of quarks" Mar. 15 , 2009 they are related to the orientation of the "W" axis of the fourth *spatial* dimension with the axis of three-dimensional space.  Therefore, each corresponding particle across the three families will have similar properties because the orientation of the "W" axis of the fourth *spatial* dimension with respect to the axis of three-dimensional space is the same for the corresponding particles in all of the families.

This explains why "The corresponding particle types across the three families having identical properties except for their mass, which grows larger in each successive family” in terms of the properties of classical resonance and the existence of four *spatial* dimensions.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2010

(In a PDF format)

We have shown throughout “The Imagineer’s Chronicles” there are many theoretical advantages to assuming the existence of four *spatial* dimensions a continuous non-quantized form of mass.

One of them is that it would provide a more logical explanation of the observed properties of Dark Matter than can be found based solely on the quantum mechanical assumption that mass exists only in particle form.

Wikipedia tells us "The first person to provide evidence and infer the presence of dark matter was Swiss astrophysicist Fritz Zwicky, of the California Institute of Technology in 1933.  He applied Newton’s law of gravity to the Coma cluster of galaxies and obtained evidence of unseen mass.   Zwicky estimated the cluster’s total mass based on the motions of galaxies near its edge and compared that estimate to one based on the number of galaxies and total brightness of the cluster.  He found that there was about 400 times more estimated mass than was visually observable.  The gravity of the visible galaxies in the cluster would be far too small for such fast orbits, so something extra was required.  This is known as the "missing mass problem".  Based on these conclusions, Zwicky inferred that there must be some non-visible form of matter which would provide enough of the mass and gravity to hold the cluster together."

Many physicists believe the vast majority of the dark matter is in a non-baryonic form such as neutrinos, and entities such as axions, supersymmetric particles, or WIMPs.

However, none of these scenarios is supported by observations.

Neutrinos because of their mass would be characterized by high random speeds in the early universe.  However, observations of the early universe indicate the matter that condensed to form galaxies was not hot enough to support the energy that would be associated with those high speeds.

The other particles, which could provide the missing mass fall into two classes: those which have been proposed for other reasons but happen to solve the dark matter problem, and those which have been proposed specifically to provide the missing dark matter.

Examples of objects in the first class are axions and the supersymmetric particles.  Their properties are defined by the theory, which predicts them, by virtue of their mass, they can solve the dark matter problem only if they exist in the correct abundance.

The second class of particles contain entities such as the WIMP or "Weakly Interacting Mass Particle" whose properties are not specified.  However, they are assumed to have properties that would allow them to explain the missing mass associated with dark matter along with other "ad hoc" ones that would explain why they have not yet been observed experimentally.

However, the existence of them along with axions and the supersymmetric particles is not based on observations so therefore there is no way to either confirm their existence or that they are responsible for the gravitational force associated with dark matter.

However, there is another theoretical possibility that is based on observations that has been overlooked by the scientific community.

In the article “What is Dark Matter?” Sept 10, 2007 it was shown that one could theoretically explain and predict the gravitational forces associated with Dark Matter in terms of a continuous non-quantized form of mass.  This non-baryonic and non-particle form of mass would have all the observed properties dark matter in that it would not interact with ordinary matter via electromagnetic forces and since it is made up of mass it would add to the gravitational force of the particle matter in the universe.

However, unlike WIMPS its existence is supported by observations.

For example, the observation that energy in itself is not quantized because a photon can have any frequency greater than zero or less than infinity and the equation defining the relationship between mass and energy, E=m*c^2 indicates that mass is completely convertible to one or more photons with energies greater than zero or less than infinity.

This means a continuous non-quantized medium must be available to support the continuous properties associated with mass energy and the electromagnetic spectrum.

Additionally the existence of a continuous non-quantized form of mass is supported by the observation of Davisson and Germer, when, in 1927 they observed that electrons along with other particles are diffracted by crystals.  This is because the only way to explain the observed wave properties of individual particles is to assume they must have the continuous non-quantized geometric substructure associated with a wave.

However, the most significant theoretical advantage to assuming its existence is that it would allow one understand, in terms laws of classical wave mechanics the quantum mechanical aspects of mass and energy.

In the article "Why is mass and energy quantized?" Oct. 4, 2007 it was shown that one could explain and predict the quantum properties of mass and energy in terms of a classically resonating system or "structure" form by a matter wave moving through a continuous non-quantized form of mass. 

Classical wave mechanics tells us a resonant phenomenon will occur with all types of vibrations or waves: there is mechanical resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR), and resonance of quantum wave functions.  Resonant systems can be used to generate vibrations of a specific frequency (e.g. musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies.

However, as mentioned earlier the observation that particles mass has the continuous non-quantized geometric substructure associated of a wave means that according to classical wave mechanics a resonant system will be established is space.  This provides observational evidence supporting the derivation of the quantum mechanical or particle aspects of mass and energy, as was done in the article "Why is mass and energy quantized?" in terms of discrete resonant systems formed by vibrations in a continuous non-quantized form of mass.

The are some who feel that the existence of substance like a continuous non-quantized form of mass permeating space validates the Newtonian concept of absolute space.  However, as was shown in the article “Reference frames” July 1, 2008 this is not true.

Briefly, in  the article "Defining energy" Nov. 26, 2007 it was shown the momentum or energy of an inertial reference frame can be derived in terms the displacement of its three dimensional volume with respect to a fourth *spatial* dimension.

Isaac Newton defined an inertial reference frame as one in which an object at rest will remain at rest, and an object in motion will remain in motion in a straight line at a constant speed.

However, as was shown in the article "Defining energy" the momentum or velocity of an inertial reference frame is defined by a constant displacement with respect to a fourth *spatial* dimension of the continuous non-quantized form of mass defining its volume. 

Therefore, the existence of a continuous non-quantized form of mass does not validate the existence of absolute space because its motion can only be measured relative to other reference frames.

This shows there is viable alternative based solely on experimentally verifiable observations to the assumption that dark matter is made up of exclusively of matter in a particular form.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2010 Jeffrey O’Callaghan

(In a PDF format)

The effort to unify all of the forces and laws of nature or find a "Theory of Everything" has primarily focused on explaining gravity in terms of the concepts contained in quantum mechanical particle theories.

However, it is unfortunate that some have not made an effort to find it by explaining the quantum mechanical properties of particles in terms of gravitational forces because if they had we may have found it by now.

We observe that all particles have mass, which is associated with gravitational force.  However, for the past century the brightest minds of the scientific community have been unable to define how this force can be propagated by a particle using the current quantum mechanical paradigms.  Additionally, even with the recent advancements in observational technologies, no one has observed the graviton or particle that many feel is responsible for the propagation of gravitational forces.

However, the fact that we have not been able to define a unifying mechanism either mathematically or conceptually for the observed quantum mechanical and gravitational properties of nature in terms of the current paradigms may not be related to their content but to how we are attempting to integrate them.

For example, the fundamental assumption of Quantum mechanics is that mass and energy is contained in discrete irreducible units or packets of energy called quarks and leptons.

However, the graviton or particle many physics associated with gravitational force has not, as mentioned earlier been observed.  Some feel that this is due to the fact our instruments are not yet advance enough to detect it but it could also be because gravitational force is not propagated by a particle but by a continuous property of mass/energy.

This conclusion is supported by the observation of Davisson and Germer, when in 1927 they observed that electrons and along with other particles are diffracted by crystals.  This means they must have a wave component because that is the only way to explain how they can generate a diffraction pattern.  However, it also indicates that each individual particle must also have the continuous geometric substructure associated of a wave.

We have shown throughout “The Imagineer’s Chronicles” that observations of our environment suggest that space is composed of four *spatial* dimensions instead of four-dimensional space-time.

In the article "Defining gravity" Dec. 15, 2007 it was shown that one can theoretically derive the relativist properties of motion, gravity and the fact that it is equivalent to an accelerated reference frame in terms of a continuous geometric property of four *spatial* dimensions in a manner that makes predictions identical to these made by both The General and Special Theories of Relativity.

While in the article "Why is mass and energy quantized?" Oct. 4, 2007 it was shown that one can derive the quantum mechanical properties of mass in terms of a classically resonating system generated by a matter wave on a continuous "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

There are four conditions required for resonance to occur in a classical Newtonian environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial.

(In a latter article "The geometry of quarks" it will be shown how and why quarks join together to form these resonant systems in terms of the geometry of four *spatial* dimensions.)

The existence of four *spatial* dimensions would give a "surface" of three dimensional space the ability to oscillate spatially with respect to it thereby fulfilling one of the requirements for classical resonance to occur.

These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital.  This would force the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.

However, these oscillations in three-dimensional space would, according to classical mechanics would generate a resonant system or "structure" in it.  These resonant systems are what defined a particle in the article "Why is mass and energy quantized?"

This cannot be done in terms of four-dimensional space-time because time or a space-time dimension is only observed to move in one direction forwards and therefore could not support the bi-directional spatial movement required to establish classical resonance.

The assumption that the quantum mechanical properties of mass are a result of a continuous properties of space is also supported by Davisson and Germer’s observation that all particles has a wave component because, as mentioned earlier it means they must have the continuous geometric substructure of a wave.

However, as mentioned earlier the properties of gravity can also be also derived in terms of the continuous property of four *spatial* dimensions. 

This suggest that we may have already found a "A Theory Everything" if, as mentioned earlier we had tried to derive the discontinuous quantum mechanical properties of mass in terms of the continuous properties of gravity instead of deriving the discontinuous properties of particles in terms of continuous properties of gravitational forces.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2008 Jeffrey O’Callaghan

(In a PDF format)

Can we justify defining the reality of an environment based on our inability to define its reality. 

The uncertainty principal of quantum mechanics tells us that we cannot know or observe the precise amount of energy contained in microscopic physical system over very short intervals of time. 

Some physicists feel that because they cannot observe it in a microscopic system, it must fluctuate around a given point even though that volume is made up of a vacuum.  They call the energy generated by this uncertainty quantum fluctuations or vacuum energy.

However, this means they are defining the reality of a vacuum in terms of their inability to define or observe the "reality" of the energy contain in that vacuum.

We have shown that it is more consistent with observations to define energy including that contained in a vacuum in terms the continuous properties of four *spatial* dimensions instead of four-dimensional space-time.

The observations many physicists associate with quantum fluctuations is one of those observations.

In the article “Why is mass and energy quantized?‘“ Oct. 4, 2007 it was shown that one can explain and predict the quantum properties of mass and energy in terms a resonant system caused by a matter wave moving on a continuous "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, this means one can define the "reality" quantum fluctuations in a vacuum based by extrapolating observations of our three dimensional environment to four spatial dimensions instead of relying, as many physicists seem to on their inability to observe them.

Wikipedia describes the Casimir or the Casimir-Polder force as a physical force arising from a quantized field, which is responsible for zero point or vacuum energy.  The typical example is of two uncharged metallic plates in a vacuum, placed a few micrometers apart, without any external electromagnetic field.  In a classical description, the lack of an external field also means that there is no field between the plates, and no force would be measured between them.  When this field is instead studied using quantum electrodynamics, it is seen that the plates do affect the virtual photons which constitute the field, and generate a net force either an attraction or a repulsion depending on the specific arrangement of the two plates.  This force has been measured, and is a striking example of an effect purely due to second quantization.  However, the treatment of boundary conditions in these calculations has led to some controversy.

Dutch physicists Hendrik B. G. Casimir and Dirk Polder first proposed the existence of the force and formulated an experiment to detect it in 1948 while participating in research at Philips Research Labs.  The classic form of the experiment, described above, successfully demonstrated the force to within 15% of the value predicted by the theory.

Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small.  On a sub micrometer scale, this force becomes so strong that it becomes the dominant force between uncharged conductors.  In fact, at separations of 10 nm—about 100 times the typical size of an atom the Casimir effect produces the equivalent of 1 atmosphere of pressure (101.3 kPa), the precise value depending on surface geometry and other factors.

In 1924, Louis de Broglie theorized all particles have a wave component.  His theories were later confirmed in 1927 by Davisson and Germer when they observed that electrons are diffracted by crystals.

Observations of waves in a classical environment indicate the number of simple harmonic oscillators that can be established in a given environment is dependent on the distance or "gap" between the "end points" of their environments.

But same concept can be applied to two uncharged metallic plates in a vacuum, because even without any external electromagnetic field the electromagnetic components of the atoms in each plate are vibrating because if they are not at absolute zero they have thermal energy.  These random vibrations of their electromagnetic components will result in a random electromagnetic field to be generated between the plates.

However, classical wave mechanics tells us these random electromagnetic vibrations would be reinforced at certain points in space.  The number of simple harmonic oscillators or quantum fields in the space between two plates formed by this reinforcing would decreases as the gap between them decreases.  In other words, the smaller the gap between the plates the fewer number of quantum fields or particles that gap could support.

This means as was shown in the article ”Why is mass and energy quantized?“ there will be a greater number simple harmonic oscillators or quantum fields impacting the plates from outside of the gap than between it.  This will cause a force that will push the plates together because the energy density associated with harmonic oscillations outside of the gap would be greater than inside of it.

We know the reality of the wave properties of particles because Davisson and Germer physically measured and observed them.

However, we cannot observe the "reality" of the quantum fluctuations physicists associate with vacuum energy because, as mentioned earlier their existence is based on the fact we cannot observe them.  

This suggests the Casimir effect may not be due to our inability to know the precise "reality" of the amount of energy contained in microscopic physical system but to the physical observable reality of the wave properties of a particle.

However, it also means the "reality" of quantum mechanics could be defined in terms of the reality of classical wave mechanics and the continuous properties of four *spatial* dimensions instead of non "reality" of the uncertainty principal.

It is not be possible to define the wave prosperities of a particle in terms of four-dimensional space-time because it cannot support the bi-directional movement required to explain the transverse wave motion Davisson and Germer observed in electrons.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2009

(In a PDF format)

We have shown throughout “The Imagineer’s Chronicles” there would be numerous theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four-dimensional space-time.

One of them is that it would allow for the theoretical definition of the "boundaries" of a quantum particle in terms of the laws of classical physics.

In the article “Embedded Dimensions” Oct. 22, 2007 it was shown that each point in three-dimensional space is embedded in a fourth *spatial* dimension and energy only propagates along a "surface" of three-dimensional space.

Classical wave mechanics tells us the oscillations generated by a wave on the surface of water is responsible for the transmission of energy.

Therefore, as was done in the article "The Photon: a matter wave?" Oct 1, 2007 one could explain how energy is propagated through three-dimensional space by extrapolating the laws classical of wave mechanics to oscillations in its ‘surface” generated by a matter wave moving on it with respect to a fourth *spatial* dimension.

However, if this were true why is it that we cannot see the fourth *spatial* dimension.

The reason is because, as was mentioned earlier all forms of energy move along a "surface" of three-dimensional space.  Therefore, electromagnetic energy or light could not penetrate and illuminate the fourth dimensional component of space.

This is analogous to how a wave on the two-dimensional surface of water can only "illuminate" or interact with objects that comes in contact with its surface.  Therefore, a fish could not "see" the three-dimensional components of objects above its surface if a wave on that surface were the only way energy or information could be transmitted through its environment.

Similarly, three-dimensional being would be unable to "see" the fourth dimensional component of objects if energy is only transmitted on a "surface" of three-dimensional space.

In Einstein’s General and Special theories of Relativity he showed that one could derive the relativistic properties of motion, gravity, space and time in terms of an interaction of a three-dimensional volume with a fourth time dimension.  But he was unable to derive a mechanism that can explain the quantum mechanical properties of mass and energy.

However, in the article “The “Relativity” of four *spatial* dimensions” Dec. 1, 2007 it was showed, one can derive the relativistic properties of gravity, motion, space and time in terms of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension in a manner that makes predictions identical to those of Einstein’s theories.

(This curvature is analogous to the space-time curvature Relativity assumes is responsible for gravity.)

While in the article “Why is mass and energy quantized?” Oct. 4, 2007 it was shown one can also derive the quantum mechanical properties of mass and energy in terms of classically resonating system generated by a matter wave moving on a “surface” of a three dimensional space manifold  with respect to a fourth *spatial* dimension.

In 1924 Louis de Broglie was the first to realize all particles have an oscillating or wavelike component.  In his paper, “Theory of the double solution“ he attempted to define a causal interpretation of their wave properties in the classical terms of space and time.  He later abandoned it in the face of the almost universal adherence of physicists to the theories presented by Born, Bohr, and Heisenberg regarding the uncertainties and probabilistic interpretation of quantum particles.

In classical physics, a point on a two-dimensional surface of paper is confined to that surface.  However, that surface can oscillate up or down with respect to three-dimensional space.

Similarly, an object occupying a volume of three-dimensional space would be confined to it.  However, as was shown in the article “Embedded Dimensions” it could, similar to the surface of the paper oscillate with respect to a fourth *spatial* dimension.

Classical wave mechanics tells us resonance phenomena will occur with all types of waves or oscillations that occur in confined volumes.

Therefore, the confinement of oscillations in a "surface" of three-dimensional space with respect to a fourth *spatial* dimension would define the geometric boundaries of the resonant system responsible for the quantum characteristics of energy in the article "Why is mass and energy quantized?"

This suggests that one could by theoretically extrapolating the laws of classical physics to a fourth *spatial* dimension define boundaries of a particle in terms of oscillations in its mass and energy components.

This defines the quantum properties of mass and energy in terms of the laws Classical Physics because it shows how extrapolating them to four *spatial* dimension would allow one to theoretically define their "boundaries" of and the reason why energy appears to be quantized.

However, it is not possible to define a classical resonant system in terms of the geometry of space-time because time is only observed to move in one direction forward and therefore could not support the bi-direction movements to define the boundary conditions required for it to occur.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2010

(In a PDF format)

or the discrepancy between the worldview implied by the accuracy of modern quantum mechanics and our Common sense (Sept. 1, 2008) understanding of reality is due to the incompleteness of its theoretical structure.  

The strangeness of the quantum mechanical worldview is exemplified by the fact that it defines reality only in terms of probabilities.

However, its absurdity arises from the fact, as Heisenberg one of the founding fathers of quantum mechanics put it "atoms or elementary particles themselves are not real but are forms of world probability or possibilities rather than things or facts".

Quantum mechanics defines the probability of finding a particle in given region in terms of Schrödinger’s wave function or equation.  However, according to the Copenhagen‎ interpretation it only defines the probability of particle being there.  Therefore, it assumes it only comes into existence after one observes it.

The reason is before it is observed it existed only as a wave function with an extended spatial environment.  Therefore, it did not have the properties of a particle.  However, after one observes it a particle appears in a compact region space while the wave function that was in that region collapses or disappears.  Therefore, quantum mechanics assumes that the act of observing a wave function is the causality of a particles existence.

However, one can define a perfectly logical reason why the act of observing the wave function results in its collapse and the formation of particle by extrapolating the laws Newtonian mechanics to four *spatial* dimensions.

The earlier article "Why is mass and energy quantized?" Oct. 4, 2007 showed that one can derive the existence of a particle in terms of a classically resonating system or "structure" formed by matter wave on a "surface" of a three-dimensional space manifold with respect to a fourth spatial dimension.  Additionally it was showed all energy exists in resonant systems and therefore must be quantized. 

This would solve the Enigma as to what is waving in the wave function by defining the medium supporting it in terms of space itself.  This is because, as was shown in that article the vibrations of three-dimensional space with respect to a fourth *spatial* dimension is responsible for the both wave and resonant structure that defines a particle’s energy and wave function.

However, according to the laws of classical wave mechanics the "concentration" of a wave’s energy is maximum at its peaks and troughs.  Therefore, one could only observe or "drain" the energy continued in a wave function at points corresponding to them because those would be the only points where its energy would be “concentrated” enough to be redirected as a resonant system to the observing instrument.  However, as was shown in the article "Why is mass and energy quantized?" a particle is defined by the energy of its resonant structure.  Therefore, when one looks at a particular spot within the confines of a wave function one would either find a particle or nothing at all because the volumes between its peaks and troughs would not contain enough energy to allow its resonant structure to be redirect towards an observer. 

This is analogous to how the potential energy of water in a sink is release by allowing it to go down the drain.  If all we could observe is the water coming out of the drain we would have to assume that it was concentrated in the region of space defined by the diameter of the drain.  However, in reality the water occupies a much larger region. 

Additionally, the fact that a particle is made up of the energy of its wave function means they both cannot exist at the same time.  Therefore, if one chooses to observe it as a particle such as an electron one would not be able to observe its wave properties because the energy associated with it would be redirected towards the observer from a specific point in space.  However if one chooses to observe its wave properties, such as the diffuse orbit of an electron around a nucleus one could not observe it as a particle because its energy is contained in the diffuse environment of the electrons orbit.

This means contrary to the Copenhagen‎ interoperation the act of observing a wave function does not create a particle but only transforms or redirects its energy from a specific point within the extended spatial environment associated its wave function. 

In other words, when we observe a wave function, we are not causing its collapse while creating a particle but only redirecting its wave energy from a compact source within the confines of that wave function.  The reason it appears as a particle is that its energy is observed to be originating from a specific point within the confines of the wave function that is observed.

This is analogous to how one can change the form of water from a solid to a liquid or vice versa by either cooling or heating it.  However, the act of cooling or heating water does not create anything that was not there before it just changes the form of the water from a liquid to a solid.

This also explains why, in classical terms a particle appears to be simultaneously in many places at one time within the confines of its wave function because it shows its position relative to that volume would be dependent on where one observed its wave function.  In other words, a wave function contains only one not multiple copies of a particle whose position within the wave function is defined by where one chooses to observe it. 

It also defines the randomness of quantum mechanics in classical terms because it shows the probability of finding a particle in a specific point in space would depend on where in the time varying environment of a wave function an observer made contact with it.  Since an observer is unable to view a wave function before his or her instruments interacted with it he or she could only determine the probably of where a particle will appear in its extended spatial environment.

Additionally, it defines a classical reason why consciousness appears to play a role in the outcome of an observation.  If one chooses to view an electron orbiting an atom from afar, so to speak he or she would only "see" the interference pattern generated by wave function.  However, if one chose to measure where it was he or she only would “see” it as a compact or point source called a particle centered within the volume defined by the peaks and troughs of its wave function.

Finally, it defines why how we chose to observe a quantum system effects what we observe.  For example, if chose to look at a specific region of space occupied by a wave function we will not be able to see the interference that it causes because, as mentioned earlier the act of observing it causes its energy to be redirected to a particle format.  Therefore, we will only be able to observe quantum or particle properties associated with its energy.  However, if we chose not to look in a specific region of space and just observe the wave function we will always observe the interference associated with its wave properties.

This shows that one can derive a classical common sense explanation of the wave particle duality of existence if one assumes, as we have done in the article "Why is mass and energy quantized?" that the quantum mechanical properties of nature are the caused by a resonant system formed by a matter wave  moving on a "surface of a three-dimensional space manifold with respect to a fourth *spatial* dimension.   Additionally, it completes the theoretical structure of quantum mechanics by providing an answer to the quantum enigma of why the conscience actions or awareness of an observer appear to affect physically reality as is demanded by the Copenhagen interpretation.

Later Jeff

The Shadows of four spatial dimensions

Copyright Jeffrey O’Callaghan 2010

(In a PDF format)

We have shown throughout "The Imagineer’s Chronicles" there would be many theoretical advantages to defining the universe in terms of a continuous non-quantized form of mass and four *spatial* dimensions instead of four dimension space-time.

One is it would allow one to define a background independent quantum theory.

One of the fundamental difficulties in unifying Quantum Mechanics with Einstein’s General Theory of Relativity is that one is background independent while the other is not.

Einstein tells us the geometry of space evolves with time and the laws of nature have to be expressed in a form that does not assume that space has a fixed geometry.  In other words, they are background independent. 

However, present quantum theories can only be formulated in a fixed geometric background. 

For example, Quantum Field Theory was unified with Maxwell’s theory of electromagnetism by assuming that electromagnetic energy does not interact with the background geometry of space.  In other words, two photons or quanta of electromagnetic energy can move through or past a fixed point in space without interacting with each other.

However, two gravitons or quanta of gravitational energy will interact with each other as they pass a fixed point in space because as mentioned earlier Einstein tells us gravity alters the geometry of space. 

This is why, as mentioned earlier one of the problems physicists encounter in defining quantum gravity is that present quantum mechanical models are background independent while Einstein tells us that gravity is not.

However, as mentioned earlier one can by assuming the existence of four *spatial* dimensions instead of four-dimensional space-time derive a background independent quantum theory with respect to gravity..

In this article "Gravity" Dec 15, 2007 it was shown that it is possible to derive gravity in terms of a curvature in a "surface" of a three-dimensional space manifold with respects to a fourth *spatial* dimensions in a manner that makes predictions identical a curvature in a space time manifold "The General Theory of Relativity assumes was responsible for gravity.

Additionally the article "Why is mass and energy quantized?" Oct 4, 2007 showed it is possible to explain and predict the quantum mechanical characteristics of mass by assuming they are a result of a classically resonating system or "structure" formed on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

There are four conditions required for resonance to occur in a classical Newtonian environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial.

(In article "The geometry of quarks" Mar. 15, 2009 it will be shown how and why quarks join together to form these resonant systems in terms of the geometry of four *spatial* dimensions.)

The existence of four *spatial* dimensions would give a continuous non-quantized form of mass (the substance in classical resonance) the ability to oscillate spatially on a "surface" between a third and fourth *spatial* dimensions thereby fulfilling one of the requirements for classical resonance to occur.

These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital.  This would force the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.

However, these oscillations in a continuous non-quantized form of mass, according to classical mechanics would generate a resonant system or "structure" on a surface of a three dimensional space.

Einstein derived the relativistic properties of motion by merging the dimensions of space and time.  However in the article “Why Space-time?” Nov. 22, 2007 it was show that one can derive the relativistic motion in terms of four independent *spatial* dimensions.

However, this also means a quantum mechanical model based on the resonant properties of four *spatial* dimensions would be background dependent with respect to its geometry while background independent relative to a curvature in that geometry.  This is because, as the article "Why is energy quantized?" showed the quantum mechanical characteristics of mass are the result of a resonating "structure" on a "surface" of three-dimensional space therefore, their interactions would be independent of the magnitude of a curvature in that "surface".  In other words, two photons or quanta of electromagnetic energy can move through or past a fixed point on curved "surface" of three-dimensional space without interacting with each other because their energy is not dependent on the geometry of space but is only dependent on the energy of associated with their resonant properties.

However, gravity would be background independent of the geometry of four *spatial* dimensions because as was shown in the article "Gravity" it is dependent on the changing geometry or curvature of a point on a "surface" of three-dimensional space with respect to a fourth *spatial* dimension.  Therefore two gravitons or quanta of gravitational energy will interact with each other as they pass a fixed point in space because of they share a property of that curvature.

Therefore, one could define a background independent quantum theory relative to gravity if one assumes as we have done the existence of  four independent *spatial* dimensions.

This is not possible if one defines the universe in terms of four-dimensional space-time because Einstein defined gravity in terms of the interdependence of space and time and therefore on could not separate gravity from its quantum mechanical properties.

Later Jeff

The "Shadows" of four *spatial* dimensions

Copyright 2009 Jeffrey O’Callaghan 

(In a PDF format)

We have shown throughout ”The Imagineer’s Chronicles” there would be many theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four-dimensional space-time.

One is that it would allow for a classical interoperation of Schrödinger’s wave equation and the wave / particle properties of mass.

In 1924 Louis de Broglie was the first to realize all particles are, in part composed of a transverse wave.  In his paper, “Theory of the double solution“ he attempted to define a causal interpretation of their wave properties in the classical terms of space and time.  He later abandoned it in the face of the almost universal adherence of physicists to the theories presented by Born, Bohr, and Heisenberg regarding the uncertainties and probabilistic interpretation of quantum particles.

One of the difficulties he may have faced in this endeavor is that he assumed along with most other scientists of his day the universe was composed of four-dimensional space-time.

This presented a problem because observations of a space-time environment indicate that a time dimension can only move in one direction, forward.  Therefore, it could not support bidirectional movement required for the propagation of a transverse wave consisting of three-dimensional space. 

However, the laws of Classical Wave Mechanics tells us transverse oscillations on any surface will form resonant systems or "structures" that, when view from distance would appear to be made up of discrete units with finite boundaries.

Summarily if the microscopic waves theorized by de Broglie were a result of oscillations in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension one could, as was shown in the article "Why is mass and energy quantized?" Oct. 4, 2007 extrapolate those laws to a fourth *spatial* dimension to derive the discrete or quantum mechanical wave / particle properties of mass and energy.

However, if a particle’s energy is a result of a matter wave on a "surface" of three dimensional space with respect to a four spatial dimensions, as we are suggesting one should be able to show why Schrödinger wave equation describes its position and momentum as it moves or oscillates through space by extrapolating the laws of Classical Wave Mechanics to four *spatial* dimensions.

Classical wave mechanics tells us a wave’s energy is instantaneously constant at its peaks and valleys or the 90 and 270-degree points as its slope changes from positive to negative while it changes most rapidly at the 180 and 360-degree points.

Therefore, the position of a particle could only be determined at the peaks and valleys of the matter wave because at those points its energy would not be moving with respect to a fourth *spatial* dimension  whereas its total momentum would only be definable with respect to where the energy change or velocity is maximum at the 180 and 360-degree points of that wave.  All points in between would only be definable in terms of a combination of its momentum and position.

Therefore, a Classical interpretation of Schrödinger’s wave equation is that it describes how the momentum and position of a quantum state changes with time in terms of a matter wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

The reason why we find a particle at one particular spot even though Schrödinger’s wave equation tells us it is spread out over an extremely wide region is because as was shown in the article Why is mass and energy quantized? Oct. 4, 2007 its energy must be packaged in terms of its resonant frequency.  However, according to the laws of classical wave mechanics the "concentration" of a wave’s energy is maximum at its peaks and valleys.  Therefore, one could only observe or "drain" the energy continued in its wave function at points corresponding to those peaks and valleys because those would be the only points in space where its energy would be “concentrated” enough to be redirected as a resonant system to the observing instrument.  However, as was shown in the article "Why is mass and energy quantized?" a particle is defined by the energy of its resonant structure.  Therefore, when one looks at a particular spot within the confines of a wave function one would either find a particle or nothing at all because the volumes between its peaks and valleys would not contain enough energy to allow its resonant structure to be redirect towards an observer. 

This is analogous to how the potential energy of water in a sink is release by allowing it to go down the drain.  If all we could observe is the water coming out of the drain we would have to assume that, it was concentrated in the region of space defined by the diameter of the drain.  However, in reality the water occupies a much larger region. 

Similarly, we assume that mass is quantized at a specific point in space represented by a particle because we can only observe where the energy of its extended environment is "drained" when observing its wave function.

Additionally the amplitude of its wave function or more precisely its square would define the probability that one would observe particle in a specific position within it confines because classical mechanics tells that defines where, with respect time how energy of a wave is concentrated.

This shows it is possible to interpret Schrödinger’s wave equation in terms of the laws of classical three-dimensional space if one extrapolates them to four *spatial* dimensions.

As mentioned earlier, this is not possible if one defines the universe in terms of four-dimensional space-time because time is observed to move in one direction forward and therefore it could not support bidirectional movement required for the propagation of a transverse wave consisting of three-dimensional space.  

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2010

(In a PDF format)

We have shown throughout "The Imagineer’s Chronicles” there would be many theoretical advantages to defining space in terms of the existence of a continuous non-quantized form of mass.

One of them is that it would allow one to define a Quantum field in terms of Classical Mechanics.

For the past 75 years physicist have used two radial different ways of defining the forces we observe in nature. 

The first, the concept of a field, was developed when physicists learned that they could simplify the calculations of the forces involved in planetary motion by assuming or imagining the existence of a continuous gravitational field.  They defined this field in such a way that if another planet were put at any point in that field the resulting force between any other planet would be exactly the Newtonian one.  This simplified the calculations of planetary motion because it allowed them to isolate and analyze the forces of one planet on another instead of trying to analyze the forces exerted on a planet by the others at the same time.

Originally, many thought this was just a trick to simplify calculations.

But Michael Faraday, while researching electromagnetism discovered that a continuous field has real physical properties and therefore was able to convince others that is was more the just a calculating device.

The second involves the Quantum Field Theory assumption that forces are a result of a distribution of discrete particles and the strength of a force can be measured by the density or quantity of field particles as any given point in space. 

However, these two definitions are mutually incompatible in that a continuous field cannot be made up of discreet or discontinuous particles.  

Nevertheless, Einstein may have given us a clue as to how to resolve this contradiction in the address "Aether and the theory of Relativity" he delivered on May 5th 1920 at the University of Leyden Germany where he indicated that The General Theory of Relativity predicts, "space is endowed with physical qualities".

"Recapitulating, we may say that according to the General Theory of Relativity space is endowed with physical qualities; in this sense, therefore, there exists an Aether.  According to the General Theory of Relativity space without Aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense.  But this Aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts, which may be tracked through time.  The idea of motion may not be applied to it."

Einstein’s statement that light or electromagnetism could not be propagated unless space is endowed with "physical qualities" suggests that an electromagnetic field must be composed of a form of mass that does not consist of parts, which can be tracked through time.  In other words, Einstein believed the propagation of light requires the existence of a continuous non-quantized form of mass.

However, the existence of a continuous non-quantized form of mass would appear to contradict the Quantum Field Theory assumption that electromagnetic fields are not continuous but are made up of discrete parts or particles.

But this may not be the case.

Classical mechanics tells us oscillations in a physical medium will form resonant systems or structures that when view from distance would appear to be made up of discrete units of with finite boundaries.

As was shown in the article "The Photon: a matter wave?" Oct. 1, 2007 one can derive the wave properties of a electromagnetic energy in terms of oscillations caused by a matter wave moving in a continuous non-quantized field of mass while deriving its particle properties in terms of a resonant system or structure formed by those same oscillations in that physical medium. 

However, this means the Quantum Field Theory assumption that a field is made up of discontinuous field particles is not inconsistent with its continuous properties observed by Faraday if one applies the concept of classical resonance to the physical medium Einstein assumed must exist to allow for the propagate of electromagnetic energy.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2010 Jeffrey O’Callaghan

(In a PDF format)