We have shown throughout "The Imagineer’s Chronicles" and its companion book "The Reality of the Fourth *Spatial* Dimension" there would be several theoretical advantages to defining the universe in term of four *spatial* dimensions instead of four-dimensional space-time.

For example, it would enable physicists to define a theoretical model that could explain and predict the uncertainty principal and probability functions of quantum mechanics in terms of the observable properties of a classical Newtonian world.

In 1924, Louis de Broglie theorized that 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 for the wave properties of particles 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.

However, his theories still served as the basis for the development of the general theory known today by the name of *wave mechanics*.

One of the difficulties he may have faced in doing this is that their wave properties are related to a classical property of space and not one of time or a space-time dimension.

In a classical world, a resonant system or "structure" will be formed when the spatial movements of a wave interact to reinforce themselves.

For example, the three-dimensional the bi-directional displacements on a two-dimensional surface of water will form a resonant system when the oscillations of the water interact to reinforce each other.

However, the transverse wave properties Louis de Broglie associated with a particle are difficult to explain in terms of four-dimensional space-time because time is only observed to move in one direction forward and therefore a universe of four-dimensional space-time could not support those bi-directional properties.

Einstein gave him the ability to do this when he used the constant velocity of light in the equation E=mc^2 to define geometric properties of a space time environment because it allows one to convert a unit of time in his four dimensional space-time universe to a unit of space in a one consisting of only four *spatial* dimensions. Additionally because the velocity of light is constant it is possible to defined a one to one correspondence between his space-time universe and one made up of four *spatial* dimensions.

But if he had assumed that space was composed of four *spatial* dimensions as is done in “The Imagineer’s Chronicles” he may have been able to define a probabilistic interpretation Born, Bohr and Heisenberg associated with particles in terms of a classical resonant "structure" formed by a transverse wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This is because as was shown in the article “Why is mass and energy quantized?” Oct. 4, 2007 one can extrapolate the properties of resonance in a three-dimensional environment to a transverse wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension which would have the wave properties Louis de Broglie theorized that all particles have.

Briefly it showed the four conditions required for resonance to occur in a classical 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 would be meet in one consisting of four.

The existence of four *spatial* dimensions would give a matter wave 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, the oscillations caused by such an event would serve as forcing function allowing a resonant system or "structure" to be established in four-dimensional space.

Classical mechanics tell us that the energy of a resonant system can only take on the discreet quantized energy associated with its fundamental or a harmonic of its fundamental frequency.

Therefore these resonant systems in four *spatial* dimensions would be responsible for the discrete quantized energy associated with quantum mechanical systems.

However, assuming the existed of four *spatial* dimensions instead of four-dimensional space-time would have also allowed him to derive a classical mechanism that could explain Born, Bohr, and Heisenberg probabilistic interpretation of quantum particles in terms of them.

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 or she 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 information left to for an instrument which was attempting to measure it. Therefore, if one was able to determine a particles exact momentum one could not say anything about its position.

This fact explains the complementary rule of quantum mechanics or the fact that cannot simultaneously measure the both the wave and particle properties of a quantum system

The reason we observe a particle as a point mass instead of an extended object such as a wave is because, as mentioned earlier the article "Why is mass and energy quantized?" showed its energy must be packaged in terms of its resonant system. Therefore, when we observe or "drain" the energy continued in its wave function, whether it be related to its position or momentum it will appear to come from a specific point in space similar how the 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 will have to pass through that opening. However, Classical Wave Mechanics tells us that as we reduce the error in our measurement by decreasing that predefine distance interference will cause its energy or momentum to be smeared our over a wider area. Similarly, to measure its momentum one must observe a portion the wavelength associated with its momentum. However, Classical wave mechanics tell us we must observe a larger portion of its wavelength to increase the accuracy of the measurement of its energy or momentum. But this means that the accuracy of its position will be reduced because the boundaries determining its position within the measurement field are greater.

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 to the other would be defined by the product of those factors.

Another way of looking at this would be to allow a photon or a particle to pass through a slit and observe where it struck a screen on the other side. One could get a more precise measurement of its position by narrowing the slit however classical wave mechanics tell us this will increase the interference of the wave properties associated with its resonant structure. However this will cause the interference pattern defining its momentum to become more spread out and therefore make it more difficult to accurately determine its value.

**Therefore, Classical wave mechanics tell us that we cannot accurately measure both the position and momentum of a particle if one assumes, as we have done in the article "Why is mass and energy quantized?" that the particle or quantum mechanical properties energy/mass are a result of a resonant system formed by a matter wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.**

Additionally, due to the time vary nature of wave motion the position or momentum of a particle will vary with respect to time within the confines of its resonant structure. Therefore, its position or momentum will be dependent on where in that time varying environment a measurement was made. Therefore ones must use probabilities to determine where particle is with the confines of its wave function because one cannot observe it before making a measurement. Therefore one must use probabilities to determine where in the confines of a wave function a particle is found.

This, as mentioned earlier would be very difficult to do if one defines the universe in term four-dimensional space-time because the spatial properties Louis de Broglie associated with the wave properties of particles is not compatible with a universe consisting space-time.

In other words, if he had assumed the existence of four *spatial* dimensions instead of four-dimensional space-time he may have been able to theoretically define the quantum mechanical properties of energy/mass in terms of a resonant system or "structure" formed by a wave which then would have allowed him to derive the uncertainties and probabilistic characteristics of their interactions found in Born, Bohr and Heisenberg theories.

Later Jeff

Copyright 2008 Jeffrey O’Callaghan

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