The quantization of energy/mass in four *spatial* dimensions.

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Neither Quantum mechanics nor Einstein theories can give us the answer.

One reason may be because quantum mechanics focuses on the mathematical properties of energy while Einstein forced on the physical properties of mass.

For example Quantum mechanics defines the position and momentum purely in terms of a mathematical construct know as the wave equation while Einstein’s theories were developed by understanding the physical interaction of mass with a space-time dimension. 

However one may find an answer to why they are quantized if we can combine them into one logically consistent theoretical model.

One of the problems in doing this is the mathematics of quantum mechanics define a particle terms of the spatial properties of position and energy while Einstein’s theories define them in terms of time or a space-time dimension.

Yet Einstein gave us a way to eliminate this problem when qualitatively and quantitatively derived the geometric properties of his space-time environment in terms of the constant velocity of light. This is because it allows one to redefine a unit of time he associated with energy in his space-time universe to unit of space in a universe consisting of only four *spatial* dimensions.

In other words by defining the geometric properties of a space-time universe in terms of the constant velocity of light he provided a qualitative and quantitative means of redefining his space-time universe in terms of the geometry of four *spatial* dimensions.
The fact that one can use Einstein’s equations to qualitatively and quantitatively to redefine the displacement in space-time he associated with the forces associated with mass in terms of four *spatial* dimensions is one bases for assuming as was done in the article “Defining energy?” Nov 27, 2007 that all forces can be derived in terms of a spatial displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This would allow one to combine the spatial properties quantum mechanics associates with particles with the space-time properties Einstein associated with their mass.

For example In an earlier article “The Photon: a matter wave?” Sept 27, 2007 it was shown the electromagnetic wave characteristics of a photon could be explained and predicted in terms of a matter wave in a continuous of field of energy/mass on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, it can be shown this matter wave would generate a classically resonating system or “structure” in space which is responsible for its quantum properties. 

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.

The existence of four *spatial* dimensions would give a continuous non-quantized field of energy/mass (the substance) 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.

Therefore, these oscillations in a continuous non-quantized field of energy/mass, would meet the requirements mentioned above for the formation of a resonant system or “structure” in space. 

Observations of a three-dimensional environment show the energy associated with resonant system can only take on the incremental or discreet values associated with a fundamental or a harmonic of the fundamental frequency of its environment.

Similarly the energy associated with resonant systems in four *spatial* dimensions could only take on the incremental or discreet values associated a fundamental or a harmonic of the fundamental frequency of its environment.

These resonant systems in four *spatial* dimensions are responsible for the incremental or discreet energy associated with quantum mechanical systems.

(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 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 on an incremental basis. 

Additionally the energy in a classically resonating system is, as mentioned earlier 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 energy/mass and when energy is either added or removed from it, its characteristics are changed incrementally.

However this also allows one to understand why some particles are stable while others are not by extrapolating the properties of classical resonance to a fourth *spatial* dimensions.

To be stable a classically resonating system must have the energy associated with the discrete values of its fundamental frequency or an integral multiple of its resonant environment.  If it does not it will either lose gain energy from until it is oscillating at that frequency.

Summarily a stable particle would be one whose three-dimensional volume is oscillating with respect to a fourth *spatial* dimension at the fundamental or harmonic of the resonant frequency associated with that volume. 

An unstable particle would be one whose three-dimensional volume is oscillating at a frequency other than the one associated with the energy of its resonant environment.  Similar to resonant systems in a classical environment, these particles will decay by losing or gaining energy from their environment until they have the stable structure associated with either the fundamental or harmonic of the resonant frequency associated with their volume.

However it also tells us in terms of the physical properties four dimensional space-time or four *spatial* dimensions why an electron cannot fall into the nucleus is because, as was shown above all energy is contained in four dimensional resonant systems. In other words the energy released by an electron “falling” into it would have to manifest itself in terms of a resonate system. Since the fundamental or lowest frequency available for a stable resonate system in either four dimensional space-time or four spatial dimension corresponds to the energy of an electron it becomes one of the fundamental energy units of the universe and could not “fall’ in the nucleus

It also allows one to define a mechanism, in terms of classical mechanics for the uncertainty principal of Quantum Mechanics which 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.

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 because those points are the only places where its energy or “position” is stationary with respect to a fourth *spatial* dimension.  Whereas it’s 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 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.

The reason we observe a particle as a point mass instead of an extended object or wave is because, as mentioned earlier its energy is propagated in quantized resonant systems. 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.

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.

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 “m”kg / s 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 above the change or uncertainty of one with respect would be defined by the product of those factors or m^2 kg / s.

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, when extrapolated to a fourth *spatial* dimension tell us the more precisely the momentum of a particle is known, the less precisely its position can be known while the more precisely its position is known, the less precisely its momentum can be determined if one assumes, as we have done here that the 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

This shows how one can answer the question as to why mass and energy are quantized by combining the spatial properties of mass Einstein relativistic theories with the position and momentum component of quantum mechanics.

It should be remember Einstein’s genius allows us to choose to define all systems in either a space-time environment or one consisting of four *spatial* dimension when he defined the geometry of space-time in terms of the constant velocity of light. This interchangeability broadens the environment encompassed by his theories thereby giving us a new perspective on the physical properties of mass and energy.

Later Jeff

Copyright Jeffrey O’Callaghan 2007

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