We have shown throughout this blog there are many theoretical advantages to assuming space is composed of four *spatial* dimensions instead of four-dimensional space-time.
One of them is that it would allow one to explain the” reality” of the probabilities associated with quantum mechanical wave function in terms of the classical laws of three-dimensional space.
Quantum mechanics assumes that one cannot define a particle in terms of its exact position or momentum but only in terms of the probabilistic values associated with its wave function. This is in stark contrast to the Classical “Newtonian” assumption that one can assign precise values of future events based on the knowledge of their past.Â
For example In a quantum system Schrödinger wave equation plays the role of Newtonian laws in that it predicts the future position or momentum of a particle in terms of a probability distribution by assuming that it simultaneously exists everywhere in three-dimensional space.Â
This accentuates the fundamental difference between quantum and classical mechanics because the latter defines the reality of future events in terms of the effects of pervious events whereas quantum mechanics defines them based on the “non-classical” reality of the sum total of all possible events that can occur. Â
However, as mentioned earlier one can define the classical “reality” of quantum probabilities by extrapolating the laws of a classical three-dimension environment to one consisting of four *spatial* dimensions.
In the article “Why is energy/mass quantized?†Oct. 4, 2007 it was shown one can define why energy/mass is quantized by extrapolating the laws of classical wave mechanics in a three-dimensional environment to a matter wave on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
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 occur in one consisting of four spatial dimensions.
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 to oscillate with the frequency associated with the energy of that event.
The oscillations caused by such an event would serve as forcing function allowing a resonant system or “structure” to be established space.
Therefore, these oscillations in a “surface” of a three-dimensional space manifold would meet the requirements mentioned above for the formation of a resonant system or “structure” in four-dimensional space if one extrapolated them to that environment.Â
Classical mechanics tells us the energy of a resonant system can only take on the discrete or quantized values associated with it fundamental or a harmonic of its fundamental frequency.
Hence, these resonant systems in four *spatial* dimensions would be responsible for the discrete quantized energy associated with the quantum mechanical systems.
This cannot be done in four-dimensional space-time because time or a space-time dimension is only observed to move in one direction forward therefore it cannot support the bidirectional movement require to define classical resonance.
(In the article “The geometry of quarks” Mar. 15, 2009Â the internal structure of quarks, a fundament component of particles was derived in terms of a resonant interaction between a continuous energy/mass component of space and the geometry of four *spatial* dimensions.)
In an earlier article “Embedded dimensions” Oct. 4, 2007 it was shown that one can derive all forms of energy including that of a quantum system in terms of displacement in a *surface* of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
However assuming its energy is result of a displacement in four *spatial* dimension allows one to derive, as mentioned earlier the probability distribution associated with its wave function by extrapolating the laws of a three-dimensional environments to a fourth *spatial* dimension.
Classical mechanics tell us that because of the continuous properties of waves the energy the article “Why is energy/mass quantized?†associated with a quantum system would be distributed throughout the entire “surface” a three-dimensional space manifold with respect to a fourth *spatial* dimension.
This would be analogous to what happens when one vibrates a rod on a rubber diaphragm. The oscillations caused by the vibrations would be felt over its entire surface while their magnitudes would be greatest at the point of contact and decreases as one moves away from it.
However, this means if one extrapolates the mechanics of the rubber diaphragm to a “surface” of a three-dimensional space manifold one must assume the oscillations associated with each individual quantum system must simultaneously exists everywhere in three-dimensional space. This also means there would be a non-zero probability they could be found anywhere in our three-dimensional environment.
As mentioned earlier the article “Why is energy/mass quantized?†shown a quantum mechanical system is a result of a resonant structure formed on the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
Yet Classical Wave Mechanics tells us a resonance would most probably occur on the surface of the rubber sheet were the magnitude of the vibrations is greatest and would diminish as one move away from that point,
Similarly a quantum system would most probably be found were the magnitude of the vibrations in a “surface” of a three-dimensional space manifold is greatest and would diminish as one move away from that point,
However as mentioned earlier this is exactly what is predicted by Quantum mechanics in that one can define a particle’s exact position or momentum only in terms of the probabilistic values associated with vibrations of its wave function.
This shows how one can define the classical “reality” of the quantum mechanical probability functions by extrapolating the laws of classical mechanics to four *spatial* dimensions.
Later Jeff
Copyright Jeffrey O’Callaghan 2011