However finding a way of conceptually integrating them has proven to be extremely difficult for two reasons

The first is that it is impossible to derive a mechanism to explain and predict the continuous properties of four dimensional space-time in terms of quantum mechanics because something that is discontinuous cannot by definition be continuous. 

However one can derive the them in terms of the continuous properties of space-time because something that is continuous by definition can be divided into smaller units. 

Yet the other reason why it is so difficult is because Quantum mechanics defines its domain in terms of the spatial properties of probabilities while Einstein theories define it in terms of the continuous geometric properties of time.

This suggest we may be able to integrate them if we could find a way defining them in the same terms.   In other words redefining the space-time environment of Relativity in terms of the spatial properties of quantum mechanics or redefine the continuous properties of space-time in terms of the quantum properties of quantum mechanics.

However the only realistic option is to redefine the space-time environment of Relativity in terms of the spatial properties of quantum mechanics because as was mentioned earlier it is impossible to derive a mechanism to explain and predict the continuous properties of four dimensional space-time in terms of quantum mechanics because something that is discontinuous cannot by definition be continuous. 

Fortunately Einstein gave us the ability to do this when he used he used the velocity of light to define its geometric properties of space time because it allows one to convert a unit of time in his space-time universe to an equivalent unit of space in four *spatial* dimensions.  Additionally because the velocity of light is constant means it is possible to defined a one to one correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words by defining the geometric properties of a space-time universe in terms of mass/energy and 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.

However as mentioned earlier doing so would also allow one to define a physical mechanism responsible for creating a quantum of space-time in terms of the existence of four *spatial* dimensions.

For example the article “Why is energy/mass quantized?” Oct. 4, 2007 showed it is possible to explain the quantum properties of energy/mass by extrapolating the laws of classical resonance in a three-dimensional environment to a matter wave on the continuous “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 be meet by a matter wave in four spatial dimensions.

The existence of four *spatial* dimensions would give a matter wave the ability to oscillate spatially on a continuous “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.

The oscillations caused by such an event would serve as forcing function allowing a resonant system or “structure” to be established in four *spatial* dimensions.

Classical mechanics tells us the energy of a resonant system can only take on the discrete or quantized values associated with its resonant or a harmonic of its resonant frequency

Therefore the discrete or quantized energy of resonant systems in a continuous four dimensional environment would be responsible for the discrete quantized energy quantum mechanics associated with energy/mass.

However, it does not explain the mechanism responsible for quantizing the space containing energy/mass

In classical physics, a point on the 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, it could, similar to the surface of the paper oscillate “up” or “down” with respect to a fourth *spatial* dimension.

The confinement of the “upward” and “downward” oscillations of a three-dimension volume with respect to a fourth *spatial* dimension is responsible for the quantization of four-dimensional space because it would result in the formation of discrete or quantized volumes associated with the observed quantum properties of energy/mass.

In other words defining space in terms of four *spatial* dimensions allows one to conceptually the integrate the discontinuous quantum mechanical properties of energy/mass into the continuous field properties four-dimensional space in terms of a resonant system created by its wave properties.

Physicists should remember it is impossible to derive a mechanism to explain and predict the continuous properties of four dimensional space-time in terms of quantum mechanics because something that is discontinuous cannot by definition be continuous.  However one can understand as was shown above the quantum mechanical properties of space in terms of the continuous properties of space-time because something that is continuous by definition can be divided  into smaller units.

Later Jeff

Copyright Jeffrey O’Callaghan 2013

The post A quantum of space-time appeared first on Unifying Quantum and Relativistic Theories.

One is that it would allow for the integration of the quantum mechanical and wave properties of energy/mass by extrapolating the classical laws of a three-dimensional environment to a fourth *spatial* dimension.
In 1924 Louis de Broglie was the first to theorize that all particles had a transverse matter wave 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.

One of the difficulties he may have faced in this endeavor is that he assume 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 time or a space-time dimension can only move in one direction, forward.  Therefore, it could not support bidirectional movement required for the propagation of a transverse wave. 

However Einstein provided a solution to this problem when he use the equation E=mc^2 and the constant velocity of light to define the geometric properties of space-time because that gave a method of converting a unit of time associated with energy to its equivalent unit of space in four *spatial* dimensions.  Additionally because the velocity of light is constant he also defined a one to one quantitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

This may have allowed Louis de Broglie to define the quantum or particle properties, as was done in the article “Why is mass and energy quantized?” Oct. 4, 2007 of the transverse wave he theorized they were made up of in terms of a spatial displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly that article showed that the four conditions required for resonance to occur in a classical three dimensional 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 satisfied by a matter wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

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 space.

As was shown in that article these resonant systems in a continuous form of mass/energy are responsible for its quantum mechanical properties.

He then many have been able defined a causal interpretation of the Quantum Mechanical equation for a particles energy or E=hv (where “h” is Planck’s constant “v” is a particles frequency and “E” is the magnitude or its energy) based on the existence of these resonant systems

Classical mechanics tells us that the energy of a resonant system is quantized in terms of multiples of the harmonics of the fundamental frequency of its environment. 

Therefore, he could have interpreted the equation E=hv as defining the quantization of a particle’s energy in terms of the incremental energies “h” associated with the fundamental or harmonic of the resonant frequency of an environment consisting of four *spatial* dimensions..

However, this would have also allowed him to define a casual mechanism responsible for the uncertainty principal and the probability functions of Quantum Mechanics, again by extrapolating the three-dimensional laws of classical resonance to four *spatial* dimensions.

Because he may have realized the causality of the uncertainty in oneâ€s ability to define the exact position or momentum of a particle was due to the fact that the resonant system that the article “Why is mass and energy quantized?” derived was responsible for the quantum mechanical properties of energy/mass is distributed over the finite volume associated with the wavelength of its resonant system.  Therefore, one could only define its specific position or momentum in terms of an uncertainty related to where relative to its finite extended volume a measurement is made.

This indicates may have been able to derive a causal interpretation of the quantum mechanical properties of energy/mass in terms of classical properties of a matter wave if he had assumed there were a result of the geometric property of four *spatial* dimensions.

However, as mentioned earlier this cannot be done if one assumes space it made up of four-dimensional space-time because its geometry cannot support the transverse wave properties Louis de Broglie associated with particles.

Later Jeff

Copyright Jeffrey O’Callaghan 2010

The post The geometry of Quantum Mechanics appeared first on Unifying Quantum and Relativistic Theories.