On the one hand quantum physics tells that the universe is made up of discrete units of energy/mass while relativistic physics tells us it is composed of a continuous field of space-time

Unfortunately these two ideas do not work well together because a continuous field by definition cannot be made up of discrete parts as is suggested by quantum mechanics.
Some have tried to merge them by defining what is has come to be called a relativistic quantum field theory.  It assumes that particles can be understood as the quanta of some quantum field which in essence elevates fields to the most fundamental objects in nature and that each type of field generates its own particular type of particle.

However, you cannot have it both ways because by definition a field is continuous and saying they can be understood in terms of some quantum field does not mean that you have connected them to the continuous properties of a relativistic space-time field or any field for that matter.  All it does is elevate its size to that of the entire universe because by definition a field is continuous throughout its entire domain.  Therefore if the fundamental component of the universe is a quantum field as quantum field theory suggests then it could only contain one quantum entity because if it contained more the continuity of the field would be broken.  In words saying one can understand the continuous properties of a field in terms of a quantum field is like saying that one can understand why a circle is round is because it is a circle. 

Another reason why it is so difficult to conceptually to integrate Quantum theory with the field properties of Einstein’s theories is because it defines space in terms of a field consisting of time or a space-time dimension while Quantum theory defines itself in terms of its spatial properties of energy/mass.  

For example Schrödinger’s wave equation only defines the probability of a particle will be located in a given volume of space without giving a reference to time while Einstein defined the geometric properties of a space-time universe in terms of a dynamic balance between mass and energy defined by the equation E=mc^2.

Yet one can overcome the difficultly by redefining the field properties of space-time Einstein associated with energy/mass to its spatial properties Quantum field theory associates with it. 

Einstein gave us the ability to do this when used the equation E=mc^2 and the velocity of light to define the geometric properties of space-time because it allows one to convert a unit of displacement he associated with energy in a four dimensional space-time universe to an equivalent spatial displacement it would  create in 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.

In other words because he defined the geometric relationship between energy and mass in terms of the constant velocity of light means that one can quantitatively and qualitatively define a one to one between the properties of energy in a space-time universe to the physical properties of space four *spatial* dimensions.

One of the theoretical advantages to assuming that the universe is made up of four *spatial* dimensions instead of four dimensional space-time is that it allows one to derive the quantum mechanical properties of energy/mass in terms of the field properties of four *spatial* dimensions instead of defining the field properties of space in terms of its quantum mechanical properties as is done in quantum field theory.

The field properties of four *spatial* dimension was developed in the article “Electromagnetism in four *spatial* dimensions” Sept 27, 2007 where it was shown the forces associated with an electromagnetic field can be explained and predicted in terms of matter wave on a continuous field consisting of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly it showed that one can derive its properties by extrapolating the laws of Classical Wave Mechanics to a field consisting of fourth *spatial* dimensions.

A wave on the two-dimensional surface of water causes a point on that surface to be become displaced or rise above or below the equilibrium point that existed before the wave was present.  A force will be developed by the differential displacement of the surfaces, which will result in the elevated and depressed portions of the water moving towards or become “attracted” to each other and the surface of the water.

Similarly a matter wave on the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension would cause a point on that “surface” to become displaced or rise above and below the equilibrium point that existed before the wave was present.

Therefore, classical wave mechanics, if extrapolated  to four *spatial* dimensions tells us the force developed by the differential displacements caused by a matter wave moving on a “surface” of three-dimensional space with respect to a fourth *spatial* dimension will result in its elevated and depressed portions moving towards or become “attracted” to each other.

This defines the causality of the attractive forces of unlike charges associated with the electromagnetic wave component of a photon in terms of a force developed by a differential displacement of a point on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, it also provides a classical mechanism for understanding why similar charges repel each other because observations of water show that there is a direct relationship between the magnitudes of a displacement in its surface to the magnitude of the force resisting that displacement.

Similarly the magnitude of a displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by two similar charges will be greater than that caused by a single one.  Therefore, similar charges will repel each other because the magnitude of the force resisting the displacement will be greater for two charges than it would be for a single charge.

One can define the causality of electrical component of electromagnetic radiation in terms of the energy associated with its “peaks” and “troughs” that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement.

However, Classical Mechanics tells us a horizontal force will be developed by that perpendicular or vertical displacement which will always be 90 degrees out of phase with it.  This force is called magnetism.

This is analogous to how the vertical force pushing up of on mountain also generates a horizontal force, which pulls matter horizontally towards the apex of that displacement.

This shows how one can explain and predict the electrical and magnetic field properties of an electromagnetic wave by extrapolate the laws of classical wave mechanics in a three dimensional environment to a matter wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, as was shown in the article “The Photon: a matter wave?” Oct. 1, 2007 the quantum field properties of four *spatial* dimensions can be explained and predicted by extrapolating the resonant properties of field in a three-dimensional environment to one consisting of four *spatial* dimension.

There are 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.

The existence of four *spatial* dimensions would give the continuous surface or field of three-dimensional space manifold (the substance) the ability to oscillate spatially with respect to a fourth *spatial* dimension 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 four *spatial* dimensions, 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 field energies associated with relativistic quantum field theories.

This shows that it is possible to logically and consistently explain and predict the quantum mechanical field properties energy/mass in a microscopic environment by assuming that space is composed of four *spatial* dimensions instead of four dimensional space-time.

However it also shows it is more logical and consistent with observations to assume that our universe is fundamentally composed of fields not quanta of energy/mass as is assumed by quantum field theory

It should be remember Einsteinâ€s genius allows us to choose whether to define the energy of all systems in either a space-time environment or one consisting of four *spatial* dimension when he defined it and the geometry of space-time in terms of the constant velocity of light. This interchangeability broadens the environment encompassed by his theories making them applicable to both the field as well as the quantum properties of our universe thereby giving us a new perspective on its causality.

Latter Jeff

Copyright 2013 Jeffrey O’Callaghan

The post Particles or fields you cannot have it both ways 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.

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 this blog 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 waveâ€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

The post Why four spatial dimensions? appeared first on Unifying Quantum and Relativistic Theories.