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Can infinities be part of our world.

Something that is infinite or the quality of having no limits or end cannot exist or be a part of the physically observable environment we live in primarily because it is finite.
 
Some might disagree by pointing out that we cannot know the full extent of our universe because the speed of light puts limits on our ability to observe parts beyond a specific point.  However some could also argue that anything beyond that point is not in our observable universe and because of that it cannot be a part of it.

Infinity: The Science of Endless

Yet even though Einstein’s theories does not mathematical rule out the possibility of an infinite universe it does not predict that one exists.

However the same cannot be said of Quantum Mechanics which mathematical defines mass, energy and forces in terms of a one dimensional point.  Infinites arise because the forces and energies associated with the integrals which define them become larger as they approach each other reaching infinity when they come in contact. 

The difference between these quantum mechanical infinites and relativistic ones is that they occur with the limits of our observable universe.  In other words it predicts existence of masses, forces, and energies that are infinite within its finite boundaries.

Some might think this indicates the basic concepts of quantum mechanics that define our observable environment in terms of the mathematical properties of a one dimensional point is incorrect because most physicists and mathematicians would agree that the infinite entity cannot exist in a finite environment.

However its proponents disagree and have devised a clever method called renormalization which alters the mathematical relationships between the parameters in the theory to make these infinites disappear.

Granted even though one may be able to use renormalization to alter the mathematical relationships between point particles to eliminate infinites they cannot change the fact the point particle responsible for those infinites still exists before those alterations take place.  In other words it assumes they exist before renormalization takes place because if they did not there would be no need for renormalization.  Therefore even though the process of renormalization solves the mathematical problem of infinities it does nothing to solve the conceptual one that exist within the framework of quantum mechanics because it relies on the existence of point particles which as mentioned earlier are responsible for the infinites. `

Why then are we still using it to explain or predict that reality?

The most probable answer is because it predicts with amazing precision the results of every experiment involving the quantum world that has ever been devised to test it: so much so that many are willing to overlook the obvious fact that as was just mentioned the conceptual arguments use to make those predictions have a fatal flaw.

However we are not going to concern ourselves with resurrecting the conceptual content of quantum mechanics as has been the focus of the past three quarters of a century but instead will define another theory that can explain the behavior of energy/mass in terms of the properties of our observable environment in a way that eliminates the need for any "adhco" procedures such as renormalization to make it consistent with that behavior.

To do this one must be able to, in a logical and consistent manner using only the physical laws of our observable environment explain the existence of the four basic components of a quantum world: the fact that energy/mass is quantized, Planck’s constant, Heisenberg’s Uncertainty Principle and the reason one can use probabilities to define a particles position.

For example in the article "Why is energy/mass quantized?" Oct. 4, 2007 it was shown it is possible to explain and predict the quantum mechanical properties of energy/mass associated with Schrödinger’s wave function by extrapolating the laws of classical resonance 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.

(Note: Einstein has already gave us a detailed mathematical description of this environment when he used the constant velocity of light to define the geometric properties of space-time 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 mathematically derive a one to one correspondence between his space-time universe and one made up of only four *spatial* dimensions.)

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 an environment of four *spatial* dimensions.

(Louis de Broglie was the first to theorize that all particles are made up of matter waves.  His theories were later confirmed by the discovery of electron diffraction by crystals in 1927 by Davisson and Germer.)

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.

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

This shows how one can conceptually derive the quantum mechanical properties energy/mass in terms of wave properties of particles observed by Davisson and Germer by assuming that they are a result of resonant properties of four *spatial* dimensions.

In other words if one assumes as is done here that its mathematical properties of Schrödinger’s wave function are representative of wave moving on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension one can form a physical image of why energy/mass is quantized in terms of the properties of our observable environment.

However it also gives one the ability to define the physical boundaries of a particle and its energy in terms of the observable properties of our environment

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 what defines the geometric boundaries or the "box" containing the wave component of Schrödinger’s wave function the article "Why is energy/mass quantized?" Oct. 4, 2007 associated with a particle. 

As mentioned earlier infinites arise in Quantum Mechanics when one applies the concept of mathematical one dimensional point to define mass, energy and forces results their integrals to become increasing larger as they approach each other reaching infinity when they come in contact. 

However the above theoretical concepts provides a solution because it shows that a particle’s energy is not confined to a one dimension point but instead exists in an extended spatial volume associated with its resonant structure.

Yet if true one must be able derive the physical meaning the other fundamental concepts of quantum mechanics like Planck’s constant or 6.626068 × 10-34 (kg*m2/s), Heisenberg’s Uncertainty Principle and the probabilities associated with Schrödinger’s wave function by extrapolating the laws of classical physics in a three-dimensional environment to a fourth *spatial* dimension.

Planck’s constant is one of fundamental components of Quantum Physics and along with Heisenberg’s Uncertainty Principle it defines the uncertainty in the ability to measure more than one quantum variable at a time.  For example attempting to measure an elementary particle’s position (▲x) to the highest degree of accuracy leads to an increasing uncertainty in being able to measure the particle’s momentum (▲p) to an equally high degree of accuracy.  Heisenberg’s Principle is typically written mathematically as ▲x▲p  ³ h / 2  where h represents Planck constant

As mentioned earlier the resonant wave that corresponds to the quantum mechanical wave function defined in the article "Why is energy/mass quantized?" Oct. 4, 2007 predicts that a particle will most likely be found in the quantum mechanical "box" whose dimensions would be defined by that resonant wave.  However quantum mechanics treats particles as a one dimensional points and because it could be anywhere in it there would be an inherent uncertainty involved in determining the exact position of a particle in that "box".

For examine the formula give above ( ▲x▲p  ³ h / 2 ) tells us that uncertainty of measuring the exact position of the point in that "box" defined by its wavefunction would be equal to ▲x▲p  ³ h / 2.   However because we are only interested in determining its exact position we can eliminate all references to its momentum.

However if we eliminate the momentum component from the uncertainty in a particle position become 6.626068 × 10-34 meters or Planck’s constant.

As mentioned earlier the uncertainty involved in determining the exact position of a particle is because it is impossible to determine were in the "box" defined earlier the quantum mechanical point representing that particle is located.  However as mentioned earlier Planck’s constant tells us that one cannot determine the position of a particle to an accuracy greater that 6.626068 × 10-34.  This suggest that Planck constant 6.626068 × 10-34 defines the physical parameters or dimensions of that "box" because it defines the parameters of where in a given volume of space a quantum particle can be found.

In other words it defines a physical interpretation of Planck’s constant or 6.626068 × 10-34 (kg*m2/s), and Heisenberg’s Uncertainty Principle by extrapolating the observable properties and laws of our three-dimensional environment to a fourth *spatial* dimension.

However it also gives one the ability to connect the probabilities associated with Schrödinger’s wave function to the observable reality of our three-dimensional environment.

As was mentioned one can conceptually derive the quantum mechanical properties of his function in terms of physical properties of a mater wave observed by Davisson and Germer by assuming that they are a result of resonant properties of four *spatial* dimensions.

Classical mechanics tell us that due to the continuous properties of waves the energy the article "Why is energy/mass quantized?" Oct. 4, 2007 associated with a quantum system would be distributed throughout the entire "surface" a three-dimensional space manifold with respect to a fourth *spatial* dimension.

For example Classical mechanics tells us that the energy of a vibrating or oscillating ball on a rubber diaphragm would be disturbed over its entire surface while the magnitude of those vibrations would decrease as one move away from the focal point of the oscillations. 

Similarly if the assumption that quantum properties of energy/mass are a result of vibrations or oscillations in a "surface" of three-dimensional space is correct then classical mechanics tell us that those oscillations would be distributed over the entire "surface" three-dimensional space while the magnitude of those vibrations would be greatest at the focal point of the oscillations and decreases as one moves away from it.

As mentioned earlier the article "Why is energy/mass quantized?" Oct. 4, 2007 shown a quantum object 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 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 particle 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.

This shows how one can eliminate infinities from our understanding of the quantum properties of energy/mass while at the same time allow one to connect those properties to the observable realities of our environment.

Later Jeff

Copyright Jeffrey O’Callaghan 2016

   

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