Quantum mechanics defines our observable environment only in terms of the probabilistic values associated with Schrödinger’s wave equation.

However it is extremely difficult to define a set of statements which explains how those probabilities are physically connected to it even though it has held up to rigorous and thorough experimental testing.

This may be the reason most physicists consider quantum mechanics only in terms of its mathematical formalization instead trying to understand the meaning of it in terms of the space-time environment we occupy. 

 

Schrödinger Equation and Material Waves

For example in 1924 Louis de Broglie was the first to realize all particles are physically composed of a matter wave as the discovery of electron diffraction by crystals in 1927 by Davisson and Germer) verified.  However in his paper, Theory of the double solution he unsuccessfully attempted to define a physical interpretation of Schrödinger equation in classical terms of space and time.

As is pointed at his biography on the nobleprize.org web site in "1951, he together with some of his younger colleagues made another attempt, one which he abandoned in the face of the almost universal adherence of physicists to the purely probabilistic mathematical interpretation of, Bohr, and Heisenberg."

However the fact that no has been able to physically connect those probabilities to our environment does not change the fact that there must be one because if there wasn’t they could not interact with it to create the physicality of observable world upon which those probabilities are based.

As mentioned earlier Louis de Broglie and his colleagues tried unsuccessfully to find a physical interpretation of Schrödinger equation in classical terms of space and time.

However the reason for their failure may be due to the fact that it is related to the spatial not time dependent properties of the wave function.

If so one may be able to establish the connection by looking at it in terms of its spatial properties instead of the space-time one Louis de Broglie and his colleagues used.

Einstein gave us the ability to do this defined the geometric properties of space-time in terms of the constant velocity of light and a dynamic balance between mass and energy because that  provided a method of converting a unit of time in a space-time environment of unit of space in four *spatial* dimensions.  Additionally because the velocity of light is constant he also defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

The fact that one can use Einstein’s equations to qualitatively and quantitatively redefine the curvature in space-time he associated with energy in terms of four *spatial* dimensions is one bases for assuming as was done in the article “Defining energy?” Nov 27, 2007 that all forms of energy 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 have allowed Louis de Broglie to physically connect the probabilities associated Schrödinger equation to the quantum properties of a matter wave in terms of a physical or spatial displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension as was done in the article  "Why is energy/mass quantized?" Oct. 4, 2007. 

Briefly that article showed that one can do this by assuming they are caused by the formation of a resonant system on a "surface" of a three-dimensional space manifold with respect to fourth "spatial" dimension.  This is because it showed the four conditions required for resonance to occur in a 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 occur in one made up 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* 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.

However, the oscillations caused by such an event would serve as forcing function allowing a resonant system or "structure" to be established on a surface of a three-dimensional space manifold.

Yet the classical laws of three-dimensional space tell us the energy of resonant systems can only take on the discontinuous or discreet energies associated with their fundamental or harmonic of their fundamental frequency.

However, these are the similar to the quantum mechanical properties of energy/mass in that they can only take on the discontinuous or discreet energies associated with the formula E=hv where "E" equals the energy of a particle "h" equal Planck’s constant "v" equals the frequency of its wave component.

In other words Louis de Broglie would have been able to physicality connect the the quantum mechanical properties of his particle waves to Schrödinger equation in terms of the discrete incremental energies associated with a resonant system in four *spatial* dimensions if he had assume space was composed of it instead of four dimensional space-time.

Yet it also would have allowed him to define the physical boundaries of a quantum system in terms of the geometric properties of four *spatial* dimensions.

For example in classical physics, a point on the two-dimensional surface of a piece 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 spatial boundaries associated with a particle in the article "Why is energy/mass quantized?" Oct. 4, 2007

As mentioned earlier in the article “Defining energy?” Nov 27, 2007 showed all forms of energy 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.

However assuming the energy associated with Louis de Broglie particle wave is result of a displacement in four *spatial* dimension instead of four dimensional space-time as was done earlier would allows one to connect the probabilities associated with Schrödinger equation to the physicality of our observable environment we all live in.

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 decease 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?” shown a quantum particle 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 physically connect the probabilities associated Schrödinger wave equation to our observable environment by redefining it in terms of four *spatial* dimensions.

It should be remember Einstein’s genius allows us to choose to define a quantum system 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 probabilistic properties of a quantum environment and how they physically connected to our observable universe.

Later Jeff

Copyright Jeffrey O’Callaghan 2015

 

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Quantum entanglement is the name that describes the way that particles can share information and interact with each other regardless of how far apart they are.

For example an electron in certain atoms will spontaneously decay after being excited by emitting pairs of polarized photons such that one is aligned horizontally the other vertically.  According to quantum mechanics these photons are entangled and act of observing one instantly affects the other no matter how far they are apart.

Quantum Entanglement

This instantaneous communication between the entangled photons is at the heart of quantum entanglement.  This is the "spooky action at a distance" Einstein believed was theoretically implausible because according to Relativistic Theories information cannot be propagated instantaneously but only at the speed of light.

To demonstrate this 1935, Einstein co-authored a paper with Podolsky and Rosen which was intended to show that Quantum Mechanics could not be a complete theory of nature.  The first thing to notice is that Einstein was not trying to disprove Quantum Mechanics in any way.  In fact, he was well aware of its power to predict the outcomes of various experiments.  What he was trying to show was that there must be a "hidden variable" that would allow Quantum Mechanics to become a complete theory of nature

The argument begins by assuming that there are two systems, A and B (which might be two free particles), whose wave functions are known.  Then, if A and B interact for a short period of time, one can determine the wave function which results after this interaction via the Schrödinger equation or some other Quantum Mechanical equation of state.  Now, let us assume that A and B move far apart, so far apart that they can no longer interact in any fashion.  In other words, A and B have moved outside of each other’s light cones and therefore are spacelike separated.

With this situation in mind, Einstein asked the question: what happens if one makes a measurement on system A?  Say, for example, one measures the momentum value for system A.  Then, using the conservation of momentum and our knowledge of the system before the interaction, one can infer the momentum of system B.  Thus, by making a momentum measurement of A, one can also measure the momentum of B.  Recall now that A and B are "spacelike" separated, and thus they cannot communicate in any way.  This separation means that B must have had the inferred value of momentum not only in the instant after one makes a measurement at A, but also in the few moments before the measurement was made.  If, on the other hand, it were the case that the measurement at A had somehow caused B to enter into a particular momentum state, then there would need to be a way for A to signal B and tell it that a measurement took place.  However, the two systems cannot communicate in any way!

If one examines the wave function at the moment just before the measurement at A is made, one finds that there is no certainty as to the momentum of B because the combined system is in a superposition of multiple momentum eigenstates of A and B.  So, even though system B must be in a definite state before the measurement at A takes place, the wave function description of this system cannot tell us what that momentum is!  Therefore, since system B has a definite momentum and since Quantum Mechanics cannot predict this momentum, Quantum Mechanics must be incomplete.

In response to Einstein’s argument about incompleteness of Quantum Mechanics, John Bell derived a mathematical formula that quantified what you would get if you made measurements of the superposition of the multiple momentum eigenstates of two particles.  If local realism was correct, the correlation between measurements made on one of the pair and those made on its partner could not exceed a certain amount, because of each particle’s limited influence.

In other words he showed there must exist inequities in the measurements made on pairs of particles that cannot be violated in any world that included both their physical reality and their separability because of the limited influence they can have on each other when they are "spacelike" separated.

When Bell published his theorem in1964 the technology to verify or reject it did not exist.  However in the early 1980s, Allen Aspect performed an experiment with polarized photons that showed that the inequities it contained were violated.

Many believed this provided experimental verification of the concept of Quantum entanglement.  Additionally it meant that science has to accept that either the reality of our physical world or the concept of separability does not exist.

However Einstein himself predicted the entanglement of particles that are moving at the velocity of light no matter how far apart they are in his Special Theory of Relativity because he showed us that separability or the distance between two points is dependent on the velocity of the observer with respect to what is being observed.

For example his theory tells the distance between the two objects A and B would be defined by their relative speed with respect to an observer

Specifically he told us that it would be defined by

 

However this tell us that the distance between two photons or any particle moving at the speed of light with respect to all observes would be zero no matter how far apart any observer might perceive them.to be because according to the concepts of relativity one could view the photons as being stationary and the observers as moving at the velocity of light.  This would be true even if the photons were moving in oppose directions because of the fact that their velocity is squared

Therefore according to Einstein’s theory all photons which are traveling at the speed of light are entangled with all other photon no matter how far apart or "spacelike" separated they may appear to be to all observer who are not traveling at the speed of light.

In other words inequities in the measurements made on pairs of photons should  be violated in a world containing the physical reality of Einstein’s theory and separability because they will influence each other even when they are "spacelike" separated when viewed from the reference frame other than a photon which is traveling at the speed of light. 

This tells us that the hidden variable that would allow Quantum Mechanics to become a complete theory of nature is the relativity properties of motion.

One method for determining if this is the reason why Allen Aspect observed that polarized photons violated bells inequities would be to see if they are also violated by particles that were traveling slower that the speed of light because they would according to the Theory of Relativity could be "spacelike" separated.

In others words if it was observed that particles which were not traveling at the speed of light did not violate Bell’s inequity then it would support Einstein perception of reality and provide a physical mechanism in terms of the existence of space-time for one of the most puzzling aspects of quantum mechanics; that of quantum entanglement.

However if it is found that bell’s inequity is violated by particles moving slower than the speed of light than Einstein’s perception of reality would be invalidated because it demands that things which are "spacelike" separated can only have a limited influence one each other

Copyright 2015 Jeffrey O’Callaghan

 

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