We think so.

Many including Albert Einstein and Erin Schrödinger, had difficulty accepting the “reality” of quantum mechanics because many of its concepts appear to contradict those of our observable universe.

For example in a quantum system Schrödinger’s wave equation defines the field properties of its environment and predicts the future distribution of a particle’s position only in terms of the abstract properties of probabilities.

However many including Einstein and Schrödinger define reality in terms of what they see or touch.
For example, Einstein used the observable “reality” of the interactions of electromagnetic energy with a photoelectric material to derive the quantum mechanical properties of energy/mass while using the observable properties of light in our three-dimensional environment to define his space-time universe.

In other words his conclusion that electromagnetic energy is quantized was based on the physical “reality” of the environment surrounding the photoelectric material and how electromagnetic energy interacted with it, not on the abstract probabilities associated with quantum fields.

However the abstract properties of probabilities share a common characteristic with Einstein space-time universe in that time or a space-time dimension have never be seen or touched and therefore they like the probability functions of quantum field theory are, by definition abstract quantities.

Fortunately they also have a common element in, as mentioned earlier the physically observable non-abstract properties of the *spatial* dimensions because the probabilities associated with Schrödinger’s wave equation are expressed in terms of the spatial properties of position.

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 unit of spatial displacement 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, mass, space and time 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.

The fact that one can use the Einstein’s equations to qualitatively and qualitatively derive the spatial properties of energy in a space-time universe 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. 

One of the theoretical advantages of a modeling the existence of energy/mass on four *spatial* dimensions instead of four dimension space-time is it allows one to derive the “reality” of a quantum fields in terms of the observable non-abstract properties of our three-dimensional environment.

The physical “reality” of the field properties energy/mass in 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 field consisting of four *spatial* dimensions.

Briefly it showed that one can derive its field properties by extrapolating the observable non-abstract properties of a three-dimensional environment to a fourth *spatial* dimension.

For example 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 observations  of our three dimensional “reality”, 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 non-abstract mechanism for understanding why similar charges repel each other because observations of wave on the surface of water tell us 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, observations of our three dimensional environment tell 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 continuous field properties of electromagnetism by extrapolating the observable non-abstract properties of our 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* dimension can also be derived by extrapolating the observable non-abstract resonant properties of a three-dimensional environment to one consisting of four *spatial* dimension.

There are 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.

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 quantum and electromagnetic field theories

This shows how one can define the “reality” of the continuous field associated with Schrödinger’s wave equation and a physical mechanism responsible for the creation of particles in that field in terms of the observable non-abstract “reality” of our three-dimensional environment.

Latter Jeff

Copyright 2013 Jeffrey O’Callaghan

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

One is that it would allow for the resolution of the conflict between the Newtonian assumption that space and time is continuous with the quantum mechanical assumption that it is composed of discrete quantized units by extrapolating observations of a three-dimensional environment to a fourth *spatial* dimension.
As Lee Smolim point out in his book “Three Roads to Quantum Gravity” these two disciplines define them differently.

“For example Newtonian physics assumes that space and time are continuous and can be arbitrary divided into smaller and smaller units.  In other words there is no smallest possible unit of space or time.

However, quantum mechanics assumes that they come in discrete irreducible quantized bits.  In other words unlike the Newtonian concept that length and time are infinitely divisible quantum theory assumes there is a finite limit to our ability to observe smaller and smaller segments of them.”

Yet one can qualitatively define the mechanism responsible for the fundamental quantum mechanical unit of length and time in terms of the Newtonian assumption that space and time are continuous and the quantum mechanical concept of the smallest measurable unit of space and time.

The smallest possible length according to quantum theory is determined defined by Planck’s constant (denoted h).  Using it along with the speed of light in a vacuum one can determine Planck’s length or the smallest length that is measurable as being equal to 1.616252(81)-35 meters. 

Using that knowledge it is possible to calculate Planck’s time or the smallest measurable unit of time as that required for light to travel, in a vacuum, a distance of 1 Planck’s length and because Planck’s length is the smallest measurable length it is the smallest measurable unit of time.

(Note those who are interested in a quantitative derivation of these units based on quantum mechanical interpretation of space and time should view the video to the right.)

In quantum mechanics Planck’s time is the limiting factor in one’s ability distinguish events because the fact that it cannot be divided in smaller units means that one could not determine if one event occur before another if they both took place within that interval.

Yet, one can derive the physicality of Planck’s or Quantum Time by extrapolating the laws of classical Newtonian space and time to a fourth *spatial* dimension.

However before we begin we must first derive Planck’s length or the smallest measurable length of in terms of four *spatial* dimensions.

In the article “Why is energy/mass quantized?” Oct. 4, 2007 it was shown that one can derive its quantum mechanical properties by extrapolating the laws of classical resonance a three-dimensional environment to a matter wave moving on a 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 Newtonian 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 the “surface” of a three-dimensional manifold the ability to oscillate with respect to it 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 a continuous “surface” of a three-dimensional space manifold, 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 physically responsible for Planck’s constant and the incremental or discreet energy associated with it and quantum mechanical systems.

However it also defines Planck’s length in terms of the length of the fundament quanta of energy/mass because it defines the smallest unit of space that exists in terms of a resonant system in four-dimensional space.  Therefore it also defines the physicality of Planck’s time as being the time required for light to travel, in a vacuum, a distance of 1 Planck length or as was just shown the wavelength of the fundamental resonant frequency of four-dimensional space.

This shows how one can resolve the conflict between the Newtonian assumption that time is continuous with the quantum mechanical assumption that it is composed of discrete quantized units because one can be derived in terms of a subset of the other

Later Jeff

Copyright Jeffrey O’Callaghan 2011

The post Quantum time as a subset of Newtonian space appeared first on Unifying Quantum and Relativistic Theories.