Introduction

Morphing fields of possibility

Psy~trance~formations

The Quantum Seeds of Revolution and Resonance

           Lets start from the beginning.  The era of quantum theory kicked off in 1900 with a discovery made by Max Plank.  Plank was studying the so-called black body radiation problem.  Classical physics predicted that black bodies should glow bright blue, a stark contradiction to the experience of steelworkers everywhere.  In order to simplify the mathematical calculations, Plank restricted the vibration of the matter particles according to the following rule: E = nhf, where E is the particle’s energy, n is any integer, f is the frequency of vibration, and h is a constant chosen by Plank.  This rule restricts the particles to energies that are certain multiples of their vibration frequency.  Plank’s intention was to let h approach zero; however, this only predicted the same blue radiation as before.  By chance, Plank discovered that if he set h to a certain value, the calculations matched the experimental results exactly.  This special value for h is now known as Plank’s constant and is also called the “quantum of action.”  Plank showed that energy can only be emitted and absorbed in tiny packets.  Each packet of energy became known as a quanta, or quantum.

           In 1905, Einstein produced three major publications that revolutionized the world of physics.  The first of these papers proposed a theory in which a beam of light behaves like a shower of tiny particles.  Picking up where Plank left off, Einstein showed that energy is not only absorbed and emitted in quantas, but energy itself comes in discrete quantum packets.  Einstein demonstrated his theory by explaining the photoelectric effect, light’s ability to knock electrons out of metal.  The fact that individual electrons could be detected as they were knocked off a metal surface seemed to imply that light was behaving like a particle.  Moreover, reducing the intensity of the light beam did not effect the energy of the ejected electron.  On the other hand, the energy of the ejected electron could be effected by changing the frequency of the light. Einstein proposed that these light particles, called photons, come in packets, each with energy given by Plank’s expression: E = hf, where h is Plank’s constant, and f is the light’s frequency.  This formula predicts that photons of high-frequency light have more energy than photons of low-frequency light.

           Einstein’s discovery was completely contradictory to the previously held scientific theories of electromagnetic radiation.  In 1864, Clark Maxwell formalized the basic equations that govern electricity and magnetism, which are both now known to be aspects of a single entity we call the electromagnetic field.  According to Maxwell’s theory, light is a wave.  In other words, light is an electromagnetic vibration at a particular frequency.  The electromagnetic field is actually the spectrum of all possible frequencies of light.   In fact, the visible light we perceive with our eyes is a tiny fraction of this spectrum.  Maxwell’s theory predicted the existence of light waves at lower and higher frequency than visible light.  Shortly thereafter, radio waves were discovered, as were X-rays, infrared waves, ultraviolet waves, microwaves, and gamma rays.  These different types of waves are just different names for light at various wavelengths.  On the other hand however, Einstein’s theory demonstrated that light behaves like a particle.

           Further evidence supporting Einstein’s quantum theory of light came in 1923 when Arthur Compton made an important discovery.  Compton’s experiment involved shining a beam of X-rays into a gas of loosely bound electrons.  Compton showed that X-rays behave like particles, which bounce off the electron.  Both the X-ray and the electron scatter at specific angles like two billiard balls colliding.  He also formulated an expression for the momentum p of the light particle given by the following expression: p = hk, where h is plank’s constant, and k is the light’s spatial frequency.  Surprisingly, in 1914 the Bragg brothers had used crystal diffraction to show that X-rays behaved like waves.  This type of experiment is known as Bragg scattering.  Physicists at this time were confronted with contradictory evidence which suggested that light behaves both like a particle and a wave!

           The plot thickened even more in 1924 when Louis de Broglie proposed that every particle of matter was associated with a wave.  De Broglie reached this conclusion by using Einstein’s two equations for energy: E = mc 2 and E = hf.  De Broglie claimed that the wavelength of a matter-wave is given by the expression λ = h / P, where h is again Plank’s constant, and P is the momentum of the particle.  De Broglie’s outlandish idea, that matter is actually a wave, was soon proven experimentally to be correct.

           In the classical view of physical reality, there was no way to reconcile the differences between waves and particles.   A wave can spread out over a large area, be split up in an infinite number of waves, and two waves can interpenetrate and emerge unchanged.  On the other hand, a particle is located in a tiny region, travels in one direction, and crashes into other particles.  Although waves and particles appear to be contradictory aspects of reality, we have discovered that all waves are also particles, and all particles are also waves.

           In order to further illustrate this peculiar wave/particle coexistence, let’s briefly consider a simple type of quantum experiment.  Imagine we have an electron gun, a device that produces a beam of electrons.  Also, our experiment will include a phosphor screen.  If an individual phosphor in the screen is struck by an electron, the phosphor gains a little energy and immediately returns to its ground state by emitting a photon of light.  Firing the electron gun at the phosphor screen produces a point of light on the screen.  In this way, we can easily observe the particle nature of the electron.

           Next, in between the gun and the screen, let’s place a card that has a small hole in the center.  If our hole is sufficiently small enough, we will observe very different pattern than before.  The image on the screen is no longer a point of light, but a series of bright and dark concentric rings resembling a bull’s eye target.  This pattern is caused by wave diffraction, and the light and dark rings are caused by wave interference.  Interference is an inherent property of all wavelike interactions.  If two waves come together that are completely in phase, the resulting wave has an amplitude which is the sum of the two original wave amplitudes.  If the two waves are completely out of phase, the original waves simply cancel each other out.  In general when waves meet, their amplitudes add. This rule is known as the wave superposition principle, and it applies to all types of waves.

           The bull’s eye pattern on the phosphor screen clearly demonstrates the wavelike nature of the electron.  This pattern is created by a large number of electrons, which individually look like little points of light on the screen.  That is, each electron is observed only as a tiny flash of light, but after a large number of electrons have hit the screen, the pattern of the bull’s eye emerges.  This can be demonstrated in the following way.  If we lower the intensity of the beam, such that only one electron can pass through the hole at a time, we would be able to observe each electron hit the phosphor screen.  The exact location of each impact is completely unpredictable.  However, if we use a photographic plate to record each impact, and allow the system to continue firing one electron at an interval of say one every ten minutes, then, when we observe the plate later on, we will see the same bull’s eye pattern as before.  This experiment seems to imply that although it appears on the screen as a particle, each electron by itself travels from the gun to the screen as though it were a wave.  It should be noted that this type of experiment could have been done using any type of charged particle, or any frequency of light such as infrared or X-rays.

           Quantum theory is the method that has been developed to analyze experiments such as the one outlined above.  This theory was created to deal with tiny creatures such as atoms, electrons, and photons.  However, quantum theory has also proved successful in dealing with the atomic nucleus as well as subatomic particles such as quarks, gluons, and leptons.  In principal, quantum theory also applies to the macroscopic world which we inhabit, as well as large scale astronomical entities such as galaxies and black holes.  To date, quantum theory has successfully predicted the results of every experiment the human mind can devise.  However, the predictive strategy of quantum theory is quite different than classical mechanics in one fundamental way: quantum theory cannot predict what will happen in a measurement situation, it can only predict the statistical probabilities of how likely an event is to occur.  For any quantum entity, quantum theory predicts the probability of each possible value of a specific physical attribute.  Depending on the nature of the measurement situation, a quantum entity may demonstrate many different types of attributes.  Quantum theory does not say anything about what happens when the quantum entity is not being measured.

           First of all, let’s discuss what we mean by quantum entity, and how the theory addresses these entities.  A quantum entity is any thing, regardless of its size, which exhibits both wave and particle characteristics.  Usually, a quantum entity will demonstrate either particle nature or wave nature depending on the type of measurement it is subjected to.  A typical quantum entity could be a photon, an atom, or an electron; but a human, a planet, or the entire universe could be considered a quantum entity as well.  In this project, we will refer to a quantum entity as a quon for simplicity.  Instead of dealing with quantum entities specifically, quantum theory represents the quon with a mathematical device called the wave function, usually labeled Ψ or psi.  The first step in any quantum experiment is to associate a particular wave function to the relevant quantum entity.

           In most ways, the wave function, Ψ, is just like any other wave we are familiar with.  Before discussing quantum waves, let’s take a look at waves in general.  A wave is typically characterized by qualities known as amplitude, wavelength, frequency, and phase.  The amplitude of a wave is a measure of the deviation from its rest state.  In general, the amplitude is the maximum height of a wave.  If the wave is cyclic, then the wavelength is the space spanned by one cycle.  The length in time of one cycle is called the period.  The number of complete cycles in a certain interval of time is called the temporal frequency.  The number of complete cycles in a certain interval of space is called the spatial frequency.  The phase of a point in a cyclic wave is a measure of how far into a cycle that point is located.

           As mentioned earlier, all waves obey the superposition principle, which states: when two waves meet, their amplitudes add.  After the two waves move through each other, each wave retains its respective amplitude, and is thus unchanged by the temporary superposition.   As we shall see later, any two waves can interact and depart each other’s company with their respective amplitudes intact, but the phases of the these two waves become entangled, and are thus phase correlated for the rest of eternity.  When any two waves meet, the superposition of amplitudes depends on the phases of the each wave.  This is characterized by constructive and destructive interference.  For example, if two waves, each with amplitude of one, meet each other completely in phase, the resulting amplitude is two.  If two waves, each with amplitude of one, meet each other completely out of phase, the resulting amplitude is zero.  If two waves, each with amplitude of one, meet each other at arbitrary phases, the resulting amplitude will be between zero and two.  Quantum waves have all the characteristics of ordinary waves that have been outlined above.

           In general, the energy of a wave is a measure of intensity, and is given by the square of the amplitude.  For example, if you double a wave’s amplitude you quadruple the wave’s energy.   Quantum waves are different from ordinary waves in one important way.  Quantum waves do not have energy.  Instead, the square of the amplitude is a measure of probability.  This idea lies at the heart of how quantum theory works.  To predict the results of an experiment, we must find the amplitudes of each possible value of the attribute we are measuring, and then we square the amplitudes to get a probability distribution which indicates how likely each possibility is to occur.

           Before we can make any type of measurement, we must first decide what attribute we want to measure.  In general, quantum entities have two kinds of attributes: static and dynamic.  The static attributes of an elementary quantum entity always have the same value.  The major static attributes are mass (M), charge (Q), and spin magnitude (S).  The values for the dynamic attributes of a quantum entity change over time.  The major dynamic attributes are position (X), momentum (P), energy (E), and spin orientation (S z ).  Before we can understand how quantum theory represents the dynamic attributes of a quantum entity, we must first discuss some more basic properties of waves in general.

           Early in the 1800’s, a man named Joseph Fourier developed a new language which could be used to express any type of wave.  Fourier showed that any wave could be decomposed into a unique recipe of sine waves.  Each sine wave has a particular value of frequency k, amplitude a, and phase p.  The process of breaking any wave up into a bunch of sine waves is known as Fourier analysis.  Conversely, any wave can be constructed by putting together a bunch of sine waves, a process known as Fourier synthesis.

           Sine waves represent one type of waveform family.  Another type of waveform family is the impulse family.  An impulse wave is an infinitely narrow spike located at a specific location.  Just as Fourier showed that any wave could be broken up into sine waves, the same wave could be broken up into impulse waves.  The basis of digital electronic music is that any wave can be constructed by putting together a bunch of these impulse waves.

           The sine waveform family and the impulse waveform family are just two examples of waveform families; in fact, there are an infinite number of waveform families.  Any imaginable wave can be decomposed into a unique recipe of particular members of any type of waveform family.  This idea is sometimes called the synthesizer theorem.  Any wave can be expressed as a unique sum of members from any particular waveform family.  This means that any wave can be taken apart in an infinite number of ways, depending on which waveform family we choose to use.  Conversely, if we choose a particular waveform family, we can create any wave imaginable.

           Quantum theory makes use of this so-called synthesizer theorem in a peculiar way.  Quantum theory represents each dynamic attribute with a particular waveform family.  In other words, every possible waveform family corresponds to some dynamic attribute of the quantum entity.  The individual members of each waveform family represent different physical values of the dynamic attribute.  To illustrate, let’s give a few well-known examples. 

           First of all, the position attribute is associated with the impulse waveform family.  Each individual impulse wave is a narrow spike, characterized by a value x which describes the position of that particular impulse wave.  Each possible position attribute value, X, of a quon is associated with the location of a specific impulse wave at position x.  The momentum attribute is associated with the spatial sine waveform family.  Each member of this waveform family is characterized by a specific value for spatial frequency, k.  A specific momentum value, P, corresponds to each member of the spatial sine waveform family according to the following rule: P = hk, where h is Plank’s constant.  The energy attribute is associated with the temporal sine waveform family.  Each individual member of this family is characterized by a specific value, f, which represents temporal frequency.  The energy value associated with each individual wave in this family is given by the following rule: E = hf, where h is again Plank’s constant.  The relationships for momentum and energy are just de Broglie’s law for the matter-wave wavelengths, and Einstein’s relation for the energy of a quantum of light.  The waveform family associated with the spin orientation attribute is known as the spherical harmonic family.  Each member of this family is distinguished by two values: m and n, such that m and n are both positive integers.  The spin orientation value, S z ,in the polar direction is given by the follow rule: S z = m 2 / (n 2 + n).

           Quantum theory works by associating each dynamic attribute with a particular waveform family.  The relationship between the values of an attribute and the individual members of a particular waveform family is given by a rule, which can be quite simple in some cases, or very complicated in others.  For the most part, physicists are concerned with the major dynamic attributes, which have been described above; however, there are an infinite number of different dynamic attributes since there are infinitely many waveform families.

           A specific waveform family has special types of relationships with other waveform families.  To understand these relationships, we must first introduce some terminology.  By the synthesizer theorem, we know that any arbitrary wave can be broken up into different sets of component waves, depending on which waveform family we choose.  Breaking up an arbitrary wave into component waves is analogous to putting the original wave through a prism.  For example, Newton showed that white light could be passed through a prism to yield a rainbow of colors, known as the spectrum of visible light.

           If we analyze an arbitrary wave with different waveform prisms, we will discover that some prisms break the wave into a small number of components while some prisms break the wave into a large number of components.  The number of waveform components which a prism spits a wave is known as wave’s spectral width, or bandwidth.  If a particular waveform prism breaks an arbitrary wave into a small bandwidth of components, we could say that the waveform family is similar to the original wave.  If a waveform prism produces a large bandwidth of components, we could say that the waveform family is not similar to the original wave.  If we take an arbitrary wave and put it through its own family prism, the resulting bandwidth will consist of only one wave component, which is the minimum spectral width.  For example, if we put any sine wave through a sine waveform prism, the result will yield only one wave, which is exactly the original sine wave.We will refer to this prism, which does not split the original wave at all, as the kin prism.  For any arbitrary wave, there exists such a kin prism, which does not decompose the wave into any components expect for itself.  Conversely, for any arbitrary wave, there exists a particular waveform prism, which breaks the original wave into the largest possible bandwidth.  This is to say that for any wave, there exists a waveform family which resembles the original wave the least.  We will refer to this prism, which yields the maximum spectral width, as the conjugate prism.  Thus, every wave belongs to a unique waveform family, and every waveform family bears a special relationship to a unique conjugate waveform family.  An example of such a conjugate relationship is found between the sine waveform family and the impulse waveform family.  Because of their mutual relationship to an arbitrary wave, we could say that these two waveform families are conjugate to each other. 

           To illustrate this relationship between a prism and its conjugate prism, let‘s consider the following experiment.  Imagine we have identified two conjugate waveform families, called A and Z.  Fist, take any arbitrary wave X, and analyze this wave by using the A prism.  The result will be a particular bandwidth ΔA of output waveforms.  If we analyze X by using the Z prism, we will get a bandwidth ΔZ of output waveforms.  Because A and Z are conjugate waveform families, if X is very similar to A, then X will not be very similar to Z.  Conversely, if X is very similar to Z, then X will not be very similar to A. Consequently, there exists a limit on how small both bandwidths of A and Z can get for the same input wave.  This limit is usually expressed by the following relation: ΔA ● ΔZ ³ C, where A and Z are conjugate waveform families, and C is some positive constant.  We will refer to this relationship as the spectral area code.  The spectral area code is a fundamental feature of all waves, including quantum waves.

           In the above example A and Z are as dissimilar as two waveform families can be.  Now suppose we have chosen another waveform family K.   Let’s assume that K is not very similar to A, but K is more similar to A than Z is.  If we analyze the original wave X using the A and K prisms, there will still be a limit on how small both bandwidths of A and K can get for the same input.  This can be expressed in the same way as above such that ΔA ● ΔK ³ C’, where C’ is another constant.  However, since A and K are more similar than A and Z, the constant C’ will be less than C.  If we were to use two waveform prisms that are very similar such as A and B, the spectral area code may yield a resolving limit that is close to zero.  In other words, if two waveform families are very similar, there is no limit on how small both bandwidths can be for the same input wave.  On the other hand, if two waveform families are strikingly different in character, the spectral area code limits the product of the two spectral widths.  In this case, a small resulting bandwidth from one prism means that the resulting bandwidth of the other prism is huge.

           In quantum theory, every dynamic attribute is represented by a particular waveform family and a specific rule, which translates how individual members of the family correspond to particular values of the physical attribute.  As a direct consequence of the spectral area code, every conceivable dynamic attribute bears special relationships to other particular types of dynamic attributes.  Each dynamic attribute has a conjugate attribute in the same sense that each waveform family has a conjugate family.  In general, if two dynamic attributes are related in this way, such that the spectral area code applies, we could say that each attribute is conjugate to the other.

           We noted earlier that the sine family and the impulse family are conjugates.  We also know that the sine family can be associated with the momentum attribute of a quon , and the impulse family can be associated with the position attribute of a quon.  The spectral area code can be translated into an expression for the physical dynamic attributes of position and momentum in the following way: ΔX ● ΔP ³ h.  Here, ΔX represents the uncertainty in our measurement of the position attribute, ΔP represents the uncertainty in our measurement of the momentum attribute, and h is Plank’s constant.  This relationship is commonly known as the Heisenberg uncertainty principle.  The result of this relation is that we can know either position or momentum with perfect accuracy; however, since position and momentum are conjugate attributes, we can not define both attributes at the same time with perfect accuracy.  In other words, if we know the exact value of one of these attributes, the value of the other attribute becomes maximally uncertain.  It is possible that two dynamic attributes are independent of each other, in which case we can know the values of both simultaneously with perfect accuracy.  The uncertainty principle applies to dynamic attributes which are not independent.  The word independent is not really used here in any rigorously defined manner; however, we shall soon see that the condition which determines whether the uncertainty principle applies actually boils down to the commutative properties of specific matrices.

           Heisenberg’s uncertainty principle directly implies that the assumptions of classical physics were incredibly naïve.  Before Quantum theory, physics was based on the formulation of deterministic physical laws, which could be used to predict the exact outcome of any system.  In general, classical systems were represented by relationships in phase space.  Every particle, or object, in phase space is characterized by a definite position and momentum.  Assuming that one knows all the laws which govern a system, as well as the position and momentum values of a particle in such a system, one should be able to predict exactly how the system will change with time.  This ideal formed the basis of classical physics and inspired the conception of a universe that operates like a giant deterministic machine.  However, according the scientific discoveries of quantum theory in the early twentieth century, it is impossible to know the exact value of an object’s position and momentum at the same time.  Thus, Heisenberg’s uncertainty principle delivered a fatal blow to the antiquated conception of long term predictive determinism in physical systems.  In general, quantum theory does not predict the result of a measurement on a physical system at all; however, quantum theory predicts the probability of each possibility in the quantum system.

           Classical physics also assumed that all objects have inherent definite attributes which exist independently of the observation of those attributes.  As we will see, the structure of quantum theory implies that the attributes of any aspect of reality are inseparable from the observation of those attributes.  In fact, it is impossible to say for sure that something possesses any type of attribute whatsoever outside the context of some measurement situation. 

           Before we delve any deeper into this mysterious theory, let’s review the basics so far.  Quantum theory represents all quantum systems with a wave function, which we call Ψ.  This wave function is not only determined by the quantum entity in question, but by the type of attribute we wish to observe as well as the measurement situation we have designed to detect such attribute values.  For simplicity, we could say that Ψ is determined by the entire measurement situation.  Granted, this description is vague, but it sufficiently expresses the fact that there can be no separation between the observer and the observed.  The   Ψ-wave represents all possibilities of the quantum system. Choosing a specific attribute to measure is analogous to choosing a waveform family prism which analyzes the Ψ-wave into component waves.  Each component wave represents a possible value of the attribute we are measuring.  Moreover, each component wave has a particular amplitude and phase.  In other words, each possibility is assigned a specific coordinate value that represents the amplitude and phase of that possibility.  The square of the amplitude at each possibility gives the probability that a particular attribute value will be observed if we were to actually make a measurement.

           The first mathematical version of quantum theory was developed by Werner Heisenberg in 1925.  In Heisenberg’s model, a quantum system is represented by a set of matrices.  Each matrix represents a specific dynamic attribute such as position, momentum, or energy.  The probability that a system has a particular attribute value is determined by the diagonal entries of the matrix.  An important property of matrices is that many types of matrices do not commute when they are multiplied together.  If two attribute matrices don’t commute, then the measurement of these attributes is limited by the uncertainty principle.   The progression of the quantum state in time is represented mathematically by certain laws of motion expressed using matrices. This first version of quantum theory is usually known as Heisenberg’s matrix mechanics.

           A few months after Heisenberg’s theory was created, another physicist, named Erwin Schrödinger, introduced a different version of quantum theory.  Schrödinger created a wave equation, which represents the evolution of a quantum system over time.  The quantum state of a system at any instant is represented by a certain field of possibilities, Ψ, such that each possibility has a certain probability of occurring.  As the quantum system evolves, the amplitudes of the Ψ-wave change continuously according to Schrödinger’s wave equation.  The time dependent Schrödinger equation is usually written in the following way: - (h / 2π i) d/dt Ψ(x, t) = ĤΨ(x, t).  In this expression, x is vector whose component values represent all possible values of any attribute X, and Ĥ is the Hamiltonian.  The Hamiltonian is a linear operator that represents the total energy of the system.  An operator is a mathematical device that transforms a given function into some other function according to a certain rule.  In the case of Schrödinger equation, the time dependent Hamiltonian operator Ĥis equal to - (h / 2π i) d/dt.  Without being too technically specific, the important thing is that Schrödinger’s wave equation defines a rule Ĥ, which describes how the Ψ-wave changes over time. 

           At about the same time as Schrödinger proposed his theory of wave mechanics, a third quantum theory was developed by Paul Dirac.  This theory was rigorously formalized a few years later by the world famous mathematician John von Neumann.  Dirac showed that the fundamental ideas of quantum theory can be represented in abstract mathematical terms by placing the theory in what is called Hilbert space.  Dirac also showed that both Heisenberg’s and Schrödinger’s theories are special cases of his own Hilbert space version of quantum theory.  Dirac’s theory is a mathematical formulation that resembles our previous description of quantum theory, which we described solely in terms of waveform families and spectrums. 

           Hilbert space is not geometrical, but is an abstract way of organizing functions.  Although it is of little relevance to the goals of this project, we will present the conditions which define Hilbert space.  Hilbert space is a vector space on which an inner product is defined, and which is complete, i.e., which is such that any Cauchy sequence of vectors in the space converge to a vector in the space.  This abstract function space provides a natural reference frame for analyzing the wave function Ψ. 

           To illustrate the idea of Hilbert space and how it applies to quantum theory, let’s take a general example.  Imagine we have a quantum system, which is composed of a quon, and a measuring device that is designed to observe a particular dynamic attribute A of the quon.  The attribute, or observable, we choose to measure is represented mathematically by a linear operator, which we can label Â.  This linear operator is analogous to the waveform family prism we used earlier.   Each possible value of our attribute A is represented by a dimension in Hilbert space, which we will call a basic ray.  Mathematically, we could also say that each dimension represents an eigenfunction of the operator Â.  Generally speaking, there are as many basic rays as there are possible values for an attribute. In three-dimensional Euclidean space, each dimension is at right angles to the others.  Similarly, each dimension, or eigenfunction, in Hilbert space is perpendicular, or orthonormal, to the other dimensions.  If our attribute has two possible values, the corresponding Hilbert space will consist of two dimensions.  If our attribute is real valued along a continuum, the corresponding Hilbert space will consist of an uncountably infinite number of dimensions.  The reference frame in Hilbert space is determined by the possible values of the attribute we have chosen to measure. 

           The wave function, Ψ, is represented by a vector in Hilbert space.  This vector, which we will call the quantum ray, is simply a direction, which passes through the origin of our given coordinate frame of reference in Hilbert space.  The quantum ray, Ψ, represents one quantum state of the system which is being analyzed.  Given our particular reference frame, the wave function assigns a specific coordinate value, or point, to each basic ray.  This coordinate point of each dimension is just the projection of the quantum ray onto each single basic ray.  However, the coordinate value is not a point on a real line, but is a point on the complex plane.  Each coordinate value is represented by a 2-dimensional complex vector which, if defined in exponential form, can be written in the following way: z = re if , such that r is the length, or magnitude, of the vector, and f is the angle of the vector.  The magnitude of a given coordinate value represents the amplitude of the wave function at a specific possibility.  The angle of a given coordinate value represents the relative phase of the wave function at a specific possibility.  Thus, for each possibility, also called a basic ray, the wave function assigns a coordinate value, which is a complex vector that represents a specific amplitude and phase.

           Each dimension, or basic ray, of Hilbert space, is associated with its own complex plane.  The projection of the Ψ-wave onto each basic ray is given by a specific complex vector.  If we let c i stand for the coordinate value along a particular basic ray, and we let Φ i stand for a particular eigenfunction, or basic ray, then we can express Ψ in the following way: Ψ(x) = ∑ c i Φ i (x), for all i.  The Ψ-wave, or quantum ray, is just the sum of all these coordinate values, or complex vectors.  Thus, Ψcan either be represented as a single entity such as a vector in Hilbert space, or by a collection of vectors in the complex plane such that each vector represents a specific possibility. The quantum wave is a field of possibilities.

           Each possibility is characterized by an amplitude and a phase.  To find the probability that a particular possibility will occur, we simply take the square of the amplitude.  If c i represents a particular complex vector, then the square of the amplitude can be expressed as  ||c i || 2 .  In order for our probability measure to make any sense, we must normalize the quantum ray, Ψ, such that ∑ ||c i || 2 = ∫ ||Ψ(x)|| 2 dx = 1.  This means that the sum of all the probabilities is equal to 1.

           As noted above, the quantum wave function is a vector, in Hilbert space, which represents one quantum state.  Without actually observing the measurement situation, we can ask how the Ψ-wave might change over time.  For simplicity, lets assume there is only one dimension of time and that it always travels in the same direction.  If the original Ψ-wave is calculated at time t o , then the Ψ-wave at time t 1 will be represented by a vector in Hilbert space which is different than the original vector.  If we assume that time is a continuum, we can show that the quantum vector changes its orientation continuously.  Thus, our spinning quantum vector, in Hilbert space, represents the continuously morphing Ψ-wave.  Therefore, Schrödinger’s equation, which describes our spinning vector, is actually a mathematical representation of a morphing field of possibilities.  As the quantum wave moves and changes direction, the magnitudes and the relative phases of all the coordinate values also change.  Note, if the quantum vector travels continuously in Hilbert space, then each projection, which determines the possibility amplitude, also changes continuously.  The probability distribution for each quantum vector also changes continuously because the probabilities are the squares of the continuously changing amplitudes.

           It is important to remember that although this theory can be used effectively to determine the probability distribution for the attribute values we are concerned with, these potential tendencies to exist are not inherent in the representation of the quantum entity.  Unless we first assume a frame of reference, such as a measurement situation designed to observe the value of a specific dynamic attribute, the quantum entity is simply a wave of infinite possibilities.  The frame of reference in Hilbert space is created based on which attribute we choose to measure.  Only after we have chosen an attribute, can we analyze, or decompose, the quantum wave into complex projections along the orthonormal basic rays.

           In order to calculate the probability distribution for the possible values of a quantum entity, we must first create the concept of an attribute, which we want to measure.  The word create is used here because, in some sense, any type of conceptual attribute such as position, momentum, energy, and spin is a construct of the imagination.  In other words, if we want to measure position we must create a unit measure of distance.  Likewise, if we want to measure momentum, we must create a unit measure of time as well as direction.  In general, these units have been created out of thin air, and bear no real connection to nature.

           Without a reference frame of observation, it is meaningless to say that the quantum entity possess any attribute whatsoever, let alone values for that attribute.  Perhaps this claim is too far out to except right off; however, it at least appears safe to say that the internal structure of quantum theory implies that the attributes of any aspect of reality are inseparable from the observation of those attributes.  For example, a  quon does not inherently possess what we call momentum; however, given a certain measurement situation, the quon will demonstrate the appearance of momentum.  In other words, momentum does not belong to the quantum entity itself, but to our interaction, or relationship, with the quantum entity.  In a similar sense, a quantum entity is neither a wave nor a particle, but if we interact with the quantum entity, it will express itself either as a wave or a particle.  This point of view is not without opposition.  For instance, many physicists still believe that the quantum entity is an ordinary object, which exists whether it is being observed or not.  Although there is a generally accepted recipe, or method, for using quantum theory, there is not much agreement amongst physicists concerning how quantum theory is actually connected to what we call reality.  Before we turn to the various interpretations of quantum theory, let’s consider a fourth version of quantum theory.

           In 1948, Richard Feynman developed a method for calculating a quon’s wave function which is called the sum-over-histories approach.  We’ve already seen that the Ψ-wave represents all the possibilities open to a given quantum system. To get more perspective, let’s consider the quon gun and phosphor screen experiment that was described earlier.  Between the quon gun and the screen, the unmeasured quon behaves like a wave of possibilities.  Feynman’s idea was that what actually happens on the screen is influenced by everything that could have happened.  Feynman’s approach to calculate Ψis to sum over the amplitudes of all possible ways a quon can get from the quon gun to the screen.  Feynman describes the unmeasured world by making two postulates: a quon takes all possible paths, and no path is better than another.  He also proposes that every path open to the quon has the same amplitude, and that each path differs from other paths only in its phase.  In quantum theory, possibilities have a wavelike nature.  Therefore, certain possibilities can cancel if they have different phases.  Feynman showed that summing up all possible paths, or histories, produces the same wave function as solving Schrödinger’s equation.

           In the context of our phosphor screen experiment, quantum theory implies that just before a flash is made on the screen we should not imagine that a tiny quon is actually heading for one particular phosphor molecule.  Before the measurement occurs, the quon is heading in all possible directions at the same time.  According to this view, an unmeasured quon exists only as a bunch of unrealized quantum potentialities.  However, every time we make a measurement, only one of these possibilities becomes an actuality.

The Battlefield of Quantum Speculation and The Meaning of It All

           The early interpretations of quantum theory reconciled this peculiar phenomenon by assuming that the world is divided into two separate parts.  The unmeasured world, it was assumed, consists only of quantum potentials.  On the other hand, the measured world consists only of classical type actualities.  This interpretation of quantum theory was primarily advanced by Niels Bohr and Werner Heisenberg and is known generally as the Copenhagen interpretation.   In addition to the view of distinct measured and unmeasured aspects of reality, the Copenhagen interpretation asserts two other fundamental assumptions.  Firstly, Copenhagenists assume that there is no reality in the absence of observation.  Secondly, this interpretation asserts that observation creates reality.

           These two assertions are based on the idea that the dynamic attributes of a quon are contextual in the sense that the attribute values are determined by which attribute we choose to measure.  For example, the position attribute of a quon is jointly determined by the quon and the measuring device.  If we take away the measuring device, we also take away the position attribute of the quon.  If we change the measurement context, then we also change the attributes of the quon.  A Copenhagenist would argue that when a quon is not being measured, it has no definite dynamic attributes.  This idea that observation creates reality is based on the so-called quantum meter option, i.e. the observer’s ability to freely select which attribute he wants to look at.  In terms of our earlier discussion of waveform languages, this quantum meter option is analogous to our freedom of choice concerning which waveform prism we will utilize to analyze an arbitrary wave.Another assumption of the Copenhagen interpretation is that all quons in the same quantum state, i.e. represented by the same wave function, are physically identical.  Furthermore, the Copenhagen interpretation asserts that the wave function tells us everything there is to know about the quantum entity.  However, Copenhagenists do not believe that the Ψ-wave is a real wave.  They view the Ψ-wave simply as a mathematical tool that can be used to determine the statistical likelihood of an event, given a specific measurement context.  It is also their position that there is absolutely no way to know which possibility will become an actuality.

           In classical mechanics, the unpredictability of an event was attributed to the ignorance of the observer.  The observer’s ignorance, in the classical sense, arose because the observer did not have a complete knowledge of all the variables in a system, or the measuring device used in the observation was technologically unable to yield perfectly accurate readings.  It was assumed that this ignorance could be overcome by making further technological improvements to the measuring devices.  However, in the Copenhagen interpretation, it is impossible to predict which possibility will become an actuality simply because the deepest form of knowledge we can have of a quantum system is purely statistical.  This type of ignorance is known as quantum ignorance, as opposed to classical ignorance.  Classically, the missing information exists, but has yet to be uncovered by the experimenter.  The idea of quantum ignorance asserts that the missing information simply does not exist. 

           Quantum ignorance is closely tied to the idea of quantum randomness.  In order to understand this idea better, let’s consider the quon gun and detector screen experiment again.  Assume that the gun fires only one quon.  The wave function, Ψ, gives us a complete description of the probabilities of each possibility.  Before the measurement, the quon assumes all possible paths.  The result of the measurement yields only one actual flash on the phosphor screen.  Now, suppose we fire a second quon at the screen.  This second quon is represented by the exact same Ψ-wave as the first quon.  The result of the second measurement again yields only one actual flash; however, this flash is most likely located in a different place on the phosphor screen relative to the first flash.  

           The Copenhagenists explain this phenomenon by appealing to what they call quantum randomness.  The basic principle of quantum randomness is that identical physical situations give rise to different outcomes.  If it is true that the Ψ -wave gives us all the information we can know, then it is impossible to predict exactly where the quon will strike the phosphor screen.  According to the Copenhagists, the occurrence of an actual event is determined by blind chance.  We shall soon get a better idea of how these quantum fields, which organize the probability distribution of our system, are extremely complex and multidimensional.  In any case, the visualization of these fields is beyond the capacity of most physicists working within the current paradigm.  At this point, we will merely note for future reference that the dynamics of a given field of possibilities may be so extraordinary chaotic that the pattern appears random to our ordinary mode of perception.

           It should be noted that the Copenhagen interpretation is based on the primary assumption that measuring devices are ordinary objects which exist and are definable in the classical sense.  Quantum theory describes neither the quantum system nor the measuring device.   The theory applies to the relationship which exists between the quantum system and the measuring device.  However, the Copenhagen interpretation asserts that a very significant and mysterious transition takes place at the boundary between the measuring device and the quantum system.  In this transition, the surreal potential existence of the unmeasured quantum entity immediately transforms into a real classical type observed actuality.  The question as to exactly how, why, and when this transition occurs is the basis of the so-called quantum measurement problem.  The Copenhagenists side step this interpretive paradox by assuming that measuring devices are real things which actually exist with definite attributes, while quantum entities are represented by a superposition of potential possibilities.

           In 1932, John von Neumann published a book called the Mathematical Foundations of Quantum Mechanics (Die Mathematische Grundlagen der Quantunmechanick), in which the ideas of quantum theory are subjected to rigorous mathematical analysis.  Von Neumann’s analysis is primarily concerned with Dirac’s Hilbert space version of quantum theory, which has been shown to be more general and complete than the Heisenberg and Schrödinger theories.  Among other things, von Neumann demonstrates that there is nothing intrinsically special about measuring devices.  Therefore, the Copenhagenist assertion that measuring devices are somehow privileged with a classical status of existence seems awkward and contrived.  In von Neumann’s theory, everything is represented by quantum Ψ-waves, even measuring devices.  Von Neumann’s interpretation is known as the all-quantum theory because there is no longer any aspect of the theory which relies on classically defined objects.

           Von Neumann showed that it is indeed possible to represent everything in the world with Ψ-waves; however, the all-quantum theory only works if we make one crucial assumption.  Before dealing with this assumption directly, let’s consider again the structure and dynamics of the wave function in Hilbert space. We know that a particular quantum state is represented by a normalized vector in Hilbert space.  The dimensionality of our frame of reference in Hilbert space is determined by the attribute operator we choose.  It is often the case that this frame of reference consists of an uncountably infinite number of dimensions.  Each dimension represents an orthonormal eigenfunction of the quantum operator that we are using.  Each of these orthonormal eigenfunctions represents a specific attribute value of the quon we are measuring.

           The amplitudes and phases of each possibility are determined by decomposing the wave function into complex vector components, which are just the projections of the wave function along each dimension.  However, a quantum system has a definite value for an observable attribute if and only if the quantum vector, Ψ, is an eigenstate of the attribute operator.  This means that the system only has a definite state if the quantum vector is parallel to a particular eigenfunction.  Since each eigenfunction of the operator is independent, or orthonormal, to all other eigenfunctions, any vector which lies along one single eigenfunction has no components along any of the other eigenfunctions.  In other words, if the quantum vector lies along one specific eigenfunction, the amplitude at that possibility is one, and the amplitudes at all other possibilities are zero.  However, in most cases, the wave function can only be expressed as a linear combination consisting of coordinates from many eigenfunctions.

           According to Feynmann’s version of quantum theory, the unmeasured quon assumes all possible values at the same time.  Contrary to this idea, in which the quon assumes all possibilities at once, is the actual fact that any type of measurement only yields one specific result.  Therefore, in order for von Neumann’s theory to be consistent, we must assume that at some point between the creation of the quantum entity in the quon gun and the observation of an experimental result, a remarkable transformation must occur.  At the exact instant the measurement occurs, the quantum entity must cease to be a superposition of possibilities, and must contract into a single possibility, corresponding to the single observed measurement result.  This mysterious and radical transformation is called the collapse of the wave function.  Von Neumann’s all-quantum theory will not work unless this collapse of the wave function actually occurs in every type of quantum measurement.  As alluded to earlier, the fundamental paradox of quantum theory is the so-called quantum measurement problem, which can be stated in the following way: how and when does the wave function collapse?

           In von Neumann’s analysis of the quantum measurement problem, he proposed that the measurement act could be broken up into a series of small steps.  In this way the entire measurement act is visualized as a chain of events stretching from the quon gun, to the phosphor screen, to the observer’s retinas, and finally to the observer’s conscious perception of the measured result.  Von Neumann’s goal was to analyze each link in this chain in order to find the most natural place to put the collapse of the Ψ-wave.  What he discovered is that we can cut the chain and insert a collapse anywhere we please, but the theory won’t work if we leave it out.  Von Neumann reasoned that the only peculiar link in the chain is the moment when the physical signals in the human brain become an actual experience in the human mind.  Based on this form of logic, von Neumann reached the conclusion that human consciousness is the only viable site for the collapse of the wave function.  Therefore, according to von Neumann, consciousness creates reality.

           This idea of consciousness-created reality is a step beyond the claims made by those who subscribe to the observer-created reality interpretation.  Observer-created reality enthusiasts simply claim that the observer is free to choose which attribute will be measured.  However, they do not claim that the observer determines what the actual result of the measurement will be.  Consciousness-created reality enthusiasts, on the other hand, claim that consciousness selects which one of the many possibilities actually becomes realized.  Granted, these claims have not been experimentally proven, yet we might still consider some general consequences of this interpretation of quantum theory.  If we assume that the basic principles of quantum theory are correct, we can easily derive two such interesting general conclusions.  Firstly, as far as the final results are concerned, there is no natural boundary line between the observer and the observed system.  Secondly, it is apparently the case that no such interpretation of quantum theory would be complete unless it successfully incorporates the function of consciousness, which seems to be inseparable from the manifestation of particular outcomes in the quantum measurement.

           There is, however, another interpretation of quantum theory, which is similar to von Neumann’s ideas, but is not dependent on the idea of a wave function collapse.  This theory, called the many-worlds interpretation, was developed by Hugh Everett in 1957.  Everett, like von Neumann, assumes that there is nothing special about measuring devices and that everything can be represented by Ψ-waves.  However, Everett leaves out the collapse of the wave function.  Instead, his theory is based on the idea every possible attribute value of a quon actually becomes realized when the quon interacts with a measuring device.  For example, if the quon can assume six possible attribute values, then all of these possibilities actually occur.  Everett claims that the entire measurement device branches into many measurement devices, each of which observe a different possible value of the chosen attribute.  Given that nobody has ever seen a measuring devices spit apart in such a way, Everett claims that each possible value is realized in its own parallel universe. 

           Everett’s quantum model implies that at every instant, the universe is a branching tree in which anything that can happen, no matter how improbable, actually does happen.  As far out as this claim might seem to our simple egos, this many-worlds interpretation actually addresses the fundamental inconsistencies of quantum theory in a satisfactory manner.  For instance, there is no such attempt to sanctify the status of measuring devices.  In addition, there is no need for the mysterious notion of the wave function collapse, which in itself, has never been detected, nor is their any a priori evidence which supports its existence other than the fact that we humans only perceive the occurrence of one event at a time.

           Up until now we have only considered the orthodox Copenhagen interpretation and its primary derivatives, i.e. von Neumann’s all quantum theory, and Everett’s many-worlds interpretation.  These theories accept as their basic premise that the fundamental level of reality, namely the quantum world, is governed solely by the statistical laws of quantum possibilities.  In addition, these theories also accept the idea that quantum entities are not ordinary objects in the classical sense.  An ordinary object possesses definite attributes independently of the observation of those attributes.  Indeed, von Neumann, in his book on the foundations of quantum mechanics, derived a proof which asserts that if quantum theory is correct, then the world cannot be made of ordinary objects.  However, despite the strong convictions amongst the majority of physicists that no such ordinary object model of reality could be consistent with the quantum facts, there is a group of physicists which believe that such a model could indeed be produced.

           The most famous of these physicists, who opposed the quantum orthodox interpretation, is Albert Einstein.  Einstein strongly believed that quantum theory was incomplete because it only gave a statistical account of elementary phenomenon.  He believed that it was possible to construct an ordinary object model of reality in which the quantum entities had definite attributes whether or not anybody was observing them.  Einstein and the other physicists who believe that an ordinary object model of reality is possible are sometimes referred to as neorealists.  The neorealist position is basically that there exists a deeper, more fundamental, level of reality, which is not described by the quantum wave function.  As we have already seen, if we assume that the Ψ-wave tells us everything there is to know about the quon, then it is impossible to predict what the actual result of a measurement will be.  The neorealists believe that the Ψ-does not tell us everything there is to know.  Indeed, they hold the position that hidden, unseen parameters exist at a deeper level, which if discovered, could be used to predict exactly what will happen in a quantum experiment.  For this reason, neorealist theories are also known as hidden-variable theories.

           As noted above, von Neumann’s proof asserts that no such theory of ordinary objects can explain the quantum facts.  However, David Bohm, a protégée of Albert Einstein, was able to develop a hidden-variable theory which is seemingly consistent with the observed quantum facts.  Bohm’s hidden-variable model of reality, which was developed in 1952, assumes that quantum entities are ordinary objects, such as real particles, which have at all times a definite position and momentum.  Whereas the Copenhagen interpretation assumes that an unmeasured quon assumes all possibilities at once, Bohm’s theory assumes that an unmeasured quon takes only one path and that this path is ultimately predictable.  However, there is a catch.  Bohm’s theory introduces a new type of wave called the “pilot wave”, which organizes the unfolding history of the quantum entity. 

           The Copenhagenists assert that the Ψ-wave is not real, but merely a fictitious mathematical device which happens to be effectively useful in calculating quantum probabilities.  Bohm, on the other hand, asserts that both the quantum entity and the pilot wave are real things which actually exist.  Although the pilot wave is supposedly a real entity, in order for Bohm’s theory to be consistent with the facts, this pilot wave must have certain remarkable characteristics which defy our conventional definitions of what is possible in reality.  For instance, this pilot wave must connect with every particle in the universe, it must be entirely invisible, and it must transfer information at superluminal speeds, i.e. faster than light. 

           Of these three, the first two properties of Bohm’s pilot wave are familiar within physics in that they are both aspects of the gravitational and electromagnetic fields.  Superluminal connections, on the other hand, seem to be the one thing most physics hate most.  This is primarily because the existence of superluminal connections would violate many fundamental assumptions of the orthodox theory on physical reality.  For example, real superluminal transfers would contradict the orthodox themata which asserts that influences can only be mediated by direct interactions.  This assumption, that object A can only effect object B via direct subluminal interactions, is called the locality assumption.  Also, faster than light connections directly imply that the past can be influenced by the future.  Most physicists, however, would like to believe that time travels in only one direction, and that what happens within each moment is solely influenced by what has already happened. 

           In Bohm’s model, each particle in the universe, it is assumed, is associated with a pilot wave.  This pilot wave is sensitive to the entire environment of the quantum entity, and the wave changes its form instantly whenever there is a change anywhere in the environment.  Conversely, this instantaneously morphing field informs the quantum entity of such changes in the environment, at which point the quantum entity alters its values of position and momentum accordingly.

           However, this theory predicts that all pilot waves of all particles are instantaneously connected across the entire universe.  This implies that the relevant environment, or measurement situation, which determines the form of the pilot wave, includes all events in the universe across all dimensions of space-time.  Understandably, most physicists abhor the idea of faster than light, let along instantaneous, connections, and consequently, many physicists consider Bohm’s theory to be absurd.  However, although it seems absurd to the quaint common sense intuitions of most physicists, it was soon proven that these superluminal connections are no accident, but a necessary condition of any theory of reality.  Big news!

EPR, Bell’s Theorem, Non-locality, and Superluminal Spaghetti

           This proof we mentioned above was devised by John Stewart Bell in 1964, and is known as Bell’s interconnectedness theorem.  Bell, while studying Bohm’s theory, was able to show how an ordinary object model of reality had been created contrary to the proof of von Neumann, which asserted that no such theory was possible.  Obviously, von Neumann’s proof contained a loophole.  Bell showed that von Neumann’s idea of an ordinary object was too limited.  Bohm was able to create such a theory by stretching the conventional idea of an ordinary object.  Most physicists would not consider any object ordinary if it can change its attributes instantaneously via resonance with some invisible, all-pervasive, superluminal field. 

           Bell’s theorem, which Bell developed after his work on Bohm and von Neumann, brings into question the assumption of a locally based version of reality, and ultimately proves that the reality which underlies our experience must be non-local.  This proof was based on the factual results of an experiment originally designed by Albert Einstein, Boris Podolsky, and Nathan Rosen.  Before taking a deeper look into Bell’s theorem and non-locality, let’s briefly discuss the logistics of Einstein’s experiment, which has since come to be known as the EPR experiment.

           As described earlier, Albert Einstein believed that quantum theory was not a complete theory of reality.  Thus, Einstein designed a specific thought experiment, which supposedly demonstrates that there are aspects of reality that are not accounted for in the quantum theory.  In brief, the EPR source emits a pair of phase entangled photons in opposite directions at the speed of light toward two spatially separated detectors.  Let’s label these detectors A and B.  In a generic form of the EPR experiment, these detectors are designed to measure the polarization attribute of the photons.  A simple form of a polarization detector can be realized by using a calcite crystal whose optic axis is pointing in a certain direction.  The crystal divides light into two beams.  The up beam consists of photons which are polarized along the optic axis, while the down beam consists of photons which are polarized at right angles to the optic axis.  Because the photons are phase entangled, the phase of each photon depends on what the other photon is doing.  Also, there is only one wave function, which describes both photons.  Before the actual measurement, quantum theory predicts that neither photon has a definite value of polarization.

           If we assume that each calcite detector is positioned at any arbitrary angle, then each detector will measure a fifty-fifty percent mixture of up / down results.  On the other hand, if we assume that each detector is orientated at the same angle, then we can measure another type of attribute called the parallel polarization attribute.  In this case, both photons are always measured to have the same polarization.  If the two detectors hold their crystals at a relative angle of ninety degrees, then the polarization value at one detector will always measure the opposite as the other detector.  As an example, let’s assume that detector A holds its crystal at zero degrees, and that this A detector is located closer to the source than B is.  This way, the polarization at A is detected first.  Also, let’s assume that at an angle of zero degrees, A measures an up value.  Quantum theory predicts that if B holds its crystal at zero degrees, it will measure up as well.  On the other hand, if B holds its crystal at ninety degrees, it will measure a down valued polarization.  If B holds its crystal at angles other than zero or ninety degrees, quantum theory gives no definite results.  For example, if B holds its crystal at forty-five degrees relative to A, then the odds are fifty-fifty that B will measure an up value.         

           Quantum theory predicts that except at certain angles, such as zero and ninety degrees, the result of B’s measurement is determined by quantum randomness.  In other words, at angles between zero and ninety degrees, the measurement at B is determined by blind chance.  However, Einstein argues that since the photons are in what can be called a twin state, if detector A is measured first at any particular angle, then the photon at the other detector must possess a definite polarization attribute value prior to its interaction with the detector, which could be set at any angle.  Einstein also argues that quantum theory only gives a statistical interpretation of attribute values which truly have a definite existence before the act of measurement.  Therefore, Einstein concludes that quantum theory is not a complete theory of reality.  The basic assumption which Einstein makes is that, after the photons have left the source, the situation at detector B is not affected by how detector A chooses to hold its crystal.  This premise is known generally as the locality assumption.  Einstein’s argument can only be refuted in two ways: either the locality assumption is violated, or there is no such thing as two spatially separated events.  This perplexing thought experiment is known as the EPR paradox.  

           While studying this thought experiment, Bell considered what would happen to the measurements of each detector if the calcite crystals vary their angles between zero and ninety degrees.  Bell incorporates a type of polarization attribute which measures how these results are correlated.  This attribute can be called the polarization correlation attribute, labeled PC(θ).  As before, if the relative angular difference between each detector is zero, then the measurements are perfectly correlated, thus PC(0) = 1.  If the crystals are set at a difference of ninety degrees, the measurements are perfectly uncorrelated, thus PC(90) = 0.  At angles between zero and ninety degrees the value of PC is some fraction between 0 and 1.

           The value of PC(θ), for angles between zero and ninety degrees, can be measured by firing many pairs of phase entangled photons and then comparing the series of measurement values recorded at each detector.   The polarization correlation attribute is a measure of the fraction of matches between two detectors over a long series of photon pair emissions.  Imagine that each list of measurements is a type of binary message.  If A and B receive exactly the same messages, then the PC(θ) value is one, and the angle between each crystal must be zero degrees.  If A and B receive exactly opposite messages, then the PC(θ) value is zero, and the relative angle must be ninety degrees.  In between these two extremes, the two messages will contain a fraction of errors.  For example, let’s assume that if the crystals are orientated at a relative angle α, then the two binary messages differ by one out of every four bits.  In other words, the error rate between the two messages is ¼.  Thus, at angle α, the polarization correlation attribute value, PC(α), is a factor of three correlated results for every four photon pairs, that is, PC(α) = ¾.  Everything presented so far concerning this type of experiment is based purely on scientific fact. 

           To understand Bell’s theorem, let’s first assume that both crystals are set vertically at zero degrees.  Now we rotate the A crystal α degrees in the clockwise direction, and we rotate the B crystal α degrees in the counterclockwise direction.  These crystals are now separated by a relative difference of 2α degrees.  Bell, like Einstein, makes only one fundamental assumption, which asserts that the situation at detector A does not effect what is happening at detector B; this is known as the locality assumption. This assumption appears reasonable since these photons are flying away from each other at the speed of light.  If we assume locality, it follows that if the error rate at angle α is equal to ¼, then the error rate at angle 2α must be less than or equal to ½.  This expression is an example of what is known as Bell’s inequality.  This inequality is a direct consequence of the locality assumption.

           However, the equation for the polarization correlation attribute can be derived mathematically such that PC(θ) = cos 2 θ.  For this equation, PC(30) = ¾, and the error rate equals ¼; that is, one error between each message for every four photon pairs.  However, at twice this angle, PC(60) = ¼, and thus, the error rate becomes ¾.    This result is a direct violation of Bell’s inequality, which predicts that the fraction of errors between the messages cannot be greater than ½.  Let’s recap Bell’s argument.  First we assume that reality is strictly local.  This assumption leads directly to a specific inequality.  Surprisingly, whenever this experiment is performed, the results violate this inequality.  Therefore, since we reached a contradiction, our original assumption must be false, i.e. reality is non-local. 

           Bell’s proof does not demonstrate any observable type of non-local interaction.  He merely proved that the correlation between two twin state photons is so strong that no version of local reality can account for the mathematically predicted violation of Bell’s inequality.  Indeed, Bell’s idea was experimentally put to the test, first by John Clauser in 1972, and later by Alain Aspect in 1982.  Both of these technologically sophisticated experiments produced results that directly violate Bell’s inequality.  Therefore, unless we resort to drastic counter-arguments, such as the claim that there is no reality at all, or that everything in the world is entirely predetermined to the infinite degree, there is no other way to save the locality assumption from its decent into the dustbin of history.  Through careful consideration of the experimental facts, it is now safe to say that locality is just as outdated and incorrect as the idea that the Earth is flat.

           Although John Bell only proved that non-locality is a necessary factor in describing a particular twin-state photon experiment, we can extend this idea to include everything that exists in reality.  We can make this type of assertion because quantum theory predicts a phenomenon known as phase entanglement.  Whenever two quantum entities interact, their phases get mixed up.  As these entities interact and then depart their separate ways, the amplitudes of each Ψ-wave come apart, but the phases of the two quons remain connected.  Indeed, the strong correlation between these two EPR photons is a direct result of the fact that they were created from the same source, and are thus, phase entangled.  This prediction of phase entanglement was recognized by physicists which preceded Bell; however, Bell was to the first to actually demonstrate that this phenomenon actually exists in the real world.

           The basic idea of phase entanglement and non-locality rests on the idea that once two entities have interacted, they are eternally connected by the correlation between their mutual phases.  An important consequence of all this is the fact that the so-called entire measurement situation, which determines the attribute values of a quon, must include situations, measurements, and events everywhere throughout the universe.  Moreover, non-locality implies that the entire measurement situation, of even a simple quantum experiment here on Earth, must include all measurements and events everywhere in the universe across all scales and dimensions of time.  Presented another way, non-locality implies that every thing is everywhere, and no thing is really separate from anything else.  Indeed, everything together is only one thing.  If we try to restrict ourselves by measuring only part of the one thing, we will inevitably encounter a limit on how accurate our predictions may be. After all, there’s nothing special about Plank’s constant h.  Perhaps the unavoidable quantum uncertainly is a direct consequence of our naïve assumption that we are separate from that which we are observing. 

Hyperdimensional Holon Attractors and the B-Sense

           Thus far, we have embarked on quite a lengthy discussion of quantum theory and it’s various interpretations.  Although many concepts have been addressed in this project, the majority of the details have been left out.  In addition, our exploration through this quantum realm has merely scratched the surface, and much of the most interesting terrain has yet to be explored.  Although there are more advanced forms of exploration which lie beyond the scope of this project, it is my hope that this preliminary exploration will form a stable foundation such that further developments and interpretations may be explored at a latter time.  For now, let’s conclude this journey with an overall survey of some ideas which might form the basis of future, more detailed, endeavors into quantum theory.  It should be obvious to the reader that the implicate seeds contained within some of the ideas to follow will certainly contradict, and bring into question, many of our presently held notions concerning the nature of reality and consciousness.

           First of all, quantum theory, in the broadest sense, is a theory of whole entities.  Any representation of a quantum entity must include a joint description of the entity itself as well as its observational context.  It must be remembered at all times that there is no real distinction between the attributes of any aspect of reality and the experience of those attributes relative to a specific observer.  Furthermore, if multiple observers are measuring the same quantum system but in different ways, the experience of these observers will also differ.  That is, the experience of reality is relative to one’s frame of reference.  In addition, regardless of whether we subscribe to an ordinary-object based interpretation, or to a statistical interpretation, the idea that the manifestations of physical reality are self-organized by abstract fields of possibility is unavoidable.

           Another unavoidable conclusion is that these fields must be interconnected in such a way that it makes absolutely no sense to speak of them as separate fields.  For example, let’s consider two seemingly separate quantum entities, each resented by its own quantum vector in its own frame of reference in Hilbert space.  If these two entities become entangled, then the composition of two Hilbert spaces, H a and H b , can be represented by the tensor product H a Ä H b , which itself forms an entirely new vector in a new Hilbert space.  In other words, entangled entities are not represented by separate quantum fields, but are represented by only one Ψ-wave.  However, it my contention that every aspect of reality is already phase entangled.  This would certainly be the case if the cosmological Big Bang theory were correct.  If it is true that everything that exists is indeed part of one phase entangled quantum system, then it might prove useful to consider the likely existence of a universal wave function.  Although this field would be incomprehensibly complex, the nature of non-locality assures us that whatever it is, it is within every thing.      

           Another interesting aspect of quantum theory is the manner in which quantum waves morph over time.  To consider a particular example, let’s measure the position attribute of a quon.  First of all, we shall assume for simplicity that we live in 3-dimensional Euclidean space.  The realm of possible values for position includes all points in a 3D continuum.  Obviously, there are an uncountably infinite number of possible positions.  Each point in 3D configuration space is represented by a single dimension in Hilbert space.  The wave function of our quon, is a vector in this space, which has a unique decomposition into complex projection components along each dimension.  To find the probability that the quon will be measured at the point (x o , y o , z o ), we simply take the square of the amplitude of that possible point at instant t o .  At a different instant in time, t 1 , the quantum state will be represented by a different quantum vector.  If we assume time is a continuum, the transition between these two states can be visualized as a spinning vector in Hilbert space. 

           However, it appears that we could modify our example to give the full picture at one glance, as opposed to watching our quantum vector spin around.  Firstly, we will expand our domain of possibilities from all points in 3D configuration space to the domain of all points in 3D configurations space for all time.  Thus, each possible value of position is now a point (x, y, z, t).  The corresponding Hilbert space is exactly what we would get if we assumed each possibility is a point in a 4D continuum.  Thus, we still have an uncountably infinite number of dimensions, albeit a much larger uncountable infinity.  Regardless, we can still represent the wave function as a vector in this new Hilbert space which decomposes into projections along each dimension.  The square of the possibility amplitude in this example will give the probability of measuring the quon at a point in a 4-dimensional continuum.  Both these examples are representations of the same thing, except that the first example required a spinning quantum vector to represent all possible instants, whereas the second example required only one quantum vector, which represents the entire quantum state from a higher dimensional perspective.  The upshot of this argument is that any particular morphing field of possibilities can be represented, alternatively, by a single stationary vector at a higher dimensional level of mathematical abstraction.

           We have also seen that one quantum vector in Hilbert space looks exactly like every other, namely a unit vector which has a absolute magnitude of one.  Indeed, it is merely our choice of which attribute we want to measure that determines the probability distribution of all unrealized possibilities.  The quantum vector can only be analyzed by choosing a specific frame of reference.  In this way a quon’s tendencies to exist in a certain state are inseparably determined by how we choose to observe the system.  Since all quantum vectors in Hilbert space are essentially the same, and since the only perceived difference is a result of different possible choices of a frame of reference, it appears safe to say that there may only be one quantum entity.  This is indeed the basic assertion of quantum theory, which utilizes one basic description for all possible quantum entities, namely an abstract wave function, Ψ, and a particular reference frame. Let’s suppose, for fun, that there is only one fundamental quantum entity.  Outside the context of a measurement situation, it is meaningless to say anything about this entity.  However, once we define a frame of reference, we then can derive the basic characteristics the Ψ-wave, which represents a specific attribute, or quality, of the one quantum entity.  I feel that it might be useful to introduce a new concept to the existing version of quantum theory. As before, we understand Ψ to be an abstract field of possibilities within a given reference frame.  Now, let us introduce a new symbol ,  which we shall define as the field of all possible reference frames.  Whereas Ψ is a representation of the quantum entity given one specific attribute reference frame,    is a broader representation of the quantum enitity given all possible attribute reference frames.  This concept is somewhat analogous to putting an arbitrary wave through all possible waveform family prisms at the same time.  I not sure if this is actually possible, or what the actual result might be; however, I feel that the basic idea could be handled simply my modifying the existing structure of quantum theory.

           For example, each possible reference frame could be represented by an independent dimension in some new type of space for which we have no name.  Obviously, there are an infinite number of possible reference frames, and thus this symbol truly represents an infinite-dimensional field, which includes all possibilities.  Whereas the coordinate values of Ψ are represented by complex vectors, the coordinate values of   could be represented by vectors in Hilbert space; that is, each dimension in our new space represents a possible Hilbert space.  If each specific frame of reference is a sense, i.e. context, then the field of all possible reference frames is truly the broadest sense.  In general, we could say that by itself,   is completely undefined, and at the same time,    assumes all possibilities at once.  Any particular  Ψ-wave is generated simply by slicing this infinite-dimensional field with a lower-dimensional reference frame.  This idea of slicing is a metaphor used for the creation of level-sets, which are lower- dimensional projections of a higher-dimensional object.  In other words, all morphing quantum fields of possibility are created by reflecting   at different angles.

           Each angle of perception constitutes it’s own frame of reference.  In reality, all possible reference frames, or dimensions, are realized simultaneously.  However, as a result of our ordinary mode of human consciousness, we specific ego-centered entities only perceive reality along one dimension at a time.  If one is able to broaden his perception to include multiple reference frames, it is possible to experience reality along more than one dimension at a time.  This is by no means a rigorous treatment of the concept in question; however, it is purposed simply because it is interesting to consider such claims given our current exploration into the unknown. 

           In addition, it will also be said that this idea, outlined above, will not work unless we assume that the ultimate source of all creation is right now.  This claim seems justified in a number of ways.  For instance, who has ever had a real experience of the future or the past anyway?  Every experience of reality is always in the now.  The past and the future can only be represented by fields of possibility; however, at every now, only one thing is actually happening.  The idea, that only now exists, is consistent with experimental fact because every type of quantum measurement yields only one actual event.   It is my claim, that from the perspective of an infinite-dimensional field of all possible reference frames, everything in space-time already exists right now.  From the perspective of a lower-dimensional reference frame, events appear to be separated by space and time.  

           Another claim, which I feel is justified, is that the existing formulation of quantum theory applies to all entities regardless of their size.  Quantum theory was discovered in the realm of atomic and sub-atomic particles because at these scales of reality, the effects of quantum waves become dramatically obvious.  It is generally assumed that at a certain limit, the quantum laws converge to the normal everyday laws of ordinary experience.  This may be the case for many types of attributes which physicists are preoccupied with, but in no way does it rule out the possibility that there may exist presently undiscovered quantum relationships between macroscopic entities such as humans, plants, star systems, or ant colonies.  

           Logically, quantum theory applies to all things primarily because everything is made from the same stuff.  It is all woven from the same fabric.  It should be obvious that there is no natural division between the realm of a super-cluster of galaxies and the realm of a bunch of quarks.  However, macroscopic entities such as humans and stars are not made of atoms, nor are they made of quarks.  In general, it seems that everything consists of frequencies of energy-mass; however, on an even deeper level, these frequencies don’t exist unless we first define a reference frame.  Therefore, it appears that the ultimate stuff of reality is simply pure infinite possibility

           As noted earlier, quantum entities are necessarily whole beings.  Obviously, physical scientists have been able to detect atomic and sub-atomic phenomenon; however, I would argue that as opposed to being made of such building-block like parts, each quantum whole is a hyperdimensional complex within which reside lower-dimensional wholes.  At the same time, each whole is embedded in a broader context of an even higher dimension.  In other words, the fields which organize individual quarks are contained within broader sense fields which organize individual atoms.  Atom fields, in turn, can be represented within molecular fields, which can be represented within cellular fields.  In this way, we can easily conceptualize human fields, collective species fields, planetary fields, star system fields, and galactic fields.

Modern travel-egg:
used primarily for interdimensional
trips to parallel universes

Bohm, David. Wholeness and the Implicate Order. Routedge London and New York, 1980.

Herbert, Nick. Quantum Reality: Beyond the New Physics.  Doubleday, 1985.

Von Neumann, John. The Mathematical Foundations of Quantum Mechanics.  Princeton       

University Press, 1955.

Zukav, Gary. The Dancing Wu Li Masters.  Bantam New Age. 1979.

http://plato.stanford.edu/entries/qt-measurement/

http://www.qedcorp.com/pcr/pcr/crowell/qmchaos.htm

http://people.deas.harvard.edu/~jones/ap216/

lectures/ls_2/ls2_u1/dirac_notation/dirac_notation.html

http://tph.tuwien.ac.at/~oemer/doc/quprog/node5.html