On the origins of the quantum theory

"Reality is that which, when you stop believing in it, doesn’t go away.”
-Philip K. Dick

The history of quantum mechanics spans a little over 100 years, with a number of notable discoveries from 1801-1900 that contributed to the foundational principles upon which the theory has been developed. For this discussion we need only highlight a small selection of some of the most notable landmark discoveries since the turn of the century (including Young’s double slit experiment) which contributed to the development of the quantum theory; A more comprehensive time line is available here.

  • Thomas Young’s double slit experiment (1801) demonstrates the wave nature of light.
  • Max Planck explains blackbody radiation (1900) by postulating that the energy of electromagnetic radiation is emitted by a blackbody in discrete packets, i.e. it is quantized.
  • Albert Einstein uses Planck’s constant to explain the Photoelectric effect (1905) stating that the energy of a beam of monochromatic light arrives in discrete packets of energy (photons).
  • Niels Bohr and Ernest Rutherford propose that the atom consists of a positively charged nucleus orbited by electrons (1913) in discretely separated orbital energy levels which are quantized.
  • Otto Stern and Walther Gerlach demonstrate that atoms possess an intrinsic spin (1922) which is quantized.
  • Samuel Goudsmit and George Uhlenbeck demonstrate that electrons possess an intrinsic spin (1926) which is quantized.

In 1935 Einstein Podolsky and Rosen challenge the completeness of quantum mechanics under the assumption that local realism is valid; this is the famous EPR thought experiment. In the same year Eriwn Schrödinger develops his Schrödinger’s cat thought experiment to challenge the Copenhagen Interpretation of quantum mechanics. For the duration of this discussion we consider the quantum theory to be composed of two parts. Quantum mechanics before 1935 which is here referred to as pre-EPR, and quantum mechanics after 1935 which is referred to as post-EPR.

Blackbody radiation and Planck's constant

The fundamental development that gave birth to the quantum theory can be traced back to 1900 when Max Planck postulated that the energy of the light field is not continuous; rather it is quantized. This discovery triggering the birth of quantum mechanics, which is today one of the central pillars of modern physics. In his discussion of blackbody radiation, Max Planck considered a large number of mono-chromatically vibrating resonators - N of frequency ν (per second). The total energy of the N resonators is E.

".... the most essential point of the whole calculation - E to be composed of a well-defined number of equal parts and use thereto the constant of nature h = 6.55 × 10 −27 erg sec. This constant multiplied by the common frequency ν of the resonators gives us the energy element in erg,”
-Max Planck

In making this statement Max Planck showed that the electromagnetic radiation emitted by a blackbody is done so in discrete intervals. These intervals are packets of energy known as photons, and their energy is quantized. Similarly the energy of the electromagnetic field is not a continuous spectrum, rather it is separated by discrete measurable intervals. As the frequency of light changes the photon energy changes in discrete steps described by Planck’s constant. This statement is often poorly quoted using the formula, E=hf, with E representing the energy, h being Planck’s constant and f is the frequency of light. In the presentations of these concepts using the E=hf formula, one might mistakenly assume that f is a continuous variable and therefore E is also a continuous variable. The essential point of Planck’s hypothesis is that neither the frequency of light f, nor its energy E are continuous variables.

In discovering the quantization of the electromagnetic field Max Planck resolved an ancient paradox known as Zeno’s paradox; which is worth exploring just for fun.

Zeno's Paradox and Planck's constant

A train traveling to its destination (figure 1) must complete the distance from A to B, which is in this example set to 100 meters. Before completing its journey it must pass the first half-way point: 50 meters. Then it must pass the second half-way point, leaving 25 meters remaining. Then again it must pass the third half-way point, leaving 12.5 meters remaining. This continues ad-infinitum. The paradox arises when we realize that in order for the train to reach its destination it must cross an infinite number of half-way points.

Zeno’s Paradox: Since there is no possible distance that cannot be divided in half, the train will never reach its destination. The logic of this argument is broken down into three parts:
(a) It is necessarily impossible to complete an infinite set of tasks.
(b) There is no possible distance that cannot be divided in half. As a result there always remains a “half-way mark” which must be passed.
(c) An infinite amount of half-way marks cannot be passed. Therefore the path can never be fully completed, and the train never reaches its destination.
If (a) and (b) are true, then (c) necessarily follows. Zeno concluded: “therefore, motion is impossible.”

From experience we inherently know that motion is possible. In fact the only thing constant is change. Zeno’s paradox demands either a reassessment of our concept of motion, or a reassessment of our concept of physical reality. Under the assumption that Planck’s hypothesis is valid, we propose that Zeno’s conclusion - motion is impossible - is unsatisfactory. It is unsatisfactory to simply state that motion is an illusion, as in that instance we require a proper definition of what motion is and furthermore why our experience of motion is only apparent.

Zeno’s Paradox resolved: Physical reality is not infinitely divisible. There is a base unit, i.e. a Planck’s constant. As the train is nearing its destination, it has to travel over the half millimeter point, half-half millimeter point, half-half-half millimeter point, etc. Since there is a base unit this division does not continue infinitely. The progression of half-way points is arrested when the base unit length is reached. As there is no meaningful “half-base unit” - at least not represented in physical reality - the train can arrive at the station without having to cross any more half-way points.

The logic of the argument for “motion is impossible” is flawed since (b) is false and it necessarily follows that (c) is false, therefore the train arrives at its destination. Applied to physical reality, the Planck scale states that time, space and energy all have a fundamental base unit; Known as the Planck time, the Planck length and the Planck energy. These are the smallest non-divisible units of physical reality.
The Universe is not infinitely small.

Reality as a symphony

Planck’s constant and the resolution of Zeno’s paradox shows that the electromagnetic field is not a continuous spectrum, rather it is discrete. This means that changes in energy, frequency or wavelength happen in discrete small steps. This can be understood in musical terms - as the musical scale is divided in intervals called semi-tones. In physical terms we say that objective physical reality is written in discrete intervals defined by the Planck scale. In music, the octave spans 12 semi-tones and there are 12 major scales. Just as the conductor conducts a symphony in a chosen key - physical reality is analogously written and conducted in a key. All that exists are geometric forms and vibrations around the number 0. The “musical key” of physical reality is written in the language of mathematics and geometry, meaning that it can be mapped. The laws of Physics can be objectively known.

As a scientific discipline “physics concerns what we can say about nature” [Niels Bohr]. Physics is concerned with uncovering the mathematical laws of nature which accurately account for our observable physical reality. Physics - as a natural science - is not concerned with the study of the symphony itself or the search for its meaning. This would be a primary function of other schools of thought. Physics seeks to unfold the mechanical laws of physical reality. As Nobel Laureate Niels Bohr explains;

EPR and hidden variables

There is a disparaging difference between quantum mechanics of today and quantum mechanics in the original meaning of the science. The word “quantum” means “quantity”, and in the original formulation of the science the name “the quantum theory” referred to the fact that measurable aspects of reality such as energy, space and time are discrete. The present theory is less concerned with this original discovery and more concerned with superposition states, non-locality, the Copenhagen interpretation of the quaternion and the infamous quantum computer - which they would have us believe is right around the corner, possibly even in time for the Christmas market!

The original difference between the two versions of the quantum theory - modern and original can be classified (to a good approximation) as quantum mechanics pre-EPR and quantum mechanics post-EPR. The acroynm EPR stands for Einstein-Podolsky-Rosen and refers to a 1935 paper written by the 3 authors entitled “Can a quantum mechanical description of reality be considered complete?” Therein the authors addressed concerns reflected by many scientists of those days with regard to the trends of the theory’s development. The essential conflicts of reason highlighted were that; “In a complete theory there is an element corresponding to each element of reality.” According to the quantum theory there is not an element corresponding to each element of reality. The description of physical reality offered by the quantum theory - both pre-EPR and post-EPR - is non-deterministic and therefore incomplete.

The quantum theory professes that there is no objective reality - that reality is observer dependent. The problems with the interpretation of the quantum theory is known as the Measurement Problem and is associated with the collapse of the wave-function - which is famously summarized in the Schrödinger’s cat paradox.

In a nutshell there is a vast difference between theoretical and experimental knowledge. The mathematics of quantum mechanics cannot deterministically account for the results of experiments in the quantum realm; on atoms, photons, electrons and so forth. The example chosen by the authors to highlight this disparaging difference is the entangled state; The entangled state is a pair of atoms or photons which have been experimentally shown to exhibit “non-local” correlations, at arbitrary distances. This means that entangled pair have an instantaneous connection which is “faster that light”. Einstein referred to this as “spooky action at a distance”. The quantum theory offers a non-deterministic description of the entangled pair. This is known as the superposition hypothesis which purports that the particles are suspended in a superposition of all possible observable states at the same time, until an observer collapses the wave-function into a single definite state through a measurement.

In the case of the entangled pair, a measurement of one particle “collapses the wave-function” to exactly determine the partner particle’s state. This is independent of spatial distance and happens instantaneously, “faster than light”. The EPR argument against quantum mechanics is that either;

  1.  the description of reality given by the wave function in quantum mechanics is not complete, or
  2. these two quantities cannot have a simultaneous reality.

Modern experiments on entangled pairs of photons and atoms have reached a level of sophistication which shows that within a reasonable level of certainty the entangled pair does have a simultaneous reality; This lends to the suggestion that point 2 above is incorrect. Therefore we are left with point 1; quantum mechanics is incomplete and there exists hidden variables unaccounted for by the theory.

Today it is known that there are hidden variables in both quantum mechanics and classical mechanics; The hidden variables were discovered by Wharton&Koch and defined by yours truly. The hidden variables are hidden dimensions. We cannot see these hidden dimension directly, but we can see their shadows i.e. their 3-dimensional projection. The entangled pair is understood as being a single multi-dimensional object. What is observed in 3-dimensions as 2 separate particles exhibiting “non-local” correlations and “spooky action at a distance” is in fact an illusion derived from observing a multi-dimensional object from a 3-dimensional perspective. The entangled pair is in fact the 3rd-dimensional shadow of one and the same multi-dimensional object. As John Stewart Bell explains;

John Stewart Bell is a Belfast born physicist who worked in the atomic energy research establishment, CERN and Standford university. The experiments in CERN come under the heading of “quantum chromodynamics” and are centered on the study of the strong nuclear interaction. This is 8-dimensional physics. Modern experiments in quantum mechanics on entangled pairs of photons and atoms is 16-dimensional physics.

FAPP and quantum computing

Academic physics is - in its natural form - the study of reality without concern for technological applications or market value. That is to say that academic physics is primarily concerned with theoretical and experimental explorations without a need for practical application. While industrial applications of physics research is certainly an important aspect of scientific development, it works optimally when existing alongside an equally strong academic field. Academic studies would be focused on the development of fundamental physics, without any need or concern for practical applications. It is a journey into the unknown.

John Bell was outspoken about the magnitude of uncertainty present in our modern under-standing of the quantum realm. This has lead many researchers to abandon the pursuit of a complete physical theory, subsequently turning their attentions to finding practical applications for their research. Notwithstanding the quantum computing rabbit holes thus far explored, current approaches have had a measure of success For All Practical Purposes (FAPP) - but perhaps there is more to the story. Perhaps we can do better. John Bell suggests that academic studies would be better served if the focus of theoretical developments were centered on the formulation of a physical theory which supersedes practical needs.

“Suppose that when formulation beyond FAPP is attempted .... Would that not be very, very interesting?”
-John Bell

While modern quantum mechanics is just fine FAPP, it claims that there is no objective physical reality. That reality is observer dependent. This reasoning is found in the quantum theory of particle spin, which rests on a Copenhagen Interpretation of the quaternion. A Schrödinger’s cat hypothesis. The confusions surrounding this logic has been compounded to the point where researchers today are scrambling to build a quantum computer - A hypothetical quantum-logic device which promises to do every possible calculation in parallel.

Discovered in 1843 by William Rowan Hamilton, the quaternion is used in industry today for all automated protocols which concern rotations in 3-dimensions; robotics, computer graphics, computer vision, virtual reality, aeronautics, kinematic measurements, etc. The quaternion is very well known to modern physics, and in no instance is there a demonstrable application of the Schrödinger’s cat hypothesis. The Copenhagen Interpretation of the quaternion is a logical fallacy.

Claims of having built a quantum computer has emerged from different quarters. These have been contested as not being genuine quantum computing, as in no instance has Shor’s algorithm or Grover’s algorithm been demonstrated. D-wave have built a quantum annealer - purchased by google for $20milion which is contested not to be a genuine quantum computer. The latest buzz word to emerge from the community is quantum supremacy, which google intend to demonstrate by the end of 2017. Whether or not something emerges from this research, one thing is now clear; no one is going to build a quantum computer using a Copenhagen Interpretation of the quaternion. Shor’s algorithm is unfeasible as it rests on a Schrödinger’s cat hypothesis of the quaternion.

The development of the modern computer was made possible through 2 key developments; 1) the gramophone (information storage) and 2) Boolean logic (information processing). It may be the case that binary computing be someday outperformed by a next generation of computing power, but this will not be the quantum computer. If binary information were to be superseded, then it would likely be derived from an information processor which is already in existence, DNA.

One might like to think that the Copenhagen Interpretation of the quaternion is the only issue with the quantum theory however this is not the case. From my keyhole perspective, some of the major issues of the quantum theory include;

  1. Everett’s many worlds interpretation is a poorly defined seductive fantasy.
  2. In the double well, the tunneling frequency is not equal to the integral overlap of 2 Gaussian functions centered on each site.
  3. The creation and annihilation operators of the spatial modes do not exist - they are not defined.
  4. The Bosonic Field Operator is just the density.
  5. Operators and observables ... why not use a more natural language of matrices; like eigenvectors and eigenvalues?
  6. The qubit is not a 2-level system; The qubit is a unit quaternion. The Copenhagen interpretation of the quaternion - and all related formalisms of the Copenhagen Interpretation - has no place in any respectable scientific discipline.
  7. Continuous variable quantum mechanics - The whole point of the quantum theory is that there are no continuous variables. Reality is discrete, coming in small steps “quantities”, this is where the word “quantum” comes from. There is a fundamental pixel; of energy, of space, of time.
  8. In the Lo Schmidt Echo, I have a question; “Why do you project on the evolved state - why not project on the initial state?”
  9. The Dirac ket is a vector with complex entries. The ket is not a magicians hat from which one can pull all forms of rabbits and cats from.
  10. 1 divided by the square root of 2. How is it possible that a 4 component complex vector, with 2 entries set equal to 0, and 2 entries set equal to 1 divided by the square root of 2, corresponds to a maximally entangled state?

The end of quantum mechanics

Quantum mechanics as a physical theory is incomplete as there are indeed hidden variables, and these hidden variables are found in the parameter space of the unit quaternion. The hidden variables of quantum mechanics are defined in “The Hopf-Fibration and Hidden Variable in Quantum and Classical Mechanics”, which has been twice rejected for review by J. Phys. A (June of this year and last); citing that the hidden variables are not of scientific interest. Given the absence of a suitable journal to publish this article in, I have taken it upon myself to source 8 referees which have reviewed the article - in minute detail - and subsequently recommended it for publication in Meditations On Geometry. The report of their review is now publicly available on YouTube and Vimeo.

In light of the fact that the qubit is a quaternion, it is the position of this journal that a complete revision of the quantum theory be carried out. A commission would be established to oversee this process. The primary function of this commission is to ascertain what claims made by the quantum theory post-EPR are feasible and what claims are not. A period of 5 to 10 years would be needed to review and re-categorize every single article that has been published by the quantum theory post-EPR. This will allow for a separation of the diamonds from the rough so to speak; and to highlight the successes of the quantum theory and to identify the failures of the theory. At a conservative estimate approximately 95% of the literature in quantum mechanics is now redundant.

The commission might proceed in the de-construction of quantum mechanics as follows; The coherent states of the harmonic oscillator are internally consistent and well defined, so they would be brought forward into the new paradigm of academic research. Whereas the orbital and spin magnetic moment operators and the related Clebsh-Gordon coefficients are artificial in their construction - as the laws of their algebra was not deduced from mathematical reasoning but rather were artificially constructed to fit with the results of experimental measures. So they would be removed, and so forth. In this way the quantum theory is de-constructed piecemeal, removing those misuses of Hamilton’s quaternions and other non-deterministic conclusions within the theory in order to pave the way for a natural mathematical theory of the fundamental processes.

The scientific method and the objective truth

Physics is a natural science which seeks to deduce the laws of nature by establishing 'objective truths' through experimenting with physical systems to test hypothesis on the nature of reality. Knowledge is achieved by uncovering the mathematical laws which account for the observable physical processes.

An observable physical process could be a state of matter such as; a solid, liquid, gas, a Bose-Einstein Condensate, a plasma, or perhaps a ferro fluid. The mathematical laws may be described by thermodynamics, fluid mechanics, classical mechanics, electrodynamics, and so on.

Scientific knowledge is in constant flux, and at any instant in time our perspective of reality has a certain resolution. The degree of which depends on current knowledge and current practices - a development of knowledge or a shift in practice the resolution of our perspective can change. "Physics concerns what we can say about nature" [Bohr].

This little note is written to draw attention to the peer review process and the future of the natural sciences. The topics of interest are the scientific method and the objective truth. At the out set it is of interest to ask the following question.

Is there any such thing as an objective truth?

This is an interesting inquiry as - in the first instance we are using language to define 'objective truth'. This is assumed to be logically consistent as it is assumed that language is itself an objective truth. But language is in constant flux - it is ever changing, and the meaning of phrases and expressions can be dependent on tone, location and context. By contrast the exact sciences, that of mathematics and geometry, are clearly defined languages which offer an unprecedented degree of precision in their expression. They permit the precise expression of an observable hypothesis -, whereas the same cannot be said for language. It is questionable to use language to define 'objective truth' because the 'objective truth' of language is on shaky foundations.

"We must be clear that when it comes to atoms [and nature], language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images and establishing mental connections. "
 - Niels Bohr

Using mathematical equations we can make objective statements about physical reality; The mathematical laws are themselves 'known with certainty' and therefore are used to form an accurate and balanced perspective. Applied to physical reality the mathematical laws are 'not necessarily absolute truths' - they would describe an aspect of reality but not the whole. The laws of physical reality are written in the language of mathematics and geometry;  "For whether I am awake or asleep, two and three added together are five and a square has no more than four sides." [Descartes]

"The world is not presented to the reflective mind as a finished product. The mind has to form its picture from innumerable sensations, experiences, communications, memories, perceptions. Hence there are probably not two thinking people whose picture of the world coincides in every respect."
 - Max Born

The scientific method

The scientific method is simply a set of guiding principles which map one possible path toward a goal; That goal is to clearly uncover the laws of nature. This pursuit requires no less than a process of removing all personal bias, and the development of critical thinking faculties. With these in place the scientific method unfolds as we;

  • 1. Observe a fundamental process in nature or some property of the universe.
  • 2. Create an hypothesis to account for the phenomena observed.
  • 3. Use the hypothesis to make predictions.
  • 4. Test those predictions by experiments to gauge and develop the accuracy of the hypothesis.
  • 5. Extract the objective truth from the account of the observed phenomena.

The goal of modern physics is - as one would think - to define the key in which reality is written. Nowadays people are scrambling to build a quantum computer. This whole quantum computing malarkey looks to me that we have an "if its not broke don't fix it" situation on our hands. What is wrong with everyday classical computers? - they work just fine. Who even cares about a quantum computer, This thing that will supposedly do every possible calculation in parallel while operating on non-deterministic logic, .... pardon my disbelief,.. I genuinely feel like the child telling the masses that the emperor has no clothes. This quantum theory you are bandying about is non-sense, the emperor is naked!

The scientific method and the ballistic equations of motion

Any line of inquiry has the potential to unfold the entire universe, and asking the right questions is an art. There are many gates through which one can enter in the process of uncovering the unknown.
A question as simple as;
How do I truthfully describe the trajectory of a stone, thrown with a momentum P at an angle ϴ to the plane on the surface of the earth?
- and our journey toward uncovering the unknown has begun.

"All direct experiences are absolutely valid but are subjective ... the exact sciences presume to aim at making objective statements but they surrender their absolute validity. ... Relative measures take the place of absolute impressions. [While sometimes] a new doctrine upsets all the ‘old facts’ ... the regions of ‘known with certainty’ are growing, [relieving] the pain which arises from solitude of the spirit, and the bridge to kindred spirits becomes built."
- Max Born

We postulate that the 3 ballistic equations of motion truthfully describe the trajectory of a stone - and we wish to test this hypothesis.
The scientific method can be used to guide our testing and to help uncover the objective truth contained within; as follows

  • 1. We observe the path of a stone thrown with a momentum P at an angle ϴ to the plane on the surface of the earth.
  • 2. We hypothesize that the 3 ballistic equations of motion truthfully describe the path of the stone (figure).
  • 3. We create prediction models for the paths of different shaped objects, with different masses, angles of incline, different heights etc.
  • 4. We experimentally test the predictions; with varying experimental conditions - different weather conditions, altitudes, atmospheric pressures, temperature etc.
  • 5. The experiments reveal that the 3 ballistic equations of motion truthfully describe the path of the stone - to a degree of accuracy.
     - There are are demonstrable deviations from prediction caused by, but not limited to - the density, rotation and spin, the geometric shape, wind and aerodynamics, etc.
     - It is concluded that the ballistic equations truthfully describe the trajectory of a stone; with the understanding that it is not the whole truth, but a half truth.

The "objective truth" of the stone's path is known, since "we can all agree" that the ballistic equations of motion truthfully describe the stone's trajectory - with the acknowledgement that they are but a half truth, as other factors are at play.

"The development of the exact sciences leads along a definite path from this state to a goal .... of creating a picture of nature .... for the purpose of depicting the sum of all experiences uniformly and without inconsistencies."
 - Max Born

The scientific method is a process which attempts to separate the bias of being an observer from the observation — letting ‘reality’ show itself in the simplest possible way.

"The world is not presented to the reflective mind as a finished product. The mind has to form its picture from innumerable sensations, experiences, communications, memories, perceptions. Hence there are probably not two thinking people whose picture of the world coincides in every respect."
 - Max Born

The scientific method and the peer review process

Some resistance is to be expected toward a work which has claimed to have defined the hidden variables of quantum mechanics, and is subsequently is calling into question the entire theory. This is no surprise.

The global phase of the qubit was first proposed as a candidate for the hidden variables of quantum mechanics in an article entitled "Unit quaternions and the bloch sphere", published in J. Phys. A. Given that the original hypothesis is published there - one would think the closed form solutions to the hidden variables would be of interest to this journal. One would think J. Phys. A would fulfill expectation and see that this article be appointed to the relevant referees for review. Alas,  the girls and boys at J. Phys. A have wavered all responsibility of meeting these claims with a judicial peer review process.

Given the absence of a suitable journal available for peer review - I have assumed the responsibility of seeing that this article is refereed myself. And I have already found a number of suitable referees among the lads. At the present moment the article is being examined and the results of the review process will be made available here shortly.

"Some subjects are so serious that one can only joke about them."
-Niels Bohr


[1] Max Born, "Einstein's Theory of Relativity" Courier Corporation, (1965).

The story so far ....

 .... and a note on the hidden variables of quantum mechanics.

Believe it or not, the hidden variables of quantum mechanics have been discovered! You'll be delighted to know they were dancing in our faces all along! Thoughts of "how did we miss it?" arise when it is learned where the hidden variables are found. There is a touch of irony to it as well, as the so-called "hidden" variables are found in that so-called "negligible" phase of the qubit! Alas we failed to pay attention - as we were caught up in our confusions trying to build a quantum computer - and all the while the "negligible" phase wryly smiled. Not to worry! The important thing is that the hidden variables have been discovered - and now we can do something about it.

And where are they? - you may ask. Why, they are found in that arbitrarily discarded phase, the one that is oft thought of as being an insignificant and negligible gauge, never before has it received much love - yes that's right the hidden variables are found in .... [drum roll please] .... the global phase!!

So what does it all mean?

It means that the famous 2-level quantum system, commonly known as a qubit, is in fact a unit quaternion. The qubit is not 2-dimensional, it is 4-dimensional.

Read all about it in: arXiv:1411.4999

Wharton and Koch, "Unit Quaternions and the Bloch Sphere" J. Phys. A: Math. Theor. 48 (2015) 235302

In a nutshell: Wharton & Koch have shown - using a projection between 4D and 3D space known as the Hopf-Fibration - that the global phase of the qubit is "curled up" as a fiber bundle in the lower 3-dimensional space. The 4th-dimension of the quaternion is encoded in the global phase of the qubit, which is currently not studied in quantum mechanics. The global phase parameterizes the unit circle which is a fiber bundle connecting the 4D space of the quaternion with the 3D space of the Bloch sphere. In this way the 4th-dimension of the quaternion (the global phase) is present in the 3D kinematics as a natural hidden variable.

"While a unit quaternion Q is effectively a point on a 3-sphere, a qubit ψ is often represented as a point on a 2-sphere (the Bloch sphere). Such dimensional reduction results from ignoring the global phase .... A more elegant method for finding the Bloch sphere unit vector R without passing through the spinor representation is to generate a unit pure quaternion Q (with no real component) and then map Q directly to R in Cartesian coordinates." [W&K]

So what's the story with the quaternion?

Historically speaking the quaternion was discovered by William Rowan Hamilton in 1843. At the time it was well known that the complex numbers describe rotations in 2-dimensions - the question on Hamilton's mind was "what extension of the complex numbers describes rotations in 3 dimensions ?" It took Hamilton some 10 years to find the solution to his question: and the answer is the quaternion. In discovering the quaternion, Hamilton made one of the greatest scientific advances since Galileo's discovery that the Earth revolves around the Sun. The shocking truth is that a 4-dimensional "complex" number is required to describe rotations in 3-dimensions. This means that our 3-dimensional reality is rooted in 4 spatial dimensions. Hamilton was so taken by this realization that he immediately carved the fundamental law of quaternion multiplication on a stone of Brougham (Broom) Bridge in Dublin. This is where he made his discovery while he was walking with his wife along the Dublin canal to a meeting of the Royal Irish Society. Today this discovery is commemorated by a plaque unveiled by Taoiseach, Eamon de Valera, in 1958.

"And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples .... An electric circuit seemed to close, and a spark flashed forth." [William Rowan Hamilton]

The quaternion is a 4-dimensional vector, which describes rotations in 3-dimensions in full generality. The quaternion has 1 "real" and 3 "imaginary" components which are parameterized by time. The 3 "imaginary" components are the ijk axes of the quaternion which describe the kinematics of our familiar 3D space. The "real" component is the fourth dimension of the quaternion which has a somewhat different quality than the 3 familiar spatial dimensions. However the "real" component of the quaternion is just as much a spatial extension in 4D space, as the remaining 3-dimensions - it is not to be confused with the parameter time. If one were to pursue a space-time extension of the quaternion, it would be the 4+1 space-time known as the Kaluza-Klein theory.

The quaternion is not to be under estimated.

W&K have discovered that the global phase is a fiber bundle connecting 4D and 3D space and therefore a natural hidden variable of quantum mechanics. In addition we have learned the qubit is a quaternion. These discoveries are very significant for modern physics and they deserve pause for acknowledgement and reflection. The quaternion does not reveal itself to just anybody. These are fantastic results and herald the end of non-deterministic science, quantum computing and the quantum theory. Personally I could not be more relieved.

It may seem like the discovery of the hidden variables is a case of not being able to see the woods for the trees - and in some respects this is exactly how it is. Kind of - I mean we talk about hidden variables and in the same breath we routinely disregard the global phase and then introduce a superposition principle. All without proper justification or logical reasoning - and God forbid you would question these motives in a room full of quantum mechanics, it simply would not work in your favour. One would be much better off hightailing it out of there. Alas, irony does have a sense of humor as the hidden variables are found in that parameter we have been habitually and confidently casting aside. Sure we have no one to blame but ourselves. 

In any other parallel universe the practice of haphazardly ignoring parameters in equations would be a cringe worthy error of modern physics, most certainly. But not in this one - here we can blame the quaternion for being far too gangster altogether. Sure the quaternion is a slippery bugger - that much has to be acknowledged - it is notoriously elusive. There is a certain aura of mystique surrounding it and even a little fear of the unknown. When it comes to the quaternion you will need a sharp sense of awareness and the full use of your logical faculties. Savage demands altogether.

We use the quaternion in many instances without actually recognizing that we are. The quaternion is known to us by many different names and titles; it is not just the qubit that is a quaternion - for example:

A quaternion by any other name would smell as sweet.

These are to highlight but a few. Often it is said that the quaternion is more efficient than the Euler angles in describing rotations in 3D as they avoid the problem of gimbal lock. However these are one and the same object - the Euler angles are quaternions.

The quaternions are 4-dimensional and they describe rotations in both 4D and 3D in full generality. In the 3-dimensional picture, the shadow of the 4th-dimension of the quaternion is present in the 3D rotation as a fiber bundle consisting of the global, geometric and dynamic phases. These are the hidden variables of quantum mechanics.

What is the meaning of the hidden variables?

Applied to the fundamental particles, the S1 fiber bundle encodes the intrinsic spin of the bosons and fermions. This may come as a surprise to many as conventional "quantum mechanical" wisdom states that

  • SU(2) is the spin-half group (fermions), and
  • SU(3) is the spin-one group (bosons).

However one must necessarily realize that

Therefore we have a contradiction - either SU(3) is the spin-boson group or it describes the quarks, we cannot have both. To resolve this conundrum let us refer to the present theory of spin, as per quantum mechanics.

What's the story with the Copenhagen Interpretation of the quaternion?

It is known from the Stern-Gerlach experiment that the spin-half particles have 2 allowed spin states, spin-up and spin-down, which are mutually exclusive. The particles are measured in either one spin state or the other, never both.

In quantum mechanics the spin-half particle is described by the qubit, which is a quaternion expressed as a complex valued vector equation whose basis states are the allowed spin-up and spin-down states. The coefficients of the qubit are known as probability amplitudes, as their magnitude is assumed to give the probability of measuring a particle in one or the other state. 

According to the Copenhagen Interpretation of the qubit, the spin-half particle is expanded in a superposition of the 2 possible spin states. That is to say that the particle exists in both the spin-up and spin-down states at the same time. Only following a measurement by an observer does the superposition collapse to return one of the spin values with a probability described below.

This is the Schrodinger's cat paradox applied to particle spin, and the current state of the art in the quantum information sciences. Of course there are other flavours of interpretation within the quantum theory but effectively they all amount to the same thing. Nature is non-deterministic, God does play dice and above all the observer is the center of the universe - without which nature could never make up her mind!

W&K have illustrated quite clearly that the qubit is a quaternion, and for any rational thinking scientist this brings the Copenhagen Interpretation of quantum mechanics clearly back into focus. One could begin with the Copenhagen Interpretation of the qubit, and move forward by removing all irrational conclusions within the quantum theory. One might even go so far as to question the entire gambit of quantum logic, or illogic as the case may be. Personally I would go for a complete overhaul of the entire theory of quantum mechanics. But that's just me - I'm thorough like that. I can understand that this is quite daunting and difficult to face but rest assured - it will be worth it in the end.

The current situation in quantum mechanics amounts to asking the following question, is there any validity in a Copenhagen Interpretation of the quaternion? Of course this is a rhetorical question - the Copenhagen Interpretation has no place in any mathematical theory, nor has it any place in differential geometry, or any other conceivable logical theory of physical reality.

The Copenhagen Interpretation of the quaternion is at the core of the measurement problem of quantum mechanics. It is a non-deterministic interpretation of deterministic equations, which has caused the appearance of probability in equations which are otherwise completely unrelated to probability theory or statistics.

"The appearance of probability is merely an expression of our ignorance of the true variables in terms of which one can find casual laws." [David Bohm]

This over sight of the quantum theory - known as the Copenhagen Interpretation - is derived from ignoring the transverse modes of the quaternion. In quantum mechanics the study of the 2-level system is the study of the longitudinal mode of the quaternion. The longitudinal mode is the principle axis of vibration and describes the rabi oscillations of the quaternion. While it is certainly important and has "worked" (for lack of a better word) to date - it is only a 2D slice of the 4D quaternion. Quantum mechanics is incomplete.

Non-determinism has been a point of contention since the birth of quantum mechanics, but it wasn't until 1935 that the hidden variables were first proposed to exist. In what has subsequently become known as the EPR paradox the first probing questions concerning whether or not nature could in fact be multi-dimensional were explored in a paper by Einstein, Podolsky and Rosen entitled: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

This question has been answered by W&K when it was shown that under the hopf-fibration the global phase of the quaternion is a fiber bundle linking the 4D and 3D spaces, and a natural hidden variable of quantum mechanics. The present theory is incomplete as there are indeed hidden variables found in the parameter space of the quaternion.

A complete and deterministic description of physical reality requires the known degrees of freedom from the quantum theory - the 2-level quantum system - in addition to the hidden degrees of freedom, which are the neglected 2-dimension of the quaternion. In total we need the existing 4-dimensions of the quaternion. That is all, and from this geometrical object we can once again pursue a complete and self-consistent picture of physical reality (which necessarily removes non-determinism and all flavours of the Copenhagen Interpretation).

"These hypothetical 'dispersion free' states would be specified by not only the quantum mechanical state vector but also by additional 'hidden' variables." [John Stewart Bell]

  • Known 2-dimensions: The quantum mechanical state vector is the 2-level quantum system known as the qubit.
  • Hidden 2-dimensions: The additional 'hidden' variables are the transverse modes of the quaternion.

Q.E.D.

Now that that one is sorted we are back on track and in pursuit of the Unified Field Theory. The quaternion will play a central role in reaching these ends. Here at Meditations on Geometry we are exploring and developing the mathematical algebra of the quaternion - this is the main focus of the technical articles. Beyond that we are interested in some philosophy and 4-dimensional physics in general.

Below is a chronology of this work to date

(abstraction tab added Feb/2017).


Article: Quantum Mechanics is a physical theory riddled with problems. At the core of these problems is the fact that the Quantum theory has not recognized that the qubit is a quaternion. Rather the qubit is thought of as a 2-level Quantum system, and the remaining degrees of freedom are ignored - giving rise to the probabilistic interpretation and indeterminism.


Video: An animated derivation of the geometric phase of the spinor. It is shown how to derive the geometric phase of the spinor by solving the equation of parallel transport for a tangent vector moving along a path on the surface of the Bloch sphere.


Meditation: The hidden variables of Quantum and Classical Mechanics are found in the parameter space of the quaternion. The quaternion is a 4-dimensional vector which describes rotations in 3 and 4 dimensions in full generality. Applied to the fundamental particles, the quaternion describes the orientation and precession of the magnetic moment of the fundamental particles. Conventionally thought of as a 3-dimensional vector the magnetic moment is in fact 4-dimensional, and the intrinsic spin of the fundamental particles, as measured in the Stern-Gerlach experiment, is understood as observing the 4th-dimensional shadow of the quaternion from a 3-dimensional perspective.


Article: The Law of the Quaternion states that the global phase is the sum of the geometric phase and the dynamic phase. These are the intrinsic parameters of the unit quaternion and are presented here in their analytic form for the first time. These parameters are the 4th-dimensional shadow of the quaternion as seen from a 3-dimensional perspective.


Article: Curious about parallel transport, quaternions, the qubit, geometric and global phases - then this article is for you. Living in an age where quantum computing technologies are promised to be right around the corner - ready to hit the market at any moment - or so we are lead to believe, ... you would not be alone thinking - '"surely we have mastered the basics" - au contraire!! This article is an account of what is known and what still remains elusive about that most fundamental piece of quantum information - the Qubit.


New Logo Design - 20th August/2015.

New logo design courtesy of barryosullivan.ca. Reminiscent of the Celtic trinity, the icon is a Möbius strip of the second kind, where it takes four complete revolutions to return to the initial point.


Article: One possible way to bridge the gap between the Quantum Mechanics and General Relativity is to utilize the mathematical language of General Relativity within the Quantum Theory itself. Differential Geometry - which is the mathematical language of General Relativity - is here used to define the geometric phase of the Qubit. The geometric phase of the Qubit is defined by solving the equation of parallel transport for all paths generated by the Unit Quaternion on the Bloch sphere.