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.

Here we focus on the origins of quantum mechanics, and begin by highlighting a small selection of some of the more notable landmark discoveries, made around the turn of the 20th century that gave birth to the quantum theory. The name "quantum" means quantity - and in the original meaning of the science, quantum mechanics concerned the study of the discrete nature of reality. 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 "quantities" 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.

Today quantum mechanics has diverged quite dramatically from the original spirit of the scientific discipline, which sought to uncover the meaning of the "quantum". Modern quantum mechanics is a theory riddled with problems, amongst which is the belief that particles exist in mutually exclusive states at the same time until they are observed by a conscious being. The underlying philosophy of the quantum theory is that there is no objective reality. That reality is observer dependent. This is contrary to the previously held destiny of the natural sciences. As Max Born eloquently puts it;

"Natural science is [the study of] the finished picture of nature. The basis of which is the idea that the ordinary world exists independently of and uninfluenced by the process of knowing."
 - Max Born

The quantum theory challenges this world view. The mathematical principle upon which this is based is interpreting the complex number as a probability amplitude and attributing it's magnitude to a probability measure. Yet it is well known that the complex number describes rotations in 2-dimensions, and the extensions of the complex number describe rotations in higher dimensional spaces. The hidden variables of quantum mechanics are found in these multidimensional rotational spaces. Real valued functions describe Rabi oscillations in 1-dimension. The complex numbers describe rotations in 2-dimensions. The unit quanternion, discovered by Hamilton in 1843, describes rotations in 3 and 4 dimensions. This is the first rotational space beyond the complex numbers. The fundamental unit of information in quantum mechanics is the qubit, and the qubit is a unit quaternion.

In 1935 Einstein Podolsky and Rosen challenged the completeness of quantum mechanics under the assumption that local realism is valid; this is the famous EPR thought experiment. The EPR argument shows that according to the quantum theory, 2 spatially separated particles exhibit a non-local connection which is instantaneous - i.e. faster than light. The entangled state is a pair of particles which exhibit non-local correlations. Measurement of one determines the partner particle's state, and action on one affects its pair instantaneously. The paradox is resolved by recognizing the entangled pair is described by the SU(4) group and is 16-dimensional. The entangled state is not two separate particles, rather they are one and the same multi-dimensional object. The appearance of 2 spatially separated particles is an illusion, derived from looking at a multi-dimensional object from a 3 dimensional perspective.

In the same year Erwin 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 resonators of total energy E, and with frequency ν, and the "the most essential point of the whole calculation" is the recognizing that the total energy E is  ".... 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]

Here Max Planck explains that the electromagnetic radiation emitted by a blackbody is done 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 reflection.

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.
Since the path can never be fully completed 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. and the only thing constant is change. Zeno’s paradox raises a reassessment of, either our concept of motion, or 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. 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, which may be referred to as 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 means 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. 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 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 applied to 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 not by 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.