.... 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
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]
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:
- The Euler angles are quaternions by another name.
- Vectors in 3D are pure quaternions.
- The Hamiltonian and the density matrix are quaternions.
- The Electromagnetic Field tensor is a sum of 2 quaternions.
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.
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
- SU(3) is used in quantum chromodynamics in the study of the strong interaction.
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.
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.
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).
Who's afraid of the Unit Quaternion ? - 11th November/2016
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.
Parallel Transport and the Geometric Phase - 15th October/2016
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.
Vectors, Spinors and Galilean Frames - 14th December/2015
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.
Parallel Transport, Quaternions and the Bloch Sphere - 28th August/2015
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.
The Geometric Phase of the Qubit - 24th July/2014
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.