Where do the Waves come from?

Structure and Continuity…

In the article It’s All Just Waves…(The Origin of Inertia) I discussed the mental walk which I took during my Undergraduate Studies, that lead me into a firm belief that every property that we ascribe to a Particle is actually an idealization of an associated Wave property. Since we never really understood what Particles were, and since we’ve almost always understood what Waves are, and we have clear mechanisms to understand them better, this is a very unburdening revelation. But it begs the question: Why Waves? Of all the things the Universe could be doing, we find that it is always waving. Why?

For me, the beginning of the answer to this question came near the end of my Undergraduate Studies. I was taking a course in Complex Analysis, and I was learning about a certain class of functions called ‘Analytic’ functions. These functions exist under rather strict conditions, and they have several properties. Among them are:

The Derivative exists. This means we can ask about how the function is changing.
They are Infinitely Differentiable. This means that the function is smooth and continuous. No sharp kinks anywhere.
The Taylor Series of the function will converge at a point, and this Series coincides with the function in a region around that point. This means that, knowing some information about a single point in the function, we can reasonably approximate the function at neighboring points as well.

These properties were kind of interesting to learn about, but when we then learned that in addition to all of these, a Complex Analytic Function will also naturally be a solution to the Wave Equation, I literally fell out of my chair. I saw this as the answer to what I believed was the most important question that I could be asking, and my mind was completely blown. I don’t remember exactly everything I did in that moment, but I do remember first falling, then jumping out of my seat, fist pumping, and victory dancing. I was definitely shouting my amazement. This behavior continued, off and on, through the remainder of the lecture. The Professor looked quizzically at me, and she started to giggle nervously. The other students were just staring confusedly at my ridiculous display. It is a matter of great sadness to me that this event occurred before the Smart Phone era, otherwise we’d likely have some enlightening video evidence.

Wave Behavior emerges from Complex Analytic Functions as a direct result of two conditions:

Complex Algebra has a certain Structure. This Structure is due mainly to the addition and multiplication rules of that Algebra. The Structure encodes something about the Geometry of the space.
An Analytic Function is Smooth and Continuous, infinitely so. This means that you can zoom in on anything that looks like a sharp boundary, or an edge, and you will see that either it is composed of smaller structures, like a fractal, or that at some level it is completely flat.

Thinking of the Universe as a thing that is Infinitely Smooth, with embedded Structure, ended up leading to many interesting consequences. The most important was this foundational, but seemingly rather obvious fact: the Universe is One Thing.

The Wave Equation which emerges as a result of Structure and Continuity takes on a form that is called Homogeneous. A Homogeneous Wave Equation represents a wave that has no ‘source terms’. Historically, physicists have used source terms to represent the presence of matter. The lack of source terms has actually been considered very problematic by physicists, for instance, this is part of what necessitates the need for the Higgs Boson in the Standard Model.

However, I was not disturbed by the Homogeneity of the emerging Wave Equation. In my Undergraduate Senior Thesis, I had already shown that mass can arise naturally from a Homogeneous wave, without the need of artificially introduced source terms. Source terms are also used to indicate the presence of electric charge and current, and at this point of my study I began to build an expectation that I would also see these ElectroMagnetic sources emerge naturally as well.

While there are some limited physics problems that can be addressed using Complex Analytic Functions, they have an immediately apparent problem that makes them otherwise useless. They are only 2-Dimensional. They don’t technically have enough legs to stand on, if you want to talk about 3 dimensions of space, and 1 dimension of time. If Analytic Functions were to ever describe the Universe, they would need at least 2 more legs in order to do so.

I decided at that moment that my primary pursuit during my Graduate Studies would be to define what the Algebraic Structure of the Universe needed to be, with the expectation that if I then imposed the constraint that it was Smooth and Continuous, I’d naturally see ‘Physics’ just sort of pop out.

Symbiotic Alliance…

I took a year off between my Undergraduate and Graduate studies, and during this time I began working in the Video Game Industry. Presumably this work was going to be temporary, but when Graduate School began, and I proposed to quit, my boss generously allowed me to work part time, while still receiving my full time pay. Thus, I continued to work as a Game Developer for the entire duration of my Graduate Studies.

In terms of my intended topic of Graduate Study, this arrangement turned out to be quite fortuitous.

Where academia relies very heavily on Linear Algebra for, basically, everything, the Game Industry additionally makes very practical use of Quaternion Algebra which is optimal for representing 3D rotations. Quaternions share many features with the Complex Numbers, such as a rich Algebraic Structure. Additionally they can be used to represent 4 Dimensions–just enough for space and time!

Many graphics packages employ the use of Quaternions to implement rigid body rotations.

During this in-between year, I took my first stab at trying to discover the Algebra of the Universe. My plan of attack was to try to define what ‘Analytic’ was supposed to mean in the Quaternion Algebra. This isn’t as clear or as straightforward, as it is with Complex Algebra. The reason is that Quaternions form a Non-Commutative Algebra, which means that there isn’t just one way to multiply things, there are two, because the order of multiplication now matters. There isn’t just one way to divide things, there are two. Defining a derivative under this circumstance means that we don’t have just one derivative, we have two.

But there was another problem.

Quaternions didn’t actually have the right Structure. Quaternion Algebra can be used to represent a 4 dimensional space, but I didn’t want just any 4 dimensional space, just any collection of 4 numbers, I needed a space that was consistent with the Relativistic concepts of Space and Time.

The kind of space I needed is called a Minkowski Space.

I started to make abstractions of the Quaternion Algebra, examining the B-*, and C-* Algebra. The Algebra that I required had some very specific needs, which ended with the creation of a new class, which for a time I called D-* Minkowski.

Unexpected Appendages…

There were several surprising concepts that emerged from this development. The first was this: in the process of creating an Algebra which could support the 4 dimensions of space and time, I discovered that 4 more dimensions automatically emerged. My Algebra ended up having 8 legs, instead of 4!

In a world where String Theory demands that we must swallow a multitude of abstract dimensions which have no clear physical meaning, it was initially difficult for me to understand these extra 4 dimensions. That is, until I eventually discovered that they actually did have a direct physical meaning, and that the meaning was very simple.

The dimensions of my Algebra can be split across 2 different axes:

Scalar/Vector. A Scalar is a single number, while a Vector is a collection of 3 numbers.
Real/Imaginary. This Algebra includes the concept of the Imaginary Number, which multiplied by itself equals -1. If a quantity, either Scalar or Vector is multiplied by the Imaginary Number, it is considered Imaginary, otherwise it is Real.

These 2 axes operate independently, and so we have 4 groups total. Each of these groups, it turns out, corresponds to a specific class of physical quantity. Here’s some specific examples:

	Scalar	Vector
Real	Time, Mass	Velocity, Electric Field
Imaginary	Volume	Rotation Axis, Magnetic Field

Each of these 4 groups have transformation properties that would prevent any practical physics equation from ever adding values together that belong to different groups. For instance, you will never see a physics equation that adds Velocity to Magnetic Field, or adds Volume to Time. But this is exactly what we mean when we are talking about Dimensions – we are talking about Linearly Independent objects and concepts.

As a species, we’ve been not adding Volume to Time, or Velocity to Magnetic Field for ages now, and there are mathematically consistent explanations, which are by no means recent, for why this is. Say it’s the mid-1800’s and I don’t know anything about Quantum Mechanics or Relativity, let alone String Theory. Even in these circumstances, I could have established and defined the 4 groups I’ve listed (which did in fact occur, though they used different names to describe them), and I could solidly conclude that the Universe actually has 8 Observable Dimensions.

During the later stages of development for this Algebra, I discovered that there were other similar developments from other folks. The Space-Time Algebra promoted by David Hestenes, and The Algebra of Physical Space promoted by W. E. Baylis. I reached out to both gentlemen, and greatly appreciated my correspondences with Baylis, whose algebra was almost identical to my own. If you want to talk about the Algebra I created, you may as well do what I do, and use the term coined by Baylis: The Algebra of Physical Space.

The Algebra I created, I learned, belonged to classes of Algebra that are called Clifford Algebra, or sometimes more appropriately Geometric Algebra. The appellate Geometric is very significant here, since the Algebraic Structure ends up directly encoding some strong statements about the Geometry of the Space.

What changes. What doesn’t…

I now had an Algebra, and it had the right structure. The problem that prevented me from moving forward to create the definition of an Analytic Function, was the fact that there was more than one plausible way to do this. It was the same trouble I hit when trying to define a Quaternion Analytic Function. Was there any tool that I could use to determine which one of these many definitions was correct?

The answer, of course, is yes. That tool is called Relativity.

One high level function of the Theory of Relativity, is that it acts as a MetaTheory, a Theory about Theories. Relativity contains some powerful statements, which can be used to qualify, or disqualify other Theories. For instance, Einstein used Special Relativity to show that Newton’s laws could only ever be a close approximation of reality. I decided I would use Relativity in order to help me determine what form of Analytic Function was correct.

Learning about Relativity is kind of an adventure. When I first started learning Relativity, things got pretty confusing, pretty fast. Most of this confusion came from the statement that there isn’t an absolute, or preferred, coordinate system. As an example, in Relativity a description of the Solar System where the Earth is revolving around the Sun is presumably just as valid as a description where the Sun revolves around the Earth. I mean, what kind of nonsense is this? Have the brightest minds of our species endured scientific and cultural revolution in order to extract from the chaos a more correct understanding of celestial mechanics, just so we could then throw it all away? Were all of those Post-Modernists right? Is there nothing meaningful in the world?

But then the confusion gives way.

No, the Post-Modernists were not right. Or perhaps they were half right — though, when it comes to Relativity, half right is the same thing as being wrong.

So, who was right? Ptolemy or Galileo? Does the Earth go around the Sun, or does the Sun go around the Earth? The Relativists doesn’t try to address the problem of which side is correct. Instead, the Relativist tries to derive understanding by accounting for the differences, not only in what each side sees, but also in where they stand.

Relativists have command of a handy tool which helps them extract Meaning from Conflict. It is called Symmetry. One way to describe Symmetry is as a relationship between how changes in what you are Observing must relate to changes in how you are Observing it. Practical implementations of Symmetry involve identifying what changed (the Transformation), and what didn’t change (the Invariant).

A simple and well known example of Symmetry is the geometric symmetry of certain English letters

These examples show a form of reflection symmetry. For this kind of symmetry, what is the Transformation, and what is the Invariant?

Transformation: Reflection about a given axis
Invariant: The Identity of the letter doesn’t change

There are always two halves to Transformation – how the Observer changed, and how what was Observed changed. Let’s take the case of reflection symmetry of letters to demonstrate this. Consider that we have the letter A painted on a small pane of glass. I can physically enact the reflection transformation in two ways

I can keep myself stationary and turn the pane around in order to see the reflected letter through the backside of the glass
I can keep the pane of glass fixed, and walk around it so that I can see the reflected letter through the backside of the glass

Transformation can always be described as a combination of those two kinds of action. The job of the Relativist is to employ Symmetry in order to Transform differences between Observers into the differences in what they Observe.

Ptolemy viewed the Solar System from the standpoint of the Earth, and observed Epicycles (spurious periodicity) in the movement of the planets through the Heavens. Galileo viewed the Solar System from the standpoint of the Sun, and observed no such Epicycles. A Relativist isn’t going to get caught up in an argument about whether Epicycles exist or not. It’s not a meaningful or even interesting conversation. What is interesting is how the transformation from a Ptolematic to a Galilean viewpoint created the differences in observation of Epicycles, and what those differences must says about the underlying Symmetry.

The existence of Epicyclic movement of planets, with no orbital motion of Earth in Ptolemy’s view maps directly to the existence of the orbital motion of Earth, with no Epicycles in Galileo’s view. Galileo viewing the Earth orbiting the Sun, is the same thing as Ptolemy viewing Epicycles in planetary motion.

Conflict Resolved. Relativity Wins.

Part of the brilliance of Einstein’s work in Relativity, was that he encoded this relationship between Observer and Observed into a mathematical form. He started off with a certain kind of Symmetry which had the following Transformation and Invariant:

Transformation: Observers can view the Universe at different relative velocities
Invariant: Physics (Initially, and specifically, the Speed of Light)

He employed this Symmetry as a tool in order to describe a relationship between how differences in the velocity of Observers relate to differences in what is Observed. The thing that surprised Einstein (and all of the rest of us), was that this relationship demanded that what must be Observed is a transformation in the definition of Space and Time itself.

In order to take such bizarre sounding results, and make them meaningful, Relativists subscribe to the Principle of Covariance. The idea behind this principle is that if you want to describe Reality, which is Invariant to Transformation, then your description must also be Invariant. Concretely, what this means, is that the very form of your mathematical equation must be composed of, and reference only those quantities whose form is Invariant under transformation. Such quantities are called Covariant Quantities.

And this became my answer.

Remember, the problem I was having was that there were multiple ways to define an Analytic Function, and the prescription provided by Relativity was to disqualify any definition whose form didn’t remain the same if I applied a transformation.

Covariant Building Blocks…

My plan to use Relativity to solve the problem where I had non-unique definitions for what an Analytic Function might be immediately hit a brick wall.

Attempting to use the Algebra of Physical Space to define a Covariant Quantity turns out to also be non-unique. There are multiple ways to do it. And here’s where I got a bit of help from W. E. Baylis. Baylis had walked this path about 10 years before I did, and he’d figured out a quite a few things. One of the things that I learned from Baylis was that the different possible ways to create Covariant Quantities were actually distinct classes of physically realizable objects.

There are 3 such classes:

Spinors
ParaVectors
BiParaVectors

The most familiar of these classes is the ParaVector. ParaVector is Baylis’ term for an Algebraically Covariant Quantity, which parallels the historic 4-Vector from standard Relativity. The kinds of things we would represent with a ParaVector would be: Position, Velocity, or the 4-Potential of EM. ParaVectors are Hermitian (they don’t have any imaginary components)

The next most familiar class is the BiParaVector. This Quantity parallels the historic Rank-2 Tensor from Relativity. The kinds of values that would be represented by a BiParaVector would be: Rotation, Lorentz Boosts, Electro-Magnetic Fields.

The final quantity is a bit weird. Spinors have a place in traditional formulations of Relativity, but they continuously remain a more esoteric aspect of the discipline. Usually, a budding physicist must wait until graduate school, or beyond, before really being introduced to the concept of a Spinor, at least under the auspices of Relativity.

If anything, most physicists will associate the concept of Spinors with Quantum Mechanics, and often (incorrectly) think that the physical significance of Spinors emerges mysteriously from QM, rather than directly from Relativity.

So, what is a Spinor?

Most physicists are first introduced to Spinors as the mathematical object which describes the Quantum Wavefunction of an Electron. The most immediately remarkable property of a Spinor is that you must rotate it through two full revolutions in order to return it back to normal. This defies our common experience, where a rotation of 360 degrees is the same as no rotation at all.

Physicists have come up with actual, realizable examples of motion, which require Spinors in order to successfully describe. P. M. Dirac described a so called ‘Belt Trick’, where a belt is given two full twists. Dirac showed that, without letting go of either end of the twisted belt, you could remove both of the twists, so that the belt was no longer twisted. However, if the belt was only twisted once, the twist couldn’t be removed without letting go of the ends of the belt

So, Spinors have something to do with two rotations. But… what do they really represent? What are they used for?

Here’s another common example, that some of my Game Developer friends might appreciate. Say you have a clock, and someone asks you to manually move the second hand from 12 o’clock to 1 o’clock. There are actually 2 ways for you to do this. Either you could rotate the hand 30 degrees Clockwise, or you could rotate the hand 330 degrees Counter-Clockwise.

The position of the hand when it is at 1 o’clock, doesn’t actually tell you anything about which one of those rotations was used in order to get there from 12. Most animation and graphics programmers are fully aware of the need to make this distinction, so how do we do it? For any given rotation, it’s possible to find two Quaternions, one which rotates the short way around an axis, and one which rotates the long way. The reason that Quaternions can make this distinction, is because they are Spinors.

The final thing that I can say about Spinors, is that they really are the fundamental Covariant Building Block. ParaVectors as well as BiParaVectors can be represented as certain combinations of Spinors.

One of the primary physical quantities that we use a Spinor in order to represent is a Lorentz Transformation. This is the Transformation that Einstein used in order to show that Space and Time are respectively compressed and stretched by the velocity of an Observer. As it turns out, the Spinorial form of a Lorentz Transformation is the most enlightening, and the easiest to manipulate (so I think).

After exploring these Covariant Building Blocks, I discovered the correct way(s) to construct an Analytic Function. As it turns out, one of the reasons that I was seeing many different ways to construct Analytic Functions was related to the fact that you can describe how each class of Covariant Quantity is Analytic.

Remember, my initial hypothesis was that if I could create an Algebra with the correct Structure, and I could then impose the condition of Continuity (which I was doing by expressing the conditions for an Analytic Function), I would see physics just sort of ‘pop out’. Secondarily, I presumed that I shouldn’t need any source terms in order to describe mass or charge. Those quantities should be wave-like properties that would emerge from Homogeneous Waves.

We’re now closing in on being able to test out these Hypotheses.

Waves and Sources…

In the case of ParaVectors, the condition for an Analytic Function results in a Homogeneous Wave Equation. This Wave Equation has 8 dimensions, which can be efficiently divided into 4 components: Real Scalar, Real Vector, Imaginary Scalar, Imaginary Vector.

Real Scalar
Imaginary Scalar
Real Vector
Imaginary Vector

As I examined these 4 components, I was surprised to discover that they almost exactly represented the 4 Maxwell Equations. These are the Equations that govern ElectroMagnetism, including ElectroMagnetic Waves. Here’s how the equations lined up:

Real Scalar – Gauss Law
Imaginary Scalar – Magnetic Gauss Law (No Magnetic Monopoles)
Real Vector – Ampere’s Law + Maxwell Current
Imaginary Vector – Faraday’s Law

My initial Hypothesis that physics was just going to ‘pop out’ was bearing some fruit.

I say that these equations almost represented Maxwell’s Equations. Maxwell’s Equations are expressed in terms of Electric Fields, Magnetic Fields, and Source Terms. When examining the equation which I had constructed from first principles, it was relatively easy to identify the Electric and Magnetic Fields.

But there was also something new.

There was a new Scalar Field, and I wasn’t immediately certain what the significance of this field might be.

Obviously, there were no Source Terms in my result (the wave was Homogeneous), but in the places where the Source Terms usually were, I now saw derivatives of this Scalar Field.

So what is this new Scalar Field? Was it really representative of a physical object? Here’s what I discovered:

One property of the physically real Electric and Magnetic fields, is that they are Covariant. Since Covariance was built in to begin with, the Scalar Field was also Covariant.

Another important property of Electric and Magnetic fields is that they are what is called Gauge Invariant. Without getting into the details of what this means, I can say that the Scalar Field is also Gauge Invariant.

Finally, I discovered that the fundamental form of this Scalar Field was something that was already present in most EM literature, and took the name of the Lorenz Gauge Condition. Historically, physicists would introduce the condition by stating that the quantity which I’m associating with this Scalar Field was identically equal to zero, in order to find solutions to the EM Equations. Coincidentally, it is usually understood that the Lorenz Gauge Condition carries an implication that we are in free space – in other words, the Source Terms are zero.

With the view that this new Scalar Field has all of the properties of known physical fields, and the fact that the derivatives of this field take the place of the Source Terms, and the fact that the historical significance of this field is that setting it identically to zero is commonly used to determine solutions where the EM sources are zero, I’ve decided to call this field the Source Field, $\phi$ .

\phi = \frac{1}{c^2}\frac{\partial V}{\partial t} + \vec{\nabla}\cdot\vec{A}

\vec{E} = -\vec{\nabla}V - \frac{\partial \vec{A}}{\partial t}

Where to next?

In my Masters Thesis, I was able to show that given the correct Structure, and the correct condition for Continuity, several physical phenomena emerge:

The physical fields that we associate with Electricity and Magnetism.
The correct relationships between these fields.
A field that we can associate with EM Sources, like Charge and Current.

The Algebra that I constructed, which I later discovered was The Algebra of Physical Space, originally discovered by W. E. Baylis, has several interesting features, relating to other fields of Physics

It naturally incorporates features we usually associate with Quantum Mechanics, such as Spinors and Commutation Relationships.
There are ready analogies for Quantum States, and Quantum Operators.
The Dirac Equation is easily represented, and only a single logical gap exists in order for that equation to emerge naturally.
Analyticity provides an immediately straightforward mechanism for introducing an entire class of Gauge Transformations, some of which I’ve never seen before. Gauge Theories are an important part of Standard Model physics.
The Algebraic Structure is determined explicitly by the SpaceTime Metric, which makes it amenable to concepts we preserve for General Relativity, such as having a non-constant metric.
Introducing a non-constant Metric, provides some very straight forward definitions for Christoffel Symbol-like objects.

The Algebra seems to be a promising language for describing Quantum as well as GR. Here are some open questions, and areas of future exploration:

Now that I understand the wave-like properties that lead to charge and mass, I’d like to try and construct a solution for a massive charged particle. Does this approach anything that looks like the Standard Model?
I’d like to understand better what the impact of having a non-constant Metric will have on the definition of Analyticity.
I’d like to close the gap with the Dirac Equation. This likely involves solving what I call the ‘Final Quantum Question’, namely, why are only certain photon energies allowed? I have a starting point here, namely that I need to construct an atemporal boundary condition.
I’d like to push the derivation of Structure closer to first principles. The Structure comes directly from the Metric, and I’m currently just stating what the Metric should be a-priori. Gravity is going to come from the Structure, so I’d like to get some insight into what is causing the Structure. I’m currently looking for clues in the direction of what I know about the relationship between Mass and Dispersion.

These are the questions that I would have attempted to answer for my PhD. If ever the path of my life takes me back into that track, where I can explore these questions full time, or at least during those moments when I have availability for part time musings, these are the questions I will try to answer.

The Fields	Modified Maxwell’s Equations	Source Relations
$\phi = \frac{1}{c^2}\frac{\partial V}{\partial t} + \vec{\nabla}\cdot\vec{A}$ $\vec{E} = -\vec{\nabla}V - \frac{\partial \vec{A}}{\partial t}$ $\vec{B} = \vec{\nabla}\times\vec{A}$	$\vec{\nabla}\cdot\vec{E}=\frac{\partial \phi}{\partial t}$ $\vec{\nabla}\cdot\vec{B}=0$ $\vec{\nabla}\times\vec{E}=-\frac{\partial B}{\partial t}$ $\vec{\nabla}\times\vec{B}=-\vec{\nabla}\phi+\frac{1}{c^2}\frac{\partial \vec{E}}{\partial t}$	$\rho = \epsilon_0\frac{\partial \phi}{\partial t}$ $\vec{j}=-\frac{1}{\mu_0}\vec{\nabla}\phi$