Euler, Gauss, Hamilton, Sweetser: The Forgotten Path to New Physics

Written by Daniel Weidele in 2024.

Please find the archived work of Doug Sweetser below.

Disclosure
Daniel (that's me, Hi! 👋) is a computer scientist (not formally trained in college-level physics!), who has enjoyed the work of Doug, and thinks that he's been up to something.
However, there's a gap that's been bugging me that Doug has not fully closed, which led me to buy this domain sadly after Doug's passing.

Therefore I have set two goals for this site:

• provide an accessible introduction leading most directly to the open gap, and
• archive Doug's work for future generations.

I. Quaternion Field Theory

While today quaternions are used computationally to calculate orbits and trajectories in spacefaring or computer graphics, the following shall provide brief intuition on how they can further generate an alternative view on fields in theoretical physics, towards the quest in the understanding of our universe. Such link has already been drawn by their inventor W.R. Hamilton in the 19th century, but has been abandoned in the early 20th century, in favor of vector calculus after J.C. Maxwell and O. Heaviside.

1. What are quaternions?

Have you ever wondered how complex numbers could work in 3-D? No? Well, not too many have, so don't feel ashamed. Quaternions can do in 3-D, what complex numbers do in 2-D. So first of all, it's an algebra. And it's especially well suited for calculating rotations. But really, quaternions are defined in 4-D. So a quaternion consists of 1 x real (a) and 3 x imaginary (b,c,d) units:

Q = a + bi + cj + dk.

The real (a) is also called the scalar, and the imaginaries (b,c,d) are often referred to as the vector component. You can refer to an introductory lecture on computer graphics to learn how to calculate rotations and orbits with this algebra. Throughout the following, however, we're on a slightly different quest: we want to understand the universe. Yet, it's somewhat reassuring that this algebra does work very well within our universe, especially where other forms of calculation are not sufficient (Gimbal Lock). Would a vector-only calculus become any better when trying to describe even more complex phenomena, like fields? Or is it only practical as long as we don't run in a lock situation? What if we used both vector and scalar components in the analysis of our universe?

To do so, let's exploit the handy fact that quaternions are 4-dimensional, just like spacetime. So we could map time to the real/scalar, and space to the imaginary/vector. That yields a spacetime number χ, where the phenomena spacetime has been projected right onto the quaternion. As a spacetime number, the quaternion now has meaning. And yes, it's okay to keep wondering why the naming scheme appears flipped overall: surely time feels more imaginary, and space more real...

Let's denote time by w in the scalar (rather than t, to not clash with the symbol for the differential of time later on). And let X be (x, y, z) our spatial vector component.
Then, instead of writing χ = w + xi + yj + zk, I prefer the more handy dense notation χ = (w, X) for spacetime numbers.

2. Quaternion multiplication rule

When multiplying quaternions in dense notation, follow this rule:

(a, b) × (c, d) = (ac - b⋅d, ad + bc + b×d)

Note that while quaternions do not commute, they are associative.

3. Spacetime differential analysis

Next let's try to derive equations that characterize the electromagnetic field (at least!). We do this by applying differential analysis, which means the following.
We let time flow a tiny little bit, and also we move the quaternion a tiny little bit in each spatial direction, only to then check how the dynamics of the system behave.
To do so, we define the differential operator ∇ for quaternions, and then apply it to our spacetime quaternion χ.

Let

    ∇ = (∂/∂t, ∇)           "nabla" operator

and

    χ = (w, X).           "spacetime" number

Then

  ∇χ = (∂/∂t, ∇) × (w, X)

        = (∂w/∂t - ∇⋅X, ∂X/∂t + ∇w + ∇×X).

The dynamics of the scalar part are described by (∂w/∂t - ∇⋅X)
The dynamics of the vector part are described by (∂X/∂t + ∇w + ∇×X).

∂/∂t is the time differential: when applied to a vector (scalar) field, it yields a vector (scalar) field representing the rate of change of the vectors (scalars) with respect to time.
∇ is the gradient: when applied to a vector (scalar) field, it yields a tensor (vector) field indicating the rate of change of the vectors (scalars).
∇⋅ is divergence: when applied to a vector field, it yields a scalar field representing the magnitude of the source or sink (think of it like a hose or drain).
∇× is the curl: when applied to a vector field, it yields a vector field measuring the tendency to rotate or circulate around a point.
(Not used here, but ∇² = ∇⋅∇ would be the Laplacian: when applied to a scalar field, it yields a scalar field representing the magnitude of the rate of change of the scalars.)

(TODO: Build a little web tool to visualize these operators.)

To sanity check our equations, let's make sure the types of the field are well-defined and compatible.

For

  ∇χ       = (∂w/∂t - ∇⋅X, ∂X/∂t + ∇w + ∇×X),

check

  type(∇χ) = (scalar - scalar, vector + vector + vector) ✔

4. Interpretation

The vector part yields fundamental force fields

• electric field E = ∂X/∂t,
• magnetic field B = ∇×X, and
• gravitational field G = ∇w.

The scalar further yields

• gravitational flux Γ = ∂w/∂t, and
• electromagnetic flux Φ = ∇⋅X.

This perspective also yields an alternative interpretation (now coming from bottom up) of our spacetime quaternion: it is the gravitoelectromagnetic (quaternional) potential, with w the gravitoelectromagnetic scalar potential, and X the gravitoelectromagnetic vector potential.

The model suggests the following fundamental relationships relevant to the three force fields (G, E and B) via this potential:

• electromagnetic flux ~ magnetic field (i.e. coupling of divergence and curl of the vector potential (ref. Helmholtz))
• electromagnetic flux ~ gravitational flux (coupling in the scalar of our model)
• gravitational flux ~ gravitational field (self-induction of the gravitation field)
• electric field ~ magnetic field (changing electric field induces magnetic field, and vice versa (ref. Faraday))

When deriving the Maxwell homogeneous equations, Doug (similar to Lorenz) orphans w as a gauge field, sets it 0, but leaves it up to the reader to explore other cases:

One enormous subject I have not looked into is what happens if one keeps this gauge term.
The resulting physics must describe thing that do not travel at the speed of light.
It is the subject of particles with a mass.

While gauging is fine when only interested in the EM-fields, gauging seems to be also what has caused us to not think harder about the meaning of these terms. When I first looked at the above equations, I asked myself:

Could not the gauge term be the gravitational field itself? I'd rather just keep it.

That's when I decided to buy this domain and write about it.

5. Deriving wave equations

The model yields the following wave equations:

(TODO: Add here how to derive these).

A. Transverse wave

∇²X = -∂(∇×X)∂t

B. Longitudinal wave

∇²(∂w/∂t) = ∇(∂(∇⋅w)/∂t)

II. Douglas Balch Sweetser (*__/__/1962 - †12/10/2022)

Doug Sweetser, who used to own this domain, has conducted extensive work on quaternions in physics, dedicating a good part of his life. I am doing my best to preserve his results. If you are missing some of his work, please let me know by email, and I will post here timely.
To the best of my knowledge, however, the interpretation of the above model is novel.

1. Doing Physics with Quaternions
   PDF-File; 30.14 MB; Nov 29, 2024; MD5 8cb9078df79a3e88e3d12d7a73685795
2. Archived works
   ZIP-Archive; 3.88 GB; Oct 6, 2023; MD5 b643b0d9deeb8c99e92fd0f27390ea71
3. Website on Github "Q"
   • Quaternion math
   • Classical physics
   • Special relativity
   • Problem sets
   • Quaternion gravity
   • Electromagnetism
   • Quantum mechanics
4. Historical Website "The World"
5. YouTube channel
6. Questions and answers "Greenspun"

III. The fringest quest

While you can find various motivational problems throughout Doug's work, I'd like for you to take a moment and, in exchange for me paying for this website and archiving the information, ponder on the following really fringe thoughts.

1. Problem

1. Can quaternions lead to an explanation for what some people experimentally perceive as a longitudinal electromagnetic wave (or scalar wave, i.e. a wave in G)?
   For example, could it be possible to manipulate the curl of X in B, in order to affect the divergence of X? Would then alternating curl(s) cause pulsation (or even resonance)? What about quickly oscillating flux? Would that affect gravitational waves?
2. Would critics' (such as, with all dear respect, Gerhard W. Bruhn's) assumptions and conclusions in standard algebra, where the scalar component is not regarded, also hold under quaternional considerations?

Please share your theoretical or experimental results by email, and I will post here timely.

2. Submissions

No submissions as of Nov. 29, 2024

3. Related materials

1. Maxwell's original quaternion theory was a unified field theory of electromagnetics and gravitation. T. Bearden, 1988
   PDF-File; 6.43 MB; Nov 28, 2024; MD5 1d5f0f2ae7742ec4108e4ee26a55081f
2. The study of electromagnetic processes in the experiments of Tesla. B. Sacco, A.K. Tomilin, 2012
   PDF-File; 2.45 MB; Nov 28, 2024; MD5 05c8bc6d5d7e1d0e8abf29b628702f14
3. Wireless power transmission. S. Jackson, 2011
   PDF-File; 23.10 MB; Nov 29, 2024; MD5 186b957117bdfa153d9d41a9ad00fdbc
4. On the Existence of K. Meyl’s Scalar Waves. G.W. Bruhn, 2001
   PDF-File; 97 KB; Nov 29, 2024; MD5 8ad3666595e29b89042490fb5f9cb052
5. Can Longitudinal Electromagnetic Waves Exist?. G.W. Bruhn, 2002
   PDF-File; 83 KB; Nov 29, 2024; MD5 1463640e50aac37d643dec324954c2f3