Written by Daniel Weidele in 2024.
Please find the archived work of Doug Sweetser below.
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:
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.
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 × real (a) and 3 × imaginary (b,c,d) units:
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.
When multiplying quaternions in dense notation, follow this rule:
Note that while quaternions do not commute, they are associative.
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
and
Then
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).
Operator definitions:
(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
check
Standard electromagnetic interpretation:
In standard electromagnetic theory, the quaternion χ = (w, X) represents electromagnetic potentials:
The electromagnetic fields are derived from these potentials:
From our quaternion differential, the vector part is:
This elegantly combines both electromagnetic fields in a single quaternion expression.
The Lorenz gauge condition:
The scalar part of our quaternion differential is:
In standard electromagnetic theory, we typically impose the Lorenz gauge condition:
This gauge choice simplifies Maxwell's equations and leads to wave equations for the potentials. It's important to note:
Beyond standard gauge theory:
When we keep Φ ≠ 0, several interesting possibilities emerge:
1. Longitudinal modes:
Standard electromagnetic waves in vacuum are purely transverse (electric and magnetic fields perpendicular to propagation). But if Φ ≠ 0, the scalar field w can support longitudinal oscillations - compression waves in the potential field itself. These would be:
2. Coupling to mass/energy:
The scalar constraint Φ = ∂w/∂t - ∇⋅X couples two fundamental quantities:
While massless photons don't couple to Φ (they satisfy the gauge condition), massive particles might interact with this scalar field differently. This could hint at:
3. Connection to gravitational coupling:
As Doug Sweetser noted:
"One enormous subject I have not looked into is what happens if one keeps this gauge term. The resulting physics must describe things that do not travel at the speed of light. It is the subject of particles with a mass."
This is profound. In General Relativity, gravitational effects propagate at the speed of light, but gravitational potential exists throughout space and couples to mass-energy. The scalar Φ might represent:
Physical predictions worth investigating:
The model yields wave equations for both the vector and scalar potentials.
A. Vector potential wave equation (transverse waves):
This describes standard electromagnetic radiation - transverse waves propagating at speed c. Sources are currents J.
B. Scalar potential wave equation (longitudinal modes):
This describes oscillations in the scalar potential. Sources are charges ρ. When coupled with the constraint Φ ≠ 0, this allows for longitudinal modes not present in standard free-space EM.
(TODO: Add detailed derivations showing how these wave equations follow from the quaternion formulation and Maxwell's equations.)
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.
The specific interpretation regarding the scalar constraint Φ and its potential connection to gravitational coupling is a novel exploration building on Doug's foundational work.
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, consider the following open questions.
Theoretical questions:
Experimental questions:
Critical analysis:
Please share your theoretical or experimental results by email, and I will post here timely.
No submissions as of Nov. 29, 2024
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