Doing Physics with Quaternions

Math (pdf)

Where quaternions fit in math, adapted from Max Tegmark, 1998.

Want to see quaternions? That's what the software for this project does: take quaternion algebra and use it to drive animations. This is Seurat at Disney: a collection of dots at different times. The image on the right shows the product of the yellow collection of dots times the blue, resulting in green which bends in spacetime. This is analytic animation, a dynamic variation on analytic geometry. These images already have a direct link to physics, specifically the Standard Model. Some of the logic behind the project is available at Dynamic graphs.

Physics

Classical Physics (pdf)

F = m a, in an inertial frame, in polar coordinates, and in a rotating reference frame

The simple harmonic oscillator and wave equation

4 tests for a conservative force

Unifying Gravity and EM

I think it is reasonable to believe I have found a way to unify gravity and electromagnetism in a way that has been quantized. Only ordinary math tools are used. The problem is a puzzle to be solved, not a math problem so difficult only a few folks at the Institute for Advanced Study to understand. At the core is the action, shown here, which requires a fun bit of twisting of the potential relative to the real current to get the spin of the current-current interaction right.

Brief summaries or calculations

The Unified GEM Action and Symmetries (pdf)

The Unified GEM Lagrangian, Field Equations, and Solutions (pdf)

The Precession of the Perihelion of Mercury Based on the Unifed GEM Proposal (pdf)

Derivation of the GEM unified field proposal using quaternions exclusively (pdf)

Slides from talks

Geometry + 4-Potentials = Unified Filed Theory, 2007 April APS talk available as html or YouTube video.

Unifying Gravity and EM, a Riddle You Can Solve (pdf), 9th Eastern Gravity Meeting, 2006.

Why a Rank 1 Unified Field Theory is Compelling and Its Background Mathematical Structure (pdf), 8th Eastern Gravity Meeting, 2005.

Unifying Gravity and EM by Generalizing EM (pdf), 7th Eastern Gravity Meeting, 2004.

Unifying gravity and EM by analogies with EM, MIT IAP course, Jan. 2003.

Lagrange Densities (pdf, video 1, video 2)
Fields and Quantum Mechanics (pdf, video 3, video 4)
Forces, Metrics, and New Physics (pdf, video 5, video 6)

Realizing Einstein's Dreams: Unifying Gravity and Light, and Why Quantum Mechanics is Weird (pdf), Joint APS/AAPT meeting, Spring 2005.

4-potential Equations for Gravity and Electromagnetism (pdf), Joint APS/AAPT meeting, Fall, 2003.

The Cosmological Consequences of the Product Rule (pdf), Joint APS/AAPT meeting, Spring, 2004. A Universe with zero dark matter is good :-)

Picture Stories of Events in Spactime (pdf), Do While Studios, Spring 2000. The relationship between Special and General relativity using quaternions.

Papers

An approaching using standard tensors and differential geometry

Deducing a Unified Field Theory from Electromagnetism (pdf).

Unifying Gravity and EM by analogies to EM (pdf) (v 4.4, 13 pages).

Geometry + 4-Potential = Unified Field Theory, submission for the 2007 Essays on Gravitation, sponsored by the Gravity Research foundation (pdf)

An approach using quaternions.

Einstein's Vision I: Classical unification of gravity and electromagnetism using Riemannian quaternions (pdf)

Einstein's Vision II: Unified force with constant velocity profiles (pdf).

Ropes and Gravity

Posters

Unifying Gravity and EM by analogies to EM (pdf) Brookhaven National Lab meeting.

Mathematica notebooks

From the Lagrange density to experimental tests (pdf, or .nb)

Videos!

will have all my latest video-education efforts. Enjoy.

Electromagnetism (pdf)

Classical electrodynamics

Electromagnetic field gauges

The Maxwell equations in the light gauge: QED?

The Lorentz force

The stress-energy-momentum 2-tensor of the electromagnetic field

Paper: Maxwell's vision: electromagnetism with Hamilton's quaternions (pdf), Second Meeting on Quaternionic Structures in Mathematics and Physics, Rome, 1999.

Let's Talk

Join the newsgroup, quaternions@TheWorld.com. This is a low traffic newsgroup where I give updates sporadically on the status of the project. To subscribe, send email to majordomo@TheWorld.com. The body (not the subject!) should just contain this: subscribe quaternions. If you ever want to leave this newsgroup, send a similar message, where the body reads "unsubscribe quaternions".
Quaternion Question and Answer, a site for chatting about quaternions.
Feedback welcome!

Get Stuff

For a fee: Books, Buttons, and T-shirts

For free: Doing Physics with Quaternions (much of this site, 156 pages), Lectures on Dynamic Graphs and Unified Field Theory (75 pages).

A few good papers:

Sudbery's first paper (memo, 1977, 44 pages) on why quaternion analysis is no good.
Sudbery's second paper (1979, 28 pages) on the topic. Please look to my work above on quaternion analysis for a much better alternative!.
C. A. Deavours paper, "The Quaternion Calculus". My critique is that using his definition of a quaternion derivative, if a function like f=q is analytic in q, f^2 is not. That indicates a better definition must be found before quaternion analysis can really begin.
Salamin's paper (1979, 9 pages) on rotations.
Howell and Lafon's paper (1975, 13 pages) on the efficiency of quaternion multiplication.
Silberstein's paper (1912, 20 pages) on using biquaternions for quaternion special relativity. Biquaternions are NOT an algebraic field, and are not used in any operations on this web site.

Video Ed