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Doing Physics with Quaternions doug <sweetser@alum.mit.edu> (.pdf)

Get a HARD COPY of this site!  (I prefer it :-)

Figuring out stuff takes time (money) and money. If I can raise $0.3 K, that will cover a local meeting, $2 K would mean I could buy Mathematica (used a student version for a decade), $5.2k would go for a Mac with Final Cut Pro (making "Stand-Up Physicist" teaching videos) and $30 K would mean a year devoted to math and physics. This stuff may be important enough for me to ask so directly for funding.

Intro, Mathematics, Classical, SR, EM, QM, G, Conclusions

Introduction

An overview

A brief history of quaternions

Mathematics (.pdf)

Multiplying quaternions the easy way

Inner and outer products

Scalars, vectors, tensors and all that

Quaternion analysis (.pdf)

Topology

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

Tools for algebra, trig, logs... and a Quaternion calculator (in Java)

An open source project QEMation to develop Quaternion Equation driven aniMation. The first result of this effort, Visualizing Constant Inertia , is more interesting from a math/physics perspective than one might guess.

(.pdf)
Doing the work of the Lorentz group with rotations and dilations
An alternative algebra for boosts

Problem set questions and solutions from MIT's 8.033, Classical and Relativistic Mechanics
Index and links to solved problems
PS 1: Kinematic effects of relativity
PS 2: More kinematic effects
PS 3: The Lorentz transformation and the addition of velocities
PS 4: The Doppler effect, 4- vector invariants, and the twin paradox
PS 5: Energy, Mass and Momentum
PS 6: The Compton effect and threshold collision problems

(.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

(.pdf)
Bracket notation & quaternions: quaternions as a complete inner product space
Multiplying polar representations (without Campbell-Hausdorff!)
Commutators and the uncertainty principle
Unifying the representations of spin and angular momentum
The Schrödinger equation
The Klein-Gordon equation
Time reversal

(.pdf)

Einstein's vision
Einstein spent the last forty years of his life trying to unify gravity and electromagnetism in a way that also lead to a new subtle understand of quantum mechanics. Just like for special relativity and general relativity, he was looking for a new logical foundation that would not change any experimental tests much at all. For a general audience, there is a slide presentation for ways of thinking about events using quaternions.

A draft paper ( pdf, v1.4) and notes for a talk (pdf) on unifying gravity and electromagnetism in a way that can be quantized are available. This method contains no quaternions, but the algebra is based on that earlier effort.

To a degree that has really surprised me, the pair of papers (last updated September 2001) is a work in progress towards that goal:

• Einstein's Vision I: Classical unification of gravity and electromagnetism using Riemannian quaternions (.pdf)
• [undergoing revision] Einstein's Vision II: Unified force with constant velocity profiles (.pdf)
• Earlier versions are still accessible.
• Most of today's research on gravity involves quantum gravity theory. I have two ideas in this direction. In the paper on the unified force, to get the algebra correct, the 4-force was normalized to the potential, which in turn required hbar/c by dimensional analysis. If that form of the force equation is correct, then if hbar goes to zero, then the unified force disappears. This is a basic characteristic of any quantum theory.
  
  What puzzled Einstein for decades was the why of quantum mechanics. I believe that quaternion analysis may answer that difficult question. I have a sketch of a third paper - Einstein's Vision III: Quantum unified field theory. This involves what I am calling the general correspondence principle. Quaternion analysis (see above) has a timelike automorphic quaternion derivative for classical physics and a spacelike automorphic quaternion derivative for quantum mechanics. The unified field equation in the first paper is classical if the automorphic quaternion derivative is timelike, and quantum if spacelike. The symmetry of this equation is none other than U(1)xSU(2)xSU(3) (I am certain about the electroweak symmetry, but not completely certain about the strong part). This may be a justification for the standard model.

A summary of physics equations written as quaternions
Conclusions
 

Stuff to get

• Check out the buttons!
• A bound Xerox copy of these web pages is available.

cost: $30 for "Doing physics with quaternions",
$40 for "Doing physics quaternions." + lecture notes on "Dynamic grpahs, quaternion analysis, and unified field theory"
And $5 for shipping (actually costs $7). Includes the button, "A brief definition of spacetime" There are two ways to pay:
1. US mail to: Douglas Sweetser, 1340 Comm Ave Apt 7, Allston, MA 02134
2. account: sweetser@world.std.com at PayPal.com
• Mathematica 3.0 notebooks used to create these pages, free to download & print
• Attended the Second Meeting on Quaternionic Structures in Mathematics and Physics in Roma, Italy, September 1999. Presented a paper with Prof. Guido Sandi of BU entitled "Maxwell's vision: electromagnetism with Hamilton's quaternions".
• 2003 Joint Fall Meeting of the New England Sections of APS and AAPT, Bates College, Maine, Oct 3-4. Delivered a 15 presentation titled: "4-Potential Equations for Gravity and Electromagnetism" (html, or as a pdf).

1. Sudbery's first paper (memo, 1977, 44 pages) on why quaternion analysis is no good.
2. Sudbery's second paper (1979, 28 pages) on the topic. Please look to my work above on quaternion analysis for a much better alternative!.
3. 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.
4. Salamin's paper (1979, 9 pages) on rotations.
5. Howell and Lafon's paper (1975, 13 pages) on the efficiency of quaternion multiplication.
6. 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.
   

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! (Would like to know why you visited, what could be added.)

Other Stuff

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Intro, Mathematics, Classical, SR, EM, QM, G, Conclusions

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