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

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


A one page summary of my unified field model for gravity and EM is available here:
pdf
A technically detailed Mathematica notebook starts from the
Lagrange density for unifying gravity and EM, and gets to two experimental
tests that can confirm or reject the proposal (available as
html,
pdf, or
.nb)
Talk entitled "Realizing Einstein's Dreams: Unifying Gravity and Light,
and Why Quantum Mechanics is Weird" for the Spring NE Section of the APS/AAPT (html or
pdf)
Talk entitled "Why a Rank 1 Unified Field Theory is Compelling and
Its Background
Mathematical Structure" for the 8th Eastern Gravity Meeting (html or
pdf)
Talk entitled "Unifying Gravity and EM by Generalizing EM" for the 7th Eastern Gravity Meeting (html or
pdf)
Poster for Brookhaven National Lab meeting, "Unifying Gravity and EM by
analogies to EM" (pdf).
A more formal presentation of the unified field model is available here:
pdf (v1.4, 18 pages).
MIT IAP course: "Unifying gravity and EM by analogies with EM"
Slides available.
Day 1: Lagrange Densities
(html or
pdf)
Day 2: Fields and Quantum Mechanics
(html or
pdf)
Day 3: Forces, Metrics, and New Physics
(html or
pdf)
The
Cosmological Consequences of the Chain Rule (pdf), slides for a 15
minute talk at joint APS/AAPT meeting, March., 2004. A Universe with zero dark matter is good :)
4potential
Equations for Gravity and Electromagnetism (pdf), slides for a 15 minute
talk at a joint APS/AAPT meeting, Oct., 2003.
Dynamic graphs, a
way to visualize complex numbers and quaternions using animations. Uses gif, so all can see for the first time
the cosine of a quaternion! This is hugely fun, honest.

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 "StandUp 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)
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
(.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 stressenergymomentum
2tensor of the electromagnetic field
(.pdf)
Bracket notation &
quaternions: quaternions as a complete inner product space
Multiplying polar representations
(without CampbellHausdorff!)
Commutators
and the uncertainty principle
Unifying the representations
of spin and angular momentum
The Schrödinger
equation
The KleinGordon
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 4force 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 paper was submitted to a peer review journal on the content of the second
paper, with all references to quaternions deleted :) You may read a summary of that
process, along with a call for papers
, the submission
, and comments from the first referee
, second referee
, and editor.
I learned, so it was good.
Strings and Relativistic
Quantum Gravity
A
slide presentation of string ideas, small movies, little math
Answering
prima facie questions in quantum gravity using quaternions (post to
s.p.r)
The length of a quaternion
in curved spacetime: a close relative of the affine parameter of general
relativity
General metrics
Gravitational redshift
experiments
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:
 US mail to: Douglas Sweetser, 1340 Comm Ave Apt 7, Allston, MA 02134

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 34. Delivered a 15 presentation titled:
"4Potential Equations for Gravity and Electromagnetism" (html, or as a pdf).
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.
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
Doug's
background in physics
Thanks to...
Intro, Mathematics, Classical, SR, EM, QM, G, Conclusions
Home Page  Quaternion Physics  Pop Science
Java  The Bike
 Lindy Hop  Contact Doug
Copyright © 1997, doug <sweetser@alum.mit.edu>
All rights reserved worldwide.
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