Papers in
Geometric Algebra and Foundations of Physics
Geometric Algebra
Did you know that the inner product space R^{n}
can be embedded in a vector space of dimension 2^{n}
which is also an associative algebra with unit, the geometric algebra?
Some members of the geometric algebra represent geometric objects in R^{n}.
Other members represent geometric operations on the geometric objects.
Geometric algebra and its extension to geometric calculus unify, simplify,
and generalize vast areas of mathematics that involve geometric ideas,
including linear algebra, multivariable calculus, real analysis, complex analysis,
and euclidean, noneuclidean, and projective geometry.
They provide a unified mathematical language for physics (classical and quantum mechanics, electrodynamics, relativity), the geometrical aspects of computer science (e.g., graphics, robotics, computer vision),
and engineering.
This textbook for the first undergraduate vector calculus course presents a unified treatment of vector calculus and geometric calculus,
while covering a majority of the usual vector calculus topics.
The link is to the book's web page.
July 23, 2014. Second printing, corrected and slightly revised.
This textbook for the first undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra,
while covering a majority of the usual linear algebra topics.
The link is to the book's web page.
Mar. 6, 2013. Second printing, corrected and slightly revised.
The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics.
No knowledge of physics is required.
The section Further Study lists many papers available on the web.
Mar. 14, 2015. Fixed discussion of Pauli equation.
Dec. 30, 2012. Several changes in Geometric Calculus section.
Oct. 15, 2012. Fixed 1.5.4h.
Adv. Appl. Cliff. Alg. 12, 1-6 (2002). (Somewhat improved.)
Presented at
The Fifth International Conference on
Clifford Algebras and their Applications in
Mathematical Physics,
Ixtapa, Mexico,
June 27-July 4, 1999.
Abstract: We give a simple, elementary, direct, and motivated construction of the geometric algebra over
R^{n}.
Adv. Appl. Cliff. Alg. 8, 5-16 (1998). (Error corrected.)
Presented at Octonions and Clifford Algebras,
Corvallis, 19-20 April 1997.
Abstract: Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus.
We assume only that the tangential derivative exists and is Lebesgue integrable.
We also give sufficient conditions that the tangential derivative exists.
Before tackling this paper you might want to read my paper Stokes' Theorem. See below.
3/13/04 Improved join and meet procedures.
Originally distributed by
Mark Ashdown
at Cambridge University.
Distributed here with permission of Ashdown.
I have fixed a few bugs and added several procedures.
You might find the software useful in conjunction with the survey paper above.
It works on Maple V and, I think, higher versions.
A 100 page book on general relativity (75 pages in the main text).
March 11, 2015. Small updates with new observations.
April 7, 2013. Used recently released cosmological parameters from the Planck spacecraft.
Sept 15, 2012. Added description of binary white dwarf system J0651.
May 3, 2009. Fixed formatting problems. Content unchanged.
Version 3.5. Added new evidence for dark matter from the bullet cluster and new results from the double pulsar.
Version 3.4. New title. Many small improvements.
From the preface:
"My purpose here is to provide,
with a minimum of mathematical machinery and in the fewest possible pages,
a clear and careful explanation of the physical principles and applications
of classical general relativity.
The prerequisites are single variable calculus,
a few basic facts about partial derivatives and line integrals,
a little matrix algebra, and some basic physics.
Only a bit of the algebra of tensors is used; it is developed in about a page of the text.
The book is for those seeking a conceptual understanding of the theory,
not computational prowess.
Despite it's brevity and modest prerequisites,
it is a serious introduction to the physics and mathematics of general relativity which demands careful study.
The book can stand alone as an introduction to general relativity
or it can be used as an adjunct to standard texts."
Found. Phys. Lett. 19, 631-631 (2006).
Abstract: The time-redshift relation of Carmeli et al. differs from that of the
standard flat ΛCDM model by more than 500 million years for 1 ≤ z ≤ 4.5.
With Ettore Minguzzi.
Found. Phys. Lett. 16, 593-604 (2003).
Abstract: This paper gives two complete and elementary proofs that if the speed of light over closed paths has a universal value c, then it is
possible to synchronize clocks in such a way that the one-way
speed of light is c.
The first proof is an elementary version of a recent proof.
The second provides high precision experimental evidence
that it is possible to synchronize clocks in such a way
that the one-way speed of light has a universal value.
We also discuss an old incomplete proof by Weyl
which is important from an historical perspective.
Much improved version of Am. J. Phys. 69, 223-225 (2001).
Abstract: In general relativity, spacetime is inseparable from a gravitational field:
no field, no spacetime. This is a lesson of Einstein's hole argument.
We use a simple transformation in a Schwartzschild spacetime to illustrate this.
March 20, 2008. Much improved from the published version.
Found. Phys. Lett. 3, 493 (1992).
Abstract: There is no indication of time dilation of clocks or of length contraction of rods
in Marzke and Wheeler's clock or in Desloge's metrosphere.
Am. J. Phys. 51, 795-797 (1983).
Abstract: This paper (i) gives necessary and sufficient conditions that clocks
in an inertial lattice can be synchronized,
(ii) shows that these conditions do not imply a universal light speed, and
(iii) shows that the terrestrial redshift experiment provides evidence that clocks
in a small inertial lattice in a gravitational field can be synchronized.
Not published.
Abstract: We outline a simple development of special and general relativity based
on the physical meaning of the spacetime interval.
The Lorentz transformation is not used.
The approach is suitable for beginning students.
John Wheeler wrote to me (79/07/31): "It makes sense to me.
After all one is dealing with a local problem, that means a local inertial Lorentz frame, that means just what you say:
'The metric postulate of general relativity rests on (2) and (3), and not the strong equivalence principle.' "
Since Wheeler wrote, I have improved the manuscript.
His reference "(2) and (3)" has become (1)-(3) on p. 8.
Slightly improved version of Int. J. Theor. Phys. 42, 943-953 (2003).
Presented at Quantum Composite Systems: theory, experiment and applications,
Ustron, Poland, September 3-7, 2002.
Abstract: Entanglement has been called the most important new feature of the quantum world.
It is expressed in the quantum formalism by the joint measurement formula.
We prove the formula for projection valued observables from a plausible assumption,
which for spacelike separated measurements is an expression of relativistic causality.
The state reduction formula is simply a way to express the joint measurement formula after one measurement has been made, and its result known.
Nov. 16, 2014. New footnote citing a result of Gisin.
Sept. 3, 2013. New footnote clarifying term "nonlocal".
Mar. 28, 2009. A few minor changes.
Oct. 19, 2008. Very minor changes in wording.
Aug. 11, 2008. A few minor changes.
July 27, 2008. Improved readability. No new content.
Much improved version of paper presented at
Quantum Theory without Collapse, Rome, April 19-21, 1989.
Abstract: There is a consistent and simple interpretation of the quantum theory of isolated systems. The interpretation suffers no measurement problem and provides a quantum explanation of state reduction, which is usually postulated. Quantum entanglement plays an essential role in the construction of the interpretation.
July 27, 2008. Improved readability. No new content.
Am. J. Phys. 67, 613-615 (1999). With Martin Barrett.
Abstract: Magnetic work takes two forms in the thermodynamics of a paramagnet as
developed in many textbooks. We observe that in the case when the lattice energy is excluded, the form dW = BdM cannot be used in a fundamental thermodynamic equation. This shows that there are thermodynamic systems with no fundamental thermodynamic equation.
Am. J. Phys. 63, 1122-1127 (1995).
Abstract: A new statement of the second law of thermodynamics is given.
The law leads almost effortlessly, for very general closed systems,
to a definition of absolute entropy S,
a demonstration that ΔS ≥ 0 in adiabatic processes,
a definition of temperature,
and a demonstration that dS ≥ δQ/T along quasistatic processes.
Entropy is given a clear physical meaning.
An abridged version appeared in Real Analysis Exchange 27, 739-747 (2002).
Abstract: We give a simple proof of Stokes' theorem on a manifold assuming only that the exterior derivative is Lebesgue integrable. The proof uses the integral definition of the exterior derivative and a generalized Riemann integral.
I give concise derivations of Price's equation and the criteria for kin and group selection,
prove that kin and group selection are equivalent,
and discuss the controversies about altruism.
Alan Macdonald
Professor Emeritus of Mathematics
Luther College
Decorah, IA 52101
macdonal at luther dot edu