Dan Schroeder said:
"But with complex numbers, it seems a little too lucky that the rule for
multiplication, "derived" using the crazy formula i^2 = -1, would turn
out to be just what we need. This issue, of course, is not unique
to quantum mechanics; I find it a bit spooky wherever it comes up in
physics. Probably I just don't understand the math deeply enough."
In David Hestenes' papers you'll find a goldmine of ideas about complex
numbers. He's a mathematical physicist and has been publishing for 35
years.
I refer you first to a 25-page long article entitled "Imaginary Numbers are
not Real - the Geometric Algebra of Spacetime", by Stephen Gull, Anthony
Lasenby, and Chris Doran (of the Cavendish Lab at Cambridge Univ) in
FOUNDATIONS OF PHYSICS 23, #9 (SEPT. 1993). This issue is dedicated to
David Hestenes, and their article is a clearly-written, easy to understand
introduction to his work.
On page 1176 the authors write, "We believe that there are two aspects of
Hestenes' work which physicists should take particularly seriously. The
first is that the geometric algebra of spacetime is the best available
mathematical tool for theoretical physics, classical or quantum. Related to
this part of the program is the claim that complex numbers arising in
physical applications usually have a natural geometric interpretation that
is hidden in conventional formulations."
As references, the authors cite the following articles by David Hestenes:
"A unified language for mathematics and physics" in J.S.R.Chisholm and A.K.
Common, eds., Clifford Algebras and Their Applications in Mathematical
Physics (Reidel, Dordrecht, 1986), p.1.
"Vectors, spinors, and complex numbers in classical and quantum physics",
Am. J. Phys. 39, 1013 (1971).
"Observables, operators, and complex numbers in the Dirac theory," J. Math.
Phys. 16(3), 556 (1973).
"The design of linear algebra and geometry," Acta Appl. Math. 23, 65 (1991).
David Hestenes has published many articles on geometric algebra, a unified
language for physics. David's articles are superb! I refer you to his web
site, where you can download several relevant insightful papers: http://modelingnts.la.asu.edu/GC_R&D.html
In their FOUNDATIONS OF PHYSICS journal article, Gull, Lasenby, and Doran
summarize David's paper entitled "A unified language for mathematics and
physics" (listed above). They write, "In summary, David said that
physicists had not learned properly how to multiply vectors and, as a
result of attempts to overcome this, had evolved a variety of mathematical
systems and notations that has come to resemble Babel."
I've read many of David's mathematical physics papers; they are models of
clarity! I find them immensely satisfying; David has destroyed the Babel
and, in the process of replacing it with a unified mathematical language
for physics, has uncovered serious misconceptions held by the entire
physics community.
Leigh Palmer, you can add to this, right? David tells me that you know
geometric algebra. Anyone else on this listserv?
It's important to realize that there is a strong connection between
Hestenes' mathematical physics work, his physics education research, and
his recent emphasis on the need for university - high school partnerships
for physics teaching reform. That's a topic for another post.
cheers,
Jane
Jane Jackson, Dir., Modeling Workshop Project
Box 871504, Dept.of Physics, ASU, Tempe, AZ 85287
480-965-8438/fax:965-7331. http://modeling.la.asu.edu
"In the matter of physics, the first lessons should
contain nothing but what is experimental and interesting
to see. A pretty experiment is in itself often more
valuable than twenty formulae extracted from our minds."
- Albert Einstein