Geometric Algebra/ Calculus for Physics

Question

I don't know if this would be a better question for physics.SE, but I'll try here first:

There is at least one good book on classical mechanics using the geometric algebra/ calculus (GA): New Foundations in Classical Mechanics by David Hestenes.

Likewise there is at least one good book on classical E&M using GA: Understanding Geometric Algebra for Electromagnetic Theory by John W. Arthur.

My question is, is there some analogous book for quantum mechanics? I know that spinors are often used in QM, and knowing that spinors are a natural part of GA, I wonder if there is any textbook which develops QM from this formalism from the beginning.

I would appreciate any suggestions. :)
Thanks.

Answer 1

To the best of my knowledge there is no such book. David Hestenes has a web page "Selected Papers on Geometric Algebra in Quantum Mechanics":

http://geocalc.clas.asu.edu/html/GAinQM.html

Answer 2

Although they are not dedicated to QM, the following two books devote a few chapters to the modeling of QM using Geometric Algebra :

• Doran & Lasenby, "Geometric algebra for physicists", CUP, 2003 (chapter 8 is about "Quantum theory and spinors", chapter 9 is relevant also)

• William Eric Baylis, "Clifford (geometric) algebras with applications to physics, mathematics, and engineering." Birkhauser, 1996. (specifically Chap.19, "Eigenspinors in Quantum Theory" ; but also Chapter 9-11, "Electron Physics" and "STA and the Interpretation of Quantum Mechanics").

These two books - at least the relevant chapters - look very much like what you're searching for, except that the second one is less of a textbook. Both are excellent references.

If you can read some French (it's got almost more equations than text anyway...), you might also look at another good reference:

• G. Casanova, "L’algèbre de Clifford et ses applications", Advances in Applied Clifford Algebras, vol. 12, no. 1, 2002. around 155 pages.

Part II delves into the solution of Hestenes' equation, among other things. A more mathematical than physical approach, I find, but after all this is a Maths forum, right ?

There are also a few theses here and there, as well as Arxiv papers if I recall correctly, but these hardly qualify as textbooks :-D

Answer 3

The lecture note of Physical Applications of Geometric Algebra is a very good review of GA in physics, including some chapters on QM.