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There is at least one good book on classical mechanics using Geometric Algebra (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, for example, that Pauli matrices can be represented as elements of a Clifford algebra. Is there a textbook that takes this tact? I have seen several short papers on the subject, but no textbook.

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  • $\begingroup$ Maybe its your time to write one? $\endgroup$ – CuriousOne Dec 17 '14 at 2:17
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    $\begingroup$ I was thinking the same thing. Give me a year or two to research and maybe I'll publish one. :) $\endgroup$ – got it--thanks Dec 17 '14 at 2:20
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    $\begingroup$ what is the advantage of this approach? $\endgroup$ – Jiang-min Zhang Dec 17 '14 at 8:22
  • $\begingroup$ @Jiang-minZhang In my study of GA in the context classical mechanics and E&M, I saw that it $1)$ provided a geometrical intuition to certain scenarios, where otherwise, I'd just be following the math without any picture in my head and $2)$ it sometimes simplifies your algebra (often, though, it really takes just about as long using GA as the standard techniques). $\endgroup$ – got it--thanks Dec 17 '14 at 17:52
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    $\begingroup$ Also, after spending several months (coming up on a year) studying GA as a mathematical object, I'd like to see exactly where it can go. It can model rotations in classical mechanics and special relativity very succinctly and it's been used to develop an alternative approach to GR. So I'm left to wonder: could we build a solid theory of quantum mechanics on GA? IDK. But I'd like to find out! P.S. To see some of the work that gives me hope, see here. $\endgroup$ – got it--thanks Dec 17 '14 at 17:58
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Geometric Algebra for Physicists by Doran and Lasenby (Cambridge University Press) has an introduction to Quantum Mechanics. The quantum section starts 250 pages into the text, and not surprising is a spin-first type introduction to quantum mechanics. It does into the detail of a spin 1/2 system and notes the similarity to to the properties of rotors (introduced in the earlier classical mechanics section), then builds up a geometric algebra based introduction to spin at the Pauli (nonrelativistic) level, then relativistically with the Dirac equation. They do examples of observables, plane waves, scattering, and the relativistic hydrogen atom before moving on to multiparticle quantum mechanics. later in the penultimate chapter of the book after they've covered some classical gauge theory they go back to quantum mechanics to look at quantum theory as a gauge theory, and in the final chapter on gravitation they again bring up quantum mechanics. I'd say you might have to read a fair amount of the whole book to get 100% of the quantum mechanics information out of it, depending on how much you already know. I think they also don't do as many examples as you'd expect in a book devoted just to quantum mechanics (only about 90 pages of the 567 pages are devoted to quantum mechanics).

The prerequisites of the book itself (assuming you start at the beginning) are mild, really the barrier is if you want to learn it. But if you want to do relativistic quantum mechanics, you'll have to learn the same mathematical objects, so really its about spending the time to learn about the different operations.

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