Geometric Algebra formulation of Quantum Mechanics

[Milsomonk]

Hi all,
I'm reading a paragraph from "Geometric Algebra for Physicists" - Chris Doran, Anthony Lasenby. I'm quite interested in applying GA to QM but I've got to a stage where I am not following part of the chapter and am wondering if someone can shed a little light for me.

The part I'm not quite sure about is the bit at the bottom where a "map" is found between the normalised spinor and the rotor. I guess I'm just not sure how I might derive this myself, which I would like to do, as I have done for the rest so far. Any guidance would be much appreciated, and wishing all a happy festive season :)

 

[FactChecker]

The only thing to prove is that there is a mapping that is 1-1. The right side of 8.20 expands to
a⁰ + a¹Iσ₁+ a²Iσ₂+ a³Iσ₃
Since the match-ups of the aⁱ on the left and right side are obvious, the 1-1 mapping follows.
The motivation of why he would want to match the two sides up that way, (with sign changes, etc.), is that the calculations of the right side are very methodically defined in geometric algebra and can be used in many situations. The GA calculations and definitions are not specific to this application, whereas the left side Pauli operators have to be specifically defined for that application.

 

[Milsomonk]

Hi, Thanks very much for your reply, it deffinately helps :) I guess I'm not really sure what map means in this context, I sort of have a vague idea but am lacking a more mathematical definition. I'm not really sure what a represents and it's superscript numbers.

 

[FactChecker]

Hi, Thanks very much for your reply, it deffinately helps :) I guess I'm not really sure what map means in this context, I sort of have a vague idea but am lacking a more mathematical definition. I'm not really sure what a represents and it's superscript numbers.

The mapping means very little on its own. The book should show that one can do routine GA calculations and get back to the same results that required specialized Pauli operators. Once that is accomplished, you should see if GA offers some routine, methodical, insight into the physical results that were less obvious without GA.

 

[Milsomonk]

Yeah basically what I'm trying to do is show that you can use the fact that the pauli matrices form a clifford algebra of space, to write the spinor wavefunction as a rotor, thus lending a little more geometric insight into the link between complex numbers and spin. This map appears to be an important component but I simply don't understand what "a" represents.

 

[FactChecker]

Yeah basically what I'm trying to do is show that you can use the fact that the pauli matrices form a clifford algebra of space, to write the spinor wavefunction as a rotor, thus lending a little more geometric insight into the link between complex numbers and spin. This map appears to be an important component but I simply don't understand what "a" represents.

It looks to me like the aⁱs come from the real and imaginary parts of equation 8.16. Beyond that, I can't help you because I don't really know anything about that physics subject.

 

[Milsomonk]

Hmm, yeah I wondered that. Thanks for your help :)