Geometric Algebra formulation of Quantum Mechanics

In summary, Chris Doran discusses how geometric algebra can be used to simplify calculations in quantum mechanics. He explains that a map exists between the normalised spinor and the rotor, and provides a brief mathematical definition of "a."
  • #1
Milsomonk
96
17
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.
upload_2016-12-19_12-15-53.png

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 :)
 
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  • #2
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.
 
  • #3
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.
 
  • #4
Milsomonk said:
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.
 
  • #5
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.
 
  • #6
Milsomonk said:
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.
 
  • #7
Hmm, yeah I wondered that. Thanks for your help :)
 

Related to Geometric Algebra formulation of Quantum Mechanics

1. What is Geometric Algebra formulation of Quantum Mechanics?

Geometric Algebra formulation of Quantum Mechanics is a mathematical framework that describes the behavior of quantum systems using geometric algebra, a branch of mathematics that combines the concepts of linear algebra and geometry. It provides a more intuitive and geometric understanding of quantum mechanics, and has applications in various fields such as quantum computing, quantum information theory, and quantum field theory.

2. How does Geometric Algebra differ from traditional approaches to Quantum Mechanics?

Geometric Algebra differs from traditional approaches to Quantum Mechanics in that it uses a different mathematical structure to describe quantum systems. While traditional quantum mechanics uses complex numbers and matrices, Geometric Algebra uses multivectors, which are composed of scalars, vectors, bivectors, and higher-dimensional objects. This allows for a more concise and elegant description of quantum phenomena.

3. What are the advantages of using Geometric Algebra in Quantum Mechanics?

One of the main advantages of using Geometric Algebra in Quantum Mechanics is that it provides a more intuitive and geometric understanding of quantum phenomena. It also allows for a more elegant and concise representation of quantum systems, making calculations and derivations simpler and more efficient. Additionally, Geometric Algebra has connections to other areas of mathematics, such as differential geometry, which can be useful in studying more complex quantum systems.

4. Are there any limitations to using Geometric Algebra in Quantum Mechanics?

While Geometric Algebra has many advantages, it also has some limitations. One of the main limitations is that it is not as widely used or understood as traditional approaches to Quantum Mechanics, so there may be a learning curve for those who are not familiar with it. Additionally, it may not be suitable for all types of quantum systems, and some problems may still require the use of traditional methods.

5. How is Geometric Algebra formulation of Quantum Mechanics being applied in current research?

Geometric Algebra formulation of Quantum Mechanics is being applied in various areas of research, such as quantum computing, quantum information theory, and quantum optics. It is also being used to study more complex quantum systems, such as quantum field theory and quantum gravity. Additionally, it is being integrated into machine learning algorithms for quantum data analysis and processing.

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