Exploring the Relationship Between Spinors and Clifford Algebras

[Gerenuk]

TL;DR Summary

Quantum mechanics of spins and the Born rule are expressed with clifford algebra differently from (most) previous works. Is that a representation for spinors?

I've used a particular clifford algebra expression for a quantum mechanics wave vector to see if the Born rule can become a simple linear inner product in a clifford algebra formulation. The expression

$$\Psi=\sum_i (e_i+Jf_i)\psi_i$$

turned out to be successful, where $J$ is the imaginary unit. For a single spin, I deduce the expression

$$\Omega=J(\Psi\Psi^\dagger-1)=T+Xx+Yy+Zz$$

where $T,X,Y,Z$ are bivectors made from $e_i,f_i$ and $x,y,z$ are the Bloch vector coordinates. This expression encodes all information about the spin and can be used in an inner product with another state to calculate probabilities of measurement. If you are interested, then the attached file explains the missing details. The short story is: Equation (2) means I can use equation (3) to get the probability of measurement and for a single spin you get equation (8) with the rotor for spatial rotations (10).

The question is: Can someone comment if this can be seen as what people mean when they talk about the square root of a vector in spinor theory?

I have an expression for $\Omega$ which has $x,y,z$ coordinates and behaves like a vector with the inner product being the Born rule. I have the wave vector $\Psi$ which is like the square root of it?

The question here is whether these particular expressions effectively represents spinor theory. I'm not looking for explanations about spinors which introduce other mathematical concepts (as I have a lot of those).

Coincidentally, as I was googling for complex clifford algebra, I found a very recent work https://arxiv.org/abs/2201.02246 that start with the same expression, but goes into a slightly different direction:

 

[otennert]

May I ask back if I understand it correctly you are asking about the meaning of "a representation of a spinor", and not a "spinor representation" of something else, like e.g. a group or an algebra? I admit I haven't read your paper yet, but I would first rule out any misunderstandings upfront and fully understand what you are asking for.

Michael Atiyah has once been quoted with speaking about the "square root of geometry" , and the connection between spinors and Clifford algebras is deep. In a certain sense it can be exemplified by the fact that you can construct vector representations from spinor representations, but not vice versa, as e.g. formally written as $\frac12\otimes\frac12 = 0\oplus 1$.

 

[Gerenuk]

May I ask back if I understand it correctly you are asking about the meaning of "a representation of a spinor", and not a "spinor representation" of something else, like e.g. a group or an algebra? I admit I haven't read your paper yet, but I would first rule out any misunderstandings upfront and fully understand what you are asking for.

Michael Atiyah has once been quoted with speaking about the "square root of geometry" , and the connection between spinors and Clifford algebras is deep. In a certain sense it can be exemplified by the fact that you can construct vector representations from spinor representations, but not vice versa, as e.g. formally written as $\frac12\otimes\frac12 = 0\oplus 1$.

Thanks for asking. I may not be using the most correct words. I'm not looking for groups which are the set of all representations? I'm looking at a particular wave vector with numbers in it which represents one or multiple qubits. This could be also represented as a complex vector. Instead I rewrite this complex vector with a Clifford Algebra expression, because it makes the maths neater.

I have the impression I'm constructing spinors and vectors all at once, but that's only a guess and I'd love to hear someone who understands my expression. It's a bunch of expressions for common things in basic QM. You may assume the algebra is right.