I haven't read through all of this, but I'm sure you took a wrong turn, because at the end you're saying quantum wavefunctions are complex-valued because quaternions are how you should represent quantities in space-time, and that all this has something to do with spinors. And that's wrong in multiple ways.
A quantity is a spinor if it transforms in certain ways under rotations. A complex-valued object in space-time doesn't have to be a spinor; it can also be a scalar, vector, or tensor. Also, on the physics side, a large part of the significance of spinors is that they necessarily become fermionic when quantized.
I suggest reading some resources that don't try to reduce everything to Clifford algebras...
I think we're agreeing on most of that, and I think you're talking about Dirac Spinors rather than spinors more generally. As far as I can tell, the more general spinors are defined based on matrix ring representations of Clifford Algebras. Dirac Spinors are the spinors associated with fermions.
I use some terminology at the end that might be confusing if you haven't read the middle section, specifically the word "states." I use that word to describe the underlying thing that gets manipulated by "spatial operations" like rotation. For spacetime in special relativity, the relevant spatial operations are given by the Lorentz Group, and so what I'm claiming there is that the underlying thing that gets manipulated by the symmetries of the Lorentz Group is a Dirac spinor. I do mostly get this from Wikipedia, so I could easily be missing something that's covered by more authoritative sources. This part in particular in almost directly stated on the Wikipedia page for Gamma Matrices. Was there some misunderstanding, or do you believe I'm incorrect on this point? If you do, can you suggest a resource that takes a different approach?
I believe spinors apply to more than just fermions. Isn't the Higgs field also spinor-valued as a pair of complex values? It's not a Dirac Spinor, but it still seems to be called a spinor. For other bosons, I guessed that they all have 1-complex valued wavefunctions rather than 2-quaternion valued wavefunctions because they follow different symmetries owing to them moving at a constant speed. Moving at a constant speed would presumably get rid of one dimension of possible symmetries, and Cl1,2 is a valid candidate with a spinor class of 1-complex values. In general, the values the wavefunction of a particle seem closely related to which symmetries it obeys, and different symmetries correspond to different spinors. I'm new to all of this, so I could easy be wrong, but I haven't found anything contradicting it. If you have, I'm curious to see it.
I suggest that you go to a Q&A site like Physics Stack Exchange, and ask, "is it true or false that all quantum wavefunctions are spinors?", and give your reasoning. Hopefully you will receive multiple replies which, taken together, will be more enlightening than just my individual perspective.
However, I can say e.g. that in a standard classification of relativistic wavefunctions due to Wigner, particles are classified by mass and spin, and we'd say that spin 0 (e.g. Higgs boson) is a scalar, spin 1/2 (e.g. electron, neutrino, quark) is a spinor, spin 1 (e.g. photon) is a vector, spin 3/2 is technically a spinor-vector, spin 2 is a tensor. These spins all correspond to different transformation properties under the Lorentz group, and only the half-integer spins are spinorial.
The two components of the Higgs field pertain to a quantum number ("weak isospin") independent of the quantum numbers associated with the Lorentz group (mass and spin). They govern the interactions of the Higgs with the weak force.
Is it a terminology issue then? The different particles under the Wigner classification transform differently under the Lorentz Group, but are their invariants are still defined based on Lorentz Group actions? If so, then wouldn't their corresponding algebras be sub-algebras of Cl1,3(R), and the scalar/spinor/vector/spinor-vector/tensors all be subgroups of C^4? That's all I'm saying, or at least that's all I intended to say.
I took your advice and posted the question to Stack Exchange. I'm pending responses.
Check https://paperclip.substack.com/p/comments-on-understanding-wavefunction for a changelist.
I haven't read through all of this, but I'm sure you took a wrong turn, because at the end you're saying quantum wavefunctions are complex-valued because quaternions are how you should represent quantities in space-time, and that all this has something to do with spinors. And that's wrong in multiple ways.
A quantity is a spinor if it transforms in certain ways under rotations. A complex-valued object in space-time doesn't have to be a spinor; it can also be a scalar, vector, or tensor. Also, on the physics side, a large part of the significance of spinors is that they necessarily become fermionic when quantized.
I suggest reading some resources that don't try to reduce everything to Clifford algebras...
Thanks for your comments. I updated this post, and I published an errata based on your comments and the ones I received from Stack Exchange.
https://paperclip.substack.com/p/comments-on-understanding-wavefunction.
I think we're agreeing on most of that, and I think you're talking about Dirac Spinors rather than spinors more generally. As far as I can tell, the more general spinors are defined based on matrix ring representations of Clifford Algebras. Dirac Spinors are the spinors associated with fermions.
I use some terminology at the end that might be confusing if you haven't read the middle section, specifically the word "states." I use that word to describe the underlying thing that gets manipulated by "spatial operations" like rotation. For spacetime in special relativity, the relevant spatial operations are given by the Lorentz Group, and so what I'm claiming there is that the underlying thing that gets manipulated by the symmetries of the Lorentz Group is a Dirac spinor. I do mostly get this from Wikipedia, so I could easily be missing something that's covered by more authoritative sources. This part in particular in almost directly stated on the Wikipedia page for Gamma Matrices. Was there some misunderstanding, or do you believe I'm incorrect on this point? If you do, can you suggest a resource that takes a different approach?
I believe spinors apply to more than just fermions. Isn't the Higgs field also spinor-valued as a pair of complex values? It's not a Dirac Spinor, but it still seems to be called a spinor. For other bosons, I guessed that they all have 1-complex valued wavefunctions rather than 2-quaternion valued wavefunctions because they follow different symmetries owing to them moving at a constant speed. Moving at a constant speed would presumably get rid of one dimension of possible symmetries, and Cl1,2 is a valid candidate with a spinor class of 1-complex values. In general, the values the wavefunction of a particle seem closely related to which symmetries it obeys, and different symmetries correspond to different spinors. I'm new to all of this, so I could easy be wrong, but I haven't found anything contradicting it. If you have, I'm curious to see it.
I suggest that you go to a Q&A site like Physics Stack Exchange, and ask, "is it true or false that all quantum wavefunctions are spinors?", and give your reasoning. Hopefully you will receive multiple replies which, taken together, will be more enlightening than just my individual perspective.
However, I can say e.g. that in a standard classification of relativistic wavefunctions due to Wigner, particles are classified by mass and spin, and we'd say that spin 0 (e.g. Higgs boson) is a scalar, spin 1/2 (e.g. electron, neutrino, quark) is a spinor, spin 1 (e.g. photon) is a vector, spin 3/2 is technically a spinor-vector, spin 2 is a tensor. These spins all correspond to different transformation properties under the Lorentz group, and only the half-integer spins are spinorial.
The two components of the Higgs field pertain to a quantum number ("weak isospin") independent of the quantum numbers associated with the Lorentz group (mass and spin). They govern the interactions of the Higgs with the weak force.
Is it a terminology issue then? The different particles under the Wigner classification transform differently under the Lorentz Group, but are their invariants are still defined based on Lorentz Group actions? If so, then wouldn't their corresponding algebras be sub-algebras of Cl1,3(R), and the scalar/spinor/vector/spinor-vector/tensors all be subgroups of C^4? That's all I'm saying, or at least that's all I intended to say.
I took your advice and posted the question to Stack Exchange. I'm pending responses.
https://physics.stackexchange.com/questions/740332/are-all-wavefunctions-on-spacetime-spinor-valued