Comments on "Understanding wavefunction values"
Mitchell Porter pointed out some important mistakes, leading to a Physics Stack Exchange discussion that (hopefully) corrected some of my misunderstandings.
I misused the word “spinor" thinking it referred to any vector acted on by the matrices associated with a Clifford Algebra. It doesn’t. A spinor is specifically acted on by a particular subset of the Clifford Algebra. I updated my post to not mention spinors and to call them vectors instead.
The Clifford Algebra might not be the most general algebra that’s relevant for describing objects in spacetime. It seems the “correct” algebra ultimately depends on which operations are used to define the invariants of the particles in question. The Clifford Algebra works for some cases, but it’s unclear to me whether it works for all cases, and there’s a good chance that it doesn’t. In any case, I updated my post to reflect this.
An editor to my Stack Exchange question implicitly suggested that the explanation is relevant for fields, and not just wavefunctions. I considered updating my post to reflect this but decided against it since (1) I didn’t want to discuss the difference between fields and wavefunctions, and (2) my guess is that most people reading haven’t heard of fields, so I would need to introduce them first. I thought that would detract too much from the original point.
Also due to Mitchell Porter’s comments, I updated a case where I said the state acted on by the Clifford Algebra for spacetime could be represented as either a 2-quaternion vector or a 4-complex vector.
Technically that’s correct, but it’s also very confusing. If you represent the Clifford Algebra as a 2x2 matrix of quaternions, then the state would be represented as a 2-quaternion vector. If you represent the Clifford Algebra as a 4x4 matrix of complex numbers, then the state would be represented as a 4-complex vector.
There’s probably a good reason why physicists use the complex representation rather than the quaternion representation. In the fully general case of the algebra of 4x4 matrices, the two may be equivalent, but the complex value representation might be necessary for identifying useful subspaces. I’m unclear on this point, so I updated the post to get rid of the 2-quaternion option.
interstice on LessWrong pointed out a way to understand spinors.
I personally don’t think it’s an intuitive explanation, but others might. Skipping the Clifford Algebra, you can look for ways to create matrices that describe the Lorentz Group, which describes the symmetries of relativistic spacetime. There happens to be a representation of this group as 2x2 complex matrices. The vectors associated with this group would be spinors.
The Lorentz Group is the same group that gives rise to the Minkowski Metric used to define the Clifford Algebra for relativistic spacetime. That makes it pretty useful for intuitively understanding the equations of the spacetime Clifford Algebra.