[Link] Wavefunctions: from Linear Algebra to Spinors

EDIT: This post is incorrect. See the reply chain [LW(p) · GW(p)] below. After correcting my misunderstanding, I agree with your explanation.

The difference you're describing between vector fields and scalar fields, mathematically, is the difference between composition and precomposition. Here it is more precisely:

• Pick a change-of-perspective function P(x). The output of P(x) is a matrix that changes vectors from the old perspective to the new perspective.
• You can apply the change-of-perspective function either before a vector field V(x) or after a vector field. The result is either V(x)P(x) or P(x)V(x).
• If you apply P(x) before, the vector field applies a flow in the new perspective, and so its arrows "tilt with your head."
• If you apply P(x) after, the vector field applies a flow in the old perspective, and so the arrows don't tilt with your head.
• You can do replace the vector field V(x) with a 3-scalar field and see the same thing.

Since both composition and precomposition apply to both vector fields and scalar fields in the same way, that can't be something that makes vector fields different from scalar fields.

As far as I can tell, there's actually no mathematical difference between a vector field in 3D and a 3-scalar field that assigns a 3D scalar to each point. It's just a choice of language. Any difference comes from context. Typically, vector fields are treated like flows (though not always), whereas scalar fields have no specific treatment.

Spinors are represented as vectors in very specific spaces, specifically spaces where there's an equivalence between matrices and spatial operations. Since a vector is something like the square root of a matrix, a spinor is something like the square root of a spatial operation. You get Dirac Spinors (one specific kind of spinor) from "taking the square root of Lorentz symmetry operations," along with scaling and addition between them.

As far as spinors go, I think I prefer your Lorentz Group explanation for the "what" though I prefer my Clifford Algebra one for the "how". The Lorentz Group explanation makes it clear how to find important spinors. For me, the Clifford Algebra makes it clear how the rest of the spinors arise from those important spinors, and it makes it clear that they're the "correct" representation when you want to sum spatial operations, as you would with wavefunctions. It's interesting that the intuition doesn't transfer as I expected. I guess the intuition transfer problem here is more difficult than I expected.

Note: Your generalization only accounts for unit vectors, and spinors are NOT restricted to unit vectors. They can be scaled arbitrarily. If they couldn't, ψ†ψ would be uniform at every point. You probably know this, but I wanted to make it explicit.

In the 2D matrix representation, the basis element corresponding to the real part of a quaternion is the identity matrix. So scaling the real part results in scaling the (real part of the) diagonal of the 2D matrix, which corresponds to a scaling operation on the spinor. It incidentally plays the same role on 3D objects: it scales them. Plus, it plays a direct role in rotations when it's -1 (180 degree rotation) or 1 (0 degree rotation). Same as with i, j, and k, the exact effect of changing the real part of the quaternion isn't obvious from inspection when it's summed with other non-zero components. For example, it's hard to tell by inspection what the 2 or the 3j is doing in the quaternion 2+3j.

In total, quaternions represent both scaling, rotating, and any mix of the two. I should have been clearer about that in the post. Spinors for quaternions do include any "state changes" resulting from the real part of the quaternion as well as any changes resulting from i, j, and k components, so the spinor does use all degrees of freedom.

The change in representation between 2-quaternion and 4-complex spinors is purely notational. It doesn't affect any of the math or underlying representations. Since a quaternion operation can be represented by a 2x2 complex matrix, you can represent a 2-quaternion operation as the tensor product of two 2x2 complex matrices, which would give you a 4x4 complex matrix. That's where 4x4 gamma matrices come from-- each is a tensor products of two 2x2 Pauli matrices. For all calculations and consequences, you get the exact same answers whether you choose to represent the operations and spinors as quaternions or complex numbers.