What interfaces are you planning to provide that other AI safety efforts can use? Blog posts? Research papers? Code? Models? APIs? Consulting? Advertisements?
Ah. Thank you, that is perfectly clear. The Wikipedia page for Scalar Field makes sense with that too. A scalar field is a function that takes values in some canonical units, and so it transforms only on the right of f under a perspective shift. A vector field (effectively) takes values both on and in the same space, and so it transforms both on the left and right of v under a perspective shift.
I updated my first reply to point to yours.
Reading the wikipedia page on scalar field, I think I understand the confusion here. Scalar fields are supposed to be invariant under changes in reference frame assuming a canonical coordinate system for space.
Take two reference frames P(x) and G(x). A scalar field S(x) needs to satisfy:
Meaning the inference of S(x) should not change with reference frame. A scalar field is a vector field that commutes with perspective transformations. Maybe that's what you meant?
I wouldn't use the phrase "transforms trivially" here since a "trivial transformation" usually refers to the identity transformation. I wouldn't use a head tilt example either since a lot of vector fields are going to commute with spatial rotations, so it's not good for revealing the differences. And I think you got the association backwards in your original explanation: scalar fields appear to represent quantities in the underlying space unaffected by head tilts, and so they would be the ones "transforming in the opposite direction" in the analogy since they would remain fixed in "canonical space".
Interesting. That seems to contradict the explanation for Lie Algebras, and it seems incompatible with commutators in general, since with commutators all operators involved need to be compatible with both composition and precomposition (otherwise AB - BA is undefined). I guess scalar fields are not meant to be operators? That doesn't quite work since they're supposed used to describe energy, which is often represented as an operator. In any case, I'll have to keep that in mind when reading about these things.
Thanks for the explanation. I found this post that connects your explanation to an explanation of the "double cover." I believe this is how it works:
EDIT: This post is incorrect. See the reply chain 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:
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.
I don't know why other people say it, but I can explain why it's nice to say it.
Logic and reason indicate the robustness of a claim, but you can have lots of robust, mutually-contradictory claims. A robust claim is one that contradicts neither itself nor other claims it associates with. The other half is how well it resonates with people. Resonance indicates how attractive a claim is through authority, consensus, scarcity, poetry, or whatever else.
Survive and spread through robustness and resonance. That's what a strong claim does. You can state that you'll only let a claim spread into your mind if it's true, but the fact that it's so common for two such people to hold contradictory claims indicates that their real metric is much weaker than truth. I'll posit that the real metric in such scenarios is robustness.
Not all disagreements will separate cleanly into true/false categorizations. Godel proved that one.
Thank you. You phrased the concerns about "integrating with a bigger picture" better than I could. To temper the negatives, I see at least two workable approaches, plus a framing for identifying more workable approaches.
As an aside, I think CogEms are a perfectly valid strategy for creating aligned AI. It doesn't matter if most humans have bad interpretability, persuadability, robustness, ethics, or whatever else. As long as it's possible for some human (or collection of humans) to be good at those things, we should expect that some subclass of CogEms (or collection of CogEms) can also be good at those things.