If you have a set of vectors, what are all the scalar functions you can make from those vectors? That is a question that Soledad Villar (JHU) and I have been working on for a few days now. Our requirements are that the scalar be rotationally invariant. That is, the scalar function must not change as you rotate the coordinate system. Today Villar proved a conjecture we had, which is that any scalar function of the vectors that is rotationally invariant can only depend on scalar products (dot products) of the vectors. That is, you can replace the vectors with all the dot products and that is just as expressive.
After that proof, we argued about vector functions of a set of vectors. Here it turns out that there are a lot more options if you want your answer to be equivariant (not invariant but equivariant) to rotations than if you wnt your answer to be equivariant to rotations and parity swaps. We still don't know what our options are, but because it's so restrictive, I think parity is a good symmetry to include.
"Our requirements are that the scalar be rotationally invariant. That is, the scalar function must not change as you rotate the coordinate system." That's the definition of a scalar
ReplyDelete@StevenG: Agreed! That's the point. But usage of the word "scalar" in the real world is very loose/sloppy, so I prefer to be explicit.
Delete(and you can define scalars with other kinds of symmetries too)
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