Thought - Euclidean Space R^(-n)

[mm2013]

While R^1, R^2, ... , R^n comes quite naturally, is it even conceivable to ponder the meaning of R^(-n)? Is this something that even can exist conceptually or is it just jibberish? This was just a random thought that rolled into my head earlier today, and it's something that I think COULD provoke some thought.

 

[gb7nash]

Rⁿ is just a fancy way of saying the space of all possible real-valued n-dimensional vectors. Not really sure what R^-n would mean.

 

[Hurkyl]

It might make sense to call R^-1 the empty set, but this wouldn't extend usefully to more negative exponents.

I'm sure there are things you can do in the sense of category theory to create new objects and morphisms between them so that it would make sense to talk about R^-n -- but I suspect that the net effect of this would wind up being that you have to view all of the Rⁿ as being indistinguishable. (in other words, you have to forget that points look different from lines, lines look different from planes, and so forth)

 

[Deveno]

i have wondered about this, too. the thought i had, is that R^-n might be some kind of "anti-vector space".

in the absence of any other space to interact with, R^-n would look (and act) pretty much like Rⁿ, but my idea is that a space of dimension -k, might "cancel" out k positive dimensions, that is:

Rⁿ "+" R^-k = R^n-k

i'm not sure how you could make this mathematically precise, or even what uses it might have, but it seems to me you could formulate it as some quotient of the tensor algebra (after all, we have annihilator spaces).

 

[chiro]

We can decompose volumes into area-slices, areas into line-slices, and lines into point-slices.

What can you decompose points into? If you have a way of defining that, you might have a definition for R^(-n) but visually I can't think of such a decomposition.

 

[disregardthat]

If you think of it: R^1 x R^-1 = R^0 would violate cardinality of sets.

 

[Deveno]

well, i did not intend it to be the standard cartesian product. that's why i put the quotes around the plus sign.

i don't think it's a matter of decomposing points. a zero-dimensional vector space is just the origin. there should be a symmetry in the construction of R^-1.

i mean think of how we construct the real projective plane from the sphere: first we construct an isometry that exchanges antipodal points. the antipodal isometry actually turns the ball of the sphere "inside out". well you can extend that to all of R³. but even though it still looks like our original space, something subtle has happened, we've reversed the chirality. some spatial relationships that held in the original space don't hold anymore, there's been a change of sign.

so two spinning objects that collide, one from the original space, and one from the antipodal one, their spins cancel.

now, normally, mathematicans will say something like: "by convention, we take e1 to be..." to indicate a choice of orientation is something of an arbitrary choice. but maybe it's not, maybe orientation is just as important as whether or not an integer is a natural number or not. not because of vector properties, but because of some "super-vector" properties that come about when one considers additional structure.

and yes, if you start reducing "anti-vector spaces" to "anti-bases", one has to consider the implication of "anti-sets" (sets with negative cardinality). obviously if one says A "U" -A = Ø, it's not ordinary union we're talking about. but hey, we have a natural structure on P(A), the power set of A, so we ought to be able to create a structure on P(A) x P(A) with a suitable equivalence.

as i pointed out before, it's not immediately clear how useful this is. but i don't think it's entirely nonsensical.

 

[Deveno]

not intending to necro-post, but i did a search for "negative sets" (because of a parenthetical comment on an article about multisets i was reading), and i found this paper:

http://www.csz.com/cyber/html/negsets.pdf