In 2d electromagnetism splits light waves into TE and TM modes that don't couple to each others (until you add a dielectric). That is unusual to our 3d picture, but doesn't break electromagnetics.
And the generalization of the cross product, the wedge product works just fine in 2d. If you take the wedge of two 1-forms (vectors) in 2d you get a 2-form. In 3d that is equivalent to a (3-2)-form or in other words a (pseudo-)vector. In 2d you get a (2-2)-form or in other words a pseudo-scalar 0-form. That is of course nothing but the z component of the resulting pseudo-vector if you had done it in 3d. So EM in 2d might get flat landers started on weird group theory a bit early, but it is not fundamentally incompatible with life.
Magnetism doesn't use a cross product; it uses a wedge product[0]. It generalizes perfectly well to 2d, you just have <<xy>> instead of <<xy,xz,yz>>, just like you have <x,y> instead of <x,y,z>.
Edit: the reason 2d doesn't have a cross product is because you can no longer misinterpret a bivector <<xy,xz,yz>> as a (1-)vector <yz,xz,xy> by confusing each basis bivector with the basis vector[1] orthogonal to it.
1: Actually one of the two (positive or negative) possible basis vectors; this is why cross products have a right-hand-rule versus left-hand-rule ambiguity.
Maxwell did not use a cross product in his original publications. In fact, the development of the cross product was a response to the longer (quaternion based) formulation of electromagnetism equations at the time.
I'm confused by GP's reasoning. Is it valid to assume that since we have an equation that requires three dimensions to show how a force reacts, that if you remove a dimension that force doesn't exist?
Isn't it just as possible that there's a different equation for how that force would work in two dimensions (or to assume our current equation is a specialized version of a general equation that works in all dimensions)?
It just seems rather odd to assume that since our understanding doesn't extend to a circumstance that means something is impossible there. Or is there some aspect of this I'm missing?
Edit: Perhaps I was misinterpreting the GP contextually. Maybe they were just asking if we have an equation for it, rather than questioning how it could exist.
It's a bit complex for a comment, but electromagnetism defined in terms of differential forms generalizes nicely to all dimensions. There certainly is a 2+1D electromagnetism.
petschge|6 years ago
And the generalization of the cross product, the wedge product works just fine in 2d. If you take the wedge of two 1-forms (vectors) in 2d you get a 2-form. In 3d that is equivalent to a (3-2)-form or in other words a (pseudo-)vector. In 2d you get a (2-2)-form or in other words a pseudo-scalar 0-form. That is of course nothing but the z component of the resulting pseudo-vector if you had done it in 3d. So EM in 2d might get flat landers started on weird group theory a bit early, but it is not fundamentally incompatible with life.
a1369209993|6 years ago
Edit: the reason 2d doesn't have a cross product is because you can no longer misinterpret a bivector <<xy,xz,yz>> as a (1-)vector <yz,xz,xy> by confusing each basis bivector with the basis vector[1] orthogonal to it.
0: https://en.wikipedia.org/wiki/Wedge_product
1: Actually one of the two (positive or negative) possible basis vectors; this is why cross products have a right-hand-rule versus left-hand-rule ambiguity.
yiyus|6 years ago
Maxwell did not use a cross product in his original publications. In fact, the development of the cross product was a response to the longer (quaternion based) formulation of electromagnetism equations at the time.
kbenson|6 years ago
Isn't it just as possible that there's a different equation for how that force would work in two dimensions (or to assume our current equation is a specialized version of a general equation that works in all dimensions)?
It just seems rather odd to assume that since our understanding doesn't extend to a circumstance that means something is impossible there. Or is there some aspect of this I'm missing?
Edit: Perhaps I was misinterpreting the GP contextually. Maybe they were just asking if we have an equation for it, rather than questioning how it could exist.
comnetxr|6 years ago
anticensor|6 years ago
empath75|6 years ago