I agree that taking quotients of the configuration space is a more natural way of doing things. But, when you say
quotienting R^6 by this action of E, you’re left with a 2-dimensional quotient space
don’t you mean you’re left with a 3-dimensional quotient space? Counting degrees of freedom: wherever we put A, that eats the translation. Wherever we put B, that eats the rotation and we’re left with the distance |AB| (one dimension). Wherever we put C, that eats reflection and we’re left with the position of C up to reflection. So, the space of triangles ends up as R x R x (R / ~), where a~b iff |a|=|b|.
But then, this space should be homeomorphic to the one Eliezer gives, with the relative distances. We’ll take a point (x,y,z) in R x R x (R / ~). Then |AB|=x, |AC|=hypot(y, z), |BC|=hypot(y-x, z), clearly this is continuous and nice, and also clearly the image doesn’t change if we replace z by -z (so the function is well-defined despite the domain being a quotient space, which generally needs to be checked). Showing that the mapping is invertible, with continuous inverse, is left as exercise for the reader.
Consider now the apparent boundary when we embed this in R³; it’s z=0, which corresponds to “A, B and C form a straight line”, which (triangle inequality) corresponds to the boundary of the subset of distance-space. But if you imagine the particles moving, it’s a lot more obvious that you should bounce off the ”/ ~” surface than that you should bounce off the “if you cross this surface you get a distance-tuple that’s un-geometric” surface. Similarly, straight lines in R x R x (R / ~) correspond to fixing any two particles and moving the third in a straight line.
I would conclude from this that the equations of physics in the quotient space are likely to be much nicer than the equivalent equations in distance-tuple space.
So why bother formulating the relational configuration space in distance-tuples? After all, with the distance-tuples, you still end up having to quotient afterwards on particle-swapping to get the quantum-mechanical picture. Isn’t it easier to just use quotients, rather than an odd mix of quotients, new bases, and subsets?
(Note 1: “g” = me; I had to change my username when Less Wrong started, but existing Overcoming Bias comments kept their existing commenter names. Note 2: I only just saw this.) Yes, I meant 3-dimensional. Sorry. (And I think we are agreed that absolute space quotiented by symmetries is likely to be a nicer thing to work with than a space parameterized by relative distances.)
I agree that taking quotients of the configuration space is a more natural way of doing things. But, when you say
don’t you mean you’re left with a 3-dimensional quotient space? Counting degrees of freedom: wherever we put A, that eats the translation. Wherever we put B, that eats the rotation and we’re left with the distance |AB| (one dimension). Wherever we put C, that eats reflection and we’re left with the position of C up to reflection. So, the space of triangles ends up as R x R x (R / ~), where a~b iff |a|=|b|.
But then, this space should be homeomorphic to the one Eliezer gives, with the relative distances. We’ll take a point (x,y,z) in R x R x (R / ~). Then |AB|=x, |AC|=hypot(y, z), |BC|=hypot(y-x, z), clearly this is continuous and nice, and also clearly the image doesn’t change if we replace z by -z (so the function is well-defined despite the domain being a quotient space, which generally needs to be checked). Showing that the mapping is invertible, with continuous inverse, is left as exercise for the reader.
Consider now the apparent boundary when we embed this in R³; it’s z=0, which corresponds to “A, B and C form a straight line”, which (triangle inequality) corresponds to the boundary of the subset of distance-space. But if you imagine the particles moving, it’s a lot more obvious that you should bounce off the ”/ ~” surface than that you should bounce off the “if you cross this surface you get a distance-tuple that’s un-geometric” surface. Similarly, straight lines in R x R x (R / ~) correspond to fixing any two particles and moving the third in a straight line.
I would conclude from this that the equations of physics in the quotient space are likely to be much nicer than the equivalent equations in distance-tuple space.
So why bother formulating the relational configuration space in distance-tuples? After all, with the distance-tuples, you still end up having to quotient afterwards on particle-swapping to get the quantum-mechanical picture. Isn’t it easier to just use quotients, rather than an odd mix of quotients, new bases, and subsets?
(Note 1: “g” = me; I had to change my username when Less Wrong started, but existing Overcoming Bias comments kept their existing commenter names. Note 2: I only just saw this.) Yes, I meant 3-dimensional. Sorry. (And I think we are agreed that absolute space quotiented by symmetries is likely to be a nicer thing to work with than a space parameterized by relative distances.)