Hrm… I have several thoughts in relation to this… One perhaps being that we need to perhaps look for a way to replace the energy term with a stress energy tensor...
But, perhaps a more specific interesting thought that I have here relates to almost an “opposite” thought I once had:
Can the space in which the quantum amplitude vectors (forget that they can be decomposed as complex numbers for a moment) move around itself be curved?
Well, this post got me thinking of that again… What would it look like for the “complex plane” to be curved? Well, one form that could take would be to allow an angle other than 2*Pi to be the angle which corresponds to the angle “spinning all the way around”
So, what’s that look like? Well, in some form of units, the arrow’s state can be represented as something like this:
e^iAx
where x is some measure of distance in configuration space and A is some coefficient...
Wait… does not energy itself show up somewhere in that A?
and if initially x was proportional in some fixed way to angle, would not that A somehow correspond to then a curvature of the complex plane, a change in the angle needed to go all the way around?
So it does look like, at least from one perspective, what energy is doing in QM directly corresponds to something sortakinda like curvature.
Except… you can always locally redefine your distance-in-configuration-space units to make A = 1 or whatever, at least locally… so maybe something along the lines of the change in A, or what A “looks like” given different subspaces factored out of the configuration space.
I have one or two mathematical “toy” ideas I want to play around with regards to that, but it’s late, and still I need to think about it a bit more, but, well.… That’s the basic notions I’ve got from the above suggested question.
Anyone else want to go somewhere with that, or smash it? :)
Hrm… I have several thoughts in relation to this… One perhaps being that we need to perhaps look for a way to replace the energy term with a stress energy tensor...
But, perhaps a more specific interesting thought that I have here relates to almost an “opposite” thought I once had:
Can the space in which the quantum amplitude vectors (forget that they can be decomposed as complex numbers for a moment) move around itself be curved?
Well, this post got me thinking of that again… What would it look like for the “complex plane” to be curved? Well, one form that could take would be to allow an angle other than 2*Pi to be the angle which corresponds to the angle “spinning all the way around”
So, what’s that look like? Well, in some form of units, the arrow’s state can be represented as something like this:
e^iAx
where x is some measure of distance in configuration space and A is some coefficient...
Wait… does not energy itself show up somewhere in that A?
and if initially x was proportional in some fixed way to angle, would not that A somehow correspond to then a curvature of the complex plane, a change in the angle needed to go all the way around?
So it does look like, at least from one perspective, what energy is doing in QM directly corresponds to something sortakinda like curvature.
Except… you can always locally redefine your distance-in-configuration-space units to make A = 1 or whatever, at least locally… so maybe something along the lines of the change in A, or what A “looks like” given different subspaces factored out of the configuration space.
I have one or two mathematical “toy” ideas I want to play around with regards to that, but it’s late, and still I need to think about it a bit more, but, well.… That’s the basic notions I’ve got from the above suggested question.
Anyone else want to go somewhere with that, or smash it? :)