A 3-sphere is the analog of a 3 dimensional Euclidean sphere in four dimensions. Our usage was toward the 3-sphere manifold, or surface, where the Euclidean space R^3 is compactified by a point at infinity.
"Just co-incidentally, in my foraging I just came across an arXiv item that might fit in here; ariv:1202.1321v1 [quant-ph] 7 Feb 2012 -- "Modified Schrodinger equation, its analysis and experimental verification" by: Isaac Shnaid. What he has done is to substitute a local time variable for 't' in the classical Schrodinger equation and also the same modification to deBroglie. It results in wavefunction propagation globally at a single magnitude of light velocity from an origin of t=0 as a continuously expanding sphere, instead of the instantaneous global propagation of conventional wavefunction."
Pretty bold conjecture. Quick scan tells me that potential energy and time are identical at the initial condition, which implies equivalence of time dependent and time independent versions of the Schrodinger equation -- the novelty is in preserving time unitarity without quantum discontinuity. Thanks for the link; I want to give it a serious read. My gut feeling is that the treatment might have to discard advanced solutions to the Schrodinger equation, and thus give up global time symmetry, and I don't think I can live with that.
"I could follow the reasoning without actually doing calculus, which is rare for me and recommends the uncluttered presentation of the abstract."
Yes, I agree that even at a glance, it looks easy to read.
"Topologically it might be another argument for Joy's framework, I think it's probably as difficult for many as for me to conceive of 'a simple pole at infinity' resulting from infinite radii reaching a maximal and then collapsing back to a point outside the maximal sphere (Riemann), which essentially makes the surface of the sphere a new origin. Correct me where I'm astray, with a straight face :-| jrc"
Well, Joy's mathematical tools are entirely different -- using the algebra of geometry generalized as topology -- while Schnaid is addressing field theory in the classical wave equation. Joy's framework is specifically designed to measure quantum correlations in a continuous measurement function, while Schrodinger's framework is for the evolution of a non-quantized wave function. Where the ideas intersect, however, is in the determinism of the result -- a statistical correlation like Joy's doesn't invoke a time parameter for the initial and end state, yet it does predict continuous covariance; the Schrodinger equation is natively covariant, dependent on boundary conditions. Joy's framework is stronger, because it eliminates the need for boundary conditions as well as a privileged coordinate system, while retaining the continuous measurement function of an analytical model, as permitted by the simple connectedness of the spherical manifold.
You have a bit of a misconception of the point at infinity. It's an analytical point, a way of saying that a curve is closed (a line is the special case of a curve). You've heard that old magician's patter -- "You'll notice that at no time does my hand ever leave my arm" -- same idea. :-)
Maybe more, later.