Remember, we are talking about our expectations, we are not talking about reality. This fact leads to the different frames of reference. The issue is that the observer may not be aware of the entire universe and/or it's impact upon his observations: i.e., if distant entities are to be ignored as insignificant, it is entirely possible that two observers may use a different coordinate system. More important, in comparing their coordinate systems, they may not agree that the sum of all momentums vanishes in the other's coordinate system. At the same time, in order to obtain valid results, they must both operate with the fundamental equation under the constraint that the sum of the momentum of the universe vanishes. It turns out that this means their coordinate systems will not be the same.

How I understand this is that if we have two different sets of elements where both are in a coordinate system that have the momentums vanish we want to compare the two sets and find a coordinate system in which the momentum of the combination of both sets of elements vanish.

If so how can we compare the two coordinate systems? Isn’t all that we can do is compare the elements in the coordinate systems? In which case isn’t it impossible to prove that the two systems are different (other then the number of elements in them). Due to not being able to prove if any element in one set is one of the elements in the other set, how are we even going to be able to determine if the two sets have a different value of 1/k.

I’m wondering at this point if the measure of the fundamental equation and the momentum of the elements is entirely defined by the arbitrary constants that were used to define the constants in your derivation of the Schrodinger equation in the first place, and in so doing defining the speed of the expanding sphere.

The issue is the form of the fundamental equation. If you can ignore the Dirac delta functions (and, since they are point interactions it is always possible to analyze the situation at a small enough scale to make them dynamically irrelevant) then the fundamental equation reduces to a collection of independent wave equations each propagating with a velocity of 1/K. That makes that propagation an expanding sphere. Everyone using that equation must see that same result. This is exactly the same conundrum presented by the Michelson-Morley results and the solution is exactly the same: their coordinate systems must display exactly the same transformations Lorentz and Fitzgerald hypothesized.

How is it that a scale transformation will cause the differentials to vanish? Are only some of the elements going to be scaled or does the scale of the measure have some effect on the value of the differentials?

I suspect that part of my problem is in understanding how the solution of the equation is the expansion of a sphere, how I understand it, any solution to the fundamental equation is describing a sphere expanding at a speed of 1/k and that this means that the term [imath]e^{it/k}[/imath] or one similar to it is multiplied through any solution to the fundamental equation, but I don’t understand how or why something like this comes about.