Alright, time to talk little bit about how Special Relativity falls out from the analysis (see the book on the right). This one is in my opinion quite a bit easier to understand than the deduction of Schrödinger, at least if the reader is familiar enough with the logical mechanisms of Special Relativity.
The first signs of relativistic relationships arose in Chapter 3, during the deduction of Schrödinger Equation (see the discussion relating to the $mc^2$ term on page 55). Chapter 4 contains a rather detailed presentation of the explicit connection between Relativity, and general epistemological requirements. If you have followed the arguments through Schrödinger, Relativity falls out almost trivially. In fact, someone who is familiar enough with the logical mechanisms behind Special Relativity, might already guess how things play out after reading just the first couple of pages of the chapter.
Unfortunately most people have a rather naive perspective towards Relativity, more revolving around ontological beliefs of space-time constructions, rather than the actual logical relationships that the theory is composed of. The space-time construction is just a specific interpretation of Special Relativity, and if the reader is under the impression that this interpretation is necessary, then the logical connection between fixed velocity wave functions to relativistic relationships might not be too obvious.
Because the space-time interpretation is so common, I think it's best to first point out few facts that everyone having any interest in understanding Relativity should be aware of.
In my opinion, one of the best sources of information on Special Relativity is still Einstein's original paper from 1905. Because unlike most modern representation, that paper was still reasonably neutral in terms of any speculative ontologies. In the paper, the relativistic relationships are viewed more as requirements of Maxwell's equations of electromagnetism (For instance, the moving magnet and conductor paradox is mentioned in the very opening) under the fact that one-way speed of light cannot be measured (something that was well understood at the time). Relativistic space-time is not mentioned anywhere, because that idea arose later, as Herman Minkowski's interpretation of the paper. Later, space-time became effectively synonymous to Relativity because the later developments, especially General Relativity, happened to be developed under that interpretation.
The history of Relativity is usually explained so inaccurately, filled with ridiculous myths and misconceptions, that it seems almost magical how Einstein came up with such a radical idea. But once you understand the context in which Einstein wrote the paper, the steps that led to Special Relativity start to look quite a bit more obvious. I would go so far as to say pretty much inevitable.
When Einstein was writing the paper, it was well understood within the physics community that the finity of the propagation speed of information (speed of light) led to the impossibility of measuring the simultaneity of any two spatially separated events. If the events look simultaneous, it is not possible to know whether they actually were simultaneous without knowing the correct time delays from the actual events. To know these delays, you would have to first know the speed of light.
But that leads directly to another fact which was also well understood back then. That one-way speed of light cannot possibly be measured. In order to measure the speed, you need two clocks that are synchronized. Which is the same thing as making sure the clocks started timing simultaneously.
That is to say, you cannot synchronize the clocks without knowing the one-way speed of light, and you can't know the one-way speed of light, without synchronizing the clocks. Note that clocks are going to be by definition electromagnetic devices, thus there must be an expectation that moving them can affect their running rate. The strength of the effect is expected to depend on the one-way speed of light in their direction of motion. Which just means you can't even synchronize the clocks first and then move them.
So here we have ran into a very trivial circular problem arising directly from finity of information speeds, that cannot be overcome without assumptions. It wasn't possible then, it's not possible now, and it will not be possible ever, by definition. This problem is not that difficult to understand, yet we still have people performing experiments where they attempt to measure one-way speed of light, without realizing the trivial fact that in order to interpret the results, we must simply assume some one-way speed of light. That is exactly the same thing as measuring whether water runs downhill, after defining downhill to be the direction where water runs.
As simple as this problem is, it does not exist in the consciousness of most people today, because they keep hearing about all kinds of accurate values for speed of light. It is almost never mentioned that these measurements are actually referring either to average two-way measurements (use a single clock), or to Einstein convention of clock synchronization (the convention established in his paper; assume C to be isotropic in all inertial frames, and synchronize the clocks under that assumption).
Next important issue to understand is how exactly Einstein's paper is related to the aether theories of the time. The physics community was working with a concept of an aether, because it yielded the most obvious interpretation of C in Maxwell's Equations, and implied some practical experiments. Long story short, the failure to produce expected experimental results led Hendrik Lorentz (and George FitzGerald) to come up with an idea of length contraction affecting moving bodies (relating to variations to the propagation speeds of internal forces), designed to explain why natural observers could not measure aether speeds.
The significance of this development is that the transformation rules Lorentz came up with survive to this day as the central components of Einstein's theory of Special Relativity; that is why the relativistic transformation is called Lorentz transformation.
In terms of pure logic, Lorentz' theory and Special Relativity were effectively both valid. The difference was philosophical. For anyone who understands relativity, it is trivial to see that Lorentz' theory can be seen as completely equivalent to Einstein's in logical sense; just choose arbitrarily any inertial frame, and treat that as the frame of a hypothetical aether. Now any object moving in that frame will follow the rules of Lorentz transformation, just like they do in Special Relativity, and all the other consequences follow similarly. For natural observer, everything looks exactly the same.
When Einstein was thinking about the problem, he had Lorentz' theory in one hand, and the fact that one-way speed of light / simultaneity of events cannot be meaningfully measured on the other hand. It doesn't take an Einstein (although it did) to put those together into a valid theory. Since the universal reference frame could be, and would have to be, arbitrarily set - as far as any natural observer goes - it is almost trivial to set C to be isotropic across inertial frames, and let the rest of the definitions play out from there.
So getting to Special Relativity from that junction is literally just a matter of defining C to be isotropic, not because you must, but because you can. Setting C as isotropic across inertial frames is exactly what the entire paper about Special Relativity is about. Note that the language used in the paper is very much about how things are measured by observers, when their measurements are interpreted under the convention defined in the paper.
While Lorentz' theory was seen as a working theory, it was also seen as containing a redundant un-observable component; the aether. Thus Einstein's theory would be more scientifically neutral; producing the same observable results, but not employing a concept of something that cannot possibly be observed by its own definition.
And just judging from the original paper itself, this is certainly true. But there is great irony in that then Einstein's theory would get a redundant un-observable component tacked to it within few years from its introduction; the relativistic space-time.
A typical knee-jerk reaction at this point is to argue that relativistic space-time is a necessary component by the time we get to general relativity, but that is actually not true. General relativity can also be expressed via different ontological interpretations; some may be less convenient than others depending on purpose, but it is certainly possible.
Another reaction is to point out that different ontological interpretations of Relativity do not fall under the category of physics at all. This is true; but it is also true when it comes to relativistic space-time.
There really are some very solid epistemological reasons for the validity of relativistic time relationships, that have nothing to do with neither aether concepts, nor relativistic space-time concepts, or any other hypothetical structure for the universe. "Epistemological reason" means purely logical reasons that have got nothing to do with the structure of the universe, but everything to do with how we understand things, and those reasons are what Chapter 4 is all about.
I will write a post more directly related to the arguments of Chapter 4 very soon.
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