Thanks. It still may fit what I got from what a physicist (Pagels) wrote though. Your description actually matches what I read once in a 700 page or so mathematical physics text called "Light" by Ditchburn.
My coments about statistics were based on what physicist Heinz Pagels wrote.
When I said statistical; although Pagels implies fluctuations by referring to 'average'; the view you give is clearly explained by Ditchburn. Whether or not the two physicists disagree; when I use "momentum is statistical" I mean at "least in so far as momentum involves a "spread"- which you confirm with the comments on wave trains.
Thus the "uncertainty principle" is a myth in so far as it is no mystery, just no more than common sense. It is as you say just a mathematical property that itself is perfectly clear- like the longer you hear a piece of music play, the more certain you can be of what piece the musician is playing, but you've sacrificed knowing exactly which note, as you have now a spread of notes.
You've got the music's 'momentum' at the cost of knowing its 'position'.
I think you'll find that in practice physicists use the principle in respect to numbers of observations and averaging of statistical fluctuations.
What I understand you're saying is that momentum is necessarily a space+time -spread/ concept; position a space+time -specific concept; so inevitably the more you know one the less you know the other. Like: your recognition of a piece of music requires an inexact position- if you had an exact position (one note) you wouldn't know what piece they were playing.
What Pagels seems to be saying is like: you take lots of little 'momenta' passages from the music; and lots of 'position' snapshots (notes); and from the statistical average of all that information you can make statistical predictions about future position (notes) and momentum (passages) within the limitations of two uncertainty principles. One uncertainty- mathematically as you say, the more you know a passage of music, the less you tie it to one note; and the other uncertainty: the statistical averaging you did.
Make an observation, and your knowledge about your prediction takes a quantum leap. You collapse the statistical averaging, and get a specific relationship (you call "Fourie transform"?) which is your specific position (or momentum) observation.
Is that right?
What about this?:
A 'photon' would be like an 'excerpt' of the music; you have a rough position and a rough music-sense (momentum) but you lose your 'music sense' (momentum) if you specify the position (get a snapshot of one note as the passage goes by). And you lose your 'position sense' (tying down an exact spot in the music) when you take in the view of the music sense (momentum) (requires noticing the whole passage- many notes).
I would say thus that a photon is a MATCH pattern, a relationship between effectively two 'musical chairs games' moving relatively with respect to each other creating a third 'musical chairs game' at the MATCH of the other two.
Suppose you draw a triangular diagram consisting of three corners each made of six little circles (spots).
Call the corners A, B, C. Now you draw three lines connecting three A spots individually to three B spots. Draw three lines connecting three different B spots to three C spots. Draw three lines connecting three different C spots to the three unconnected A spots. No spot has more than one line attached; all spots are connected in this triangle.
Measuring 'position' or 'momentum' is like connecting all the A spots with all the B spots- the more exactly you do this, the more you use up your availability of spots to make any connections with C spots, thus making the C connections uncertain.
Since 'space' and 'time' are examples of 'change' or 'distinction' (to exist a quality must be distinct- if it was in every way the same as another quality it would be that other quality);
and this existence requires three-way self-reference; the diagram maps anything in physics when you allow the spots in each corner to play musical chairs with alternative spots in that corner. (Free will: God gives us the musical chairs; we create the matches through Him).
Thus the connecting lines play 'musical chairs' too, to create a new set of connections- so there you have three 'musical chairs' fields connected by a fourth 'musical chairs' field, a MATCH = a phenomenon. (Or two fields, a match gives third field gives phenomenon. gives possibility of 4th field gives new phenomenon. That gives you a new field which can interact with another giving a third = new phenomenon. And so countless fields all creating new fields, new phenomena. Haven't figured it all out I know).
('Speed' is distance/time is distance/reference distance (maybe why Julian Barbour says time is an illusion?) is available space-matches/actual matches / available time (reference space) matches/ actual matches.
Speed of light a constant means the 'match frequency' (the joining of the dots mentioned above) between two space-time reference frames is constant. So light is like a third inertial reference frame (match system) connecting two other inertial reference frames (match systems).