A wave function is the probability (density) of something before a measurement. You don't need a matrix for defining this. Matrices are overall questionable if the universe is observer dependent, as your matrix only can be mine, agreeing on a measurement and position in space and time, when us being at absolute rest with each other, (or) equivalent in all circumstances and respects. If there is a matrix to the universe, it will not be a measurable one, only theoretical. Doesn't mean it can't exist, but I don't think we will be able to measure it from where we are.

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what I might be able to argue is that as we find some principles to be common 'laws', no matter what we measure relative each other, 'locality' should be the most objective description of what a measurement, and the universe, is. And as locality has a direct coupling to scaling, and as QM also is a question of at what scale you are measuring? Then again, to assume some 'objectively possible' common to us all matrix to exist at a very small plane, should crave you to create a split between macroscopic observer dependencies and this 'objective microscopic reality'. Either we can magnify and shrink, using scales, and smoothly so, a universe. Or we will need 'emergences' taking place, a little like black body radiation coming in discrete energies 'photons', as well as creating discrete 'orbitals'.

It makes sense to me, assuming that scaling takes us closer to some 'commonly shared universe', as it is converging on what 'locality' should mean, practically. But it is also a view in where you should define what is common from what a repeatable experiment tells you, locally measured. Then again, all of this is assuming that there is a logic existing, and that our ideas and mathematics are able to describe it.

Also, defining it from 'locality' is not necessarily the same as what we have used to think of as our 'common universe', as proved by 'repeatable experiments'. What locality demands, as I think or hope

, Is that there is a way to define our 'common' universe macroscopically as coming from some simple principles, constants, valid locally. If you define it this way, the universe we see is a 'mental construct' we agree on, although finding observer dependencies measuring, the 'reality' being those constants becoming a property of all frames of reference, even when not measurable 'locally'. Relativity uses two frames, at least, and comparisons, but what I'm wondering about is one frame, and what you might define it as consisting of.

(One more thing, by 'locality' I'm not defining it as of actions and reactions, as some linear processes, proceeding from some center. What I mean by it is that all experiments done are local, any measurements created done from your 'frame', relative some other. That's relativity. It's not a statement of a frozen geometry on that small scale, just think of entanglements. And if you look at relativity you will find that it can't be a frozen geometry relativistically either. Using 'locality' as a description of particles interacting only from adjoining positions in space and time is wrong, in my thoughts. There are too many experiments questioning that.

Instead I lean to the idea of 'gestalts' as in configurations where you are inseparable from your experiment (Copenhagen definition), and in a wider context also Mach's ideas of everything being connected. What makes us differ it, is the arrow we perceive.

But it is definitely local, the arrow you find, as well as everything else, and the question should be how to minimize what 'local' should mean. That's why one frame of reference is interesting, and if it is possible to relate such a frame to some mathematical object, constants, or 'quantum grain' if you like.)