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In elementary QM courses we always consider that components of momentum vector form a complete set of commuting observables.

I am confused whether this is an input to our theory or whether we somehow derive this? As far as I can recall we take this as an intuition that the components of momentum vector will form a complete set because momentum measured in one direction cannot affect momentum measured in another direction. On the other hand, since both momentum and position are promoted to operators (as compared to classical mechanics) we "expect" them to defy the intuition from classical mechanics.

If so, then I wonder what modern quantum theory says about entanglement in such scenarios. Say I prepare a state with two entangled electrons such that one of them goes in x direction and the other goes in y direction. Do the components of momentum operator still form a complete set of commuting variables? How?

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Good question. The supposition that the components of four vector momentum form a complete set of commuting observables is taken as an input.

The trivial commutation relations for four momentum vector components is equivalent (after a Fourier transform) to the relation $$[\hat{x_{a}},\hat{x_{b}}]=0.$$ That is equivalent to say that the underlying spacetime geometry is taken to be commutative. A reasonable hypothesis in light of the fact that we haven't observed (yet) spacetime "fuzziness".

The point is that something like $$[\hat{x_{a}},\hat{x_{b}}]= i\hbar\delta_{ab}$$ is perfectly fine, at least in principle and in the non-relativistic setup.

It's worth to say that maybe noncommutative geometry appear in some relevant physical situations(example), but nothing too relevant for elementary purposes.

Completeness of the set of four momentum operators is also an input because classically any system has momentum, so it is reasonable to expect that quantum mechanically any state can be labbeled with a reasonable expectation value for the momentum operator.

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    $\begingroup$ Yes exactly. I had a suspicion that this was going to connect with commutativity of space-time. Could you please also give a response to the second question I asked? $\endgroup$
    – self.grassmanian
    Commented Jun 10, 2020 at 18:36
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    $\begingroup$ One more thing. You mention that we take the underlying space-time geometry to be commutative. Basically this also implies space-time translation invariance, right? So, if I am considering an example of quantum theory with chaos then time translation invariance is broken (I think). In that case, how good is this assumption about components of momentum? Furthermore, I have seen that people study QFT in curved background and of course they must talk about the inflationary period. I strongly doubt if this assumption is going to hold true there. $\endgroup$
    – self.grassmanian
    Commented Jun 10, 2020 at 18:40
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    $\begingroup$ Wonderful observation! Yes, my argument is just good in ordinary quantum mechanics but its extension to QFT is subtle (at most) because space and time are on equal footing in any relativistic theory and non-commutativity in time seems to violate locality and unitarity (the purely time coordinates behave as if they were tachyons because its norm squared is negative). There is no simple way to surpass your objection in QFT upt to now, at least as far as I know. On the other hand ... $\endgroup$
    – Ramiro Hum-Sah
    Commented Jun 10, 2020 at 19:02
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    $\begingroup$ General covariant and noncommutative theories exist within string theory, the BFSS matrix model and matrix string theory are examples of this. But is important to remark that those models are supersymmetric quantum mechanical models and not honest quantum field theories (I urge to read the blog post I've shared to understand better how it is possible that an ordinary quantum mechanical model can be general covariant.) $\endgroup$
    – Ramiro Hum-Sah
    Commented Jun 10, 2020 at 19:06
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    $\begingroup$ Some tries to define relativistic noncommutative quantum field theories have been developed, but I'm unsure of the state of the art of those tries ( see this talk). Also you can find reviews on noncommutative field theories like this one arxiv.org/abs/hep-th/0109162, but notice that all of them are non-renormalizable (just valid up to a cut-off) or consistent as seen as part of a bigger commutative system (as in the noncommutative Yang-Mills case). $\endgroup$
    – Ramiro Hum-Sah
    Commented Jun 10, 2020 at 19:12

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