Meaning of inertial frames in quantum mechanics In quantum mechanics where it is difficult to speak about the precise position and momentum of a particle for a macroscopic observer, does it still make sense to speak of inertial frames or even rest frames?
 A: I am not an expert on this topic but I would try to give a naively sensible answer.

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*At least for a limited class of systems who admit a large action classical-limit, there is a sense in which we can give meaning to the concept of inertial frames in quantum mechanics using the insight from Ehrenfest theorem that the equations of motion for expectation values of Hermitian observables are isomorphic to classical equations of motion for the corresponding canonical quantities.

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*Namely, inertial frames are those in which free particles evolve according to such laws that produce $\frac{d}{dt}{\langle p\rangle}=0$.



*As to the definition of rest-frame, the only meaningful definition ought to be that the frame in which the system is in a momentum-eigenstate with the eigenvalue $0$ is the rest-frame of the said system.

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*However, as one can imagine, it is not guaranteed that one can always find a rest-frame for a given system because a given system need not be in its momentum-eigenstate (in fact, it is guaranteed for most systems that this would never happen because the momentum-eigenstates for most systems are non-normalizable and thus, physically unrealizable).



I admit that this discussion does not address the question of what physical configuration would realize an inertial observer in the way one imagines such a system in relativity (i.e., an imagined grid of clocks and rulers).
A: Here is another way to look at it. Let us assume a quantum system $\mathcal O$ whose pure state is denoted by the ray $[\Psi]$. The fundamental assumption of implementing any symmetry (and in particular the Galilean symmetry) in Quantum Mechanics is that there are two macroscopic observers $O$ and $\bar{O}$, so that each of them has its own version of $[\Psi]$. They define what is known as Galilean/inertial reference frames which should be familiar from Galilean/special relativity. This is the starting point of any symmetry (typically known as the Weyl-Wigner approach to Quantum Physics = "Gruppenpest") analysis of Quantum Mechanics. Replace Galilean with Lorentzian and you have the fundamentals of Relativistic Quantum Field Theory, chapter 2 of the 1st volume of Weinberg's QFT text.
The thing about these macroscopic observers is that their kinematics is well known and the "Heisenberg uncertainty" does not really apply to them. Why? Well, let us say that there is a small uncertainty in the position and momentum of the observers and that $\delta x \cdot \delta p \approx \hbar $. But for Galilean/Lorenzian frames we have a macroscopic speed $v$ which connects $O$ and $\bar{O}$ and then $\delta v =\frac{\delta p}{M_{\text{OBSERVER}}}$. The observer's mass can be taken arbitrarily high so that $\delta v$ is literally $0$. This argument is also used by Fonda and Ghirardi in their famous text.
So it always makes sense to have $[\Psi] ({x_{O},v_{O},t})$ and $[\Psi] ({x_{\bar{O}},v_{\bar{O}},t})$ and $x_{O} = x_{\bar{O}} + v_{O\rightarrow \bar{O}} t$, for example.
