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As we know, we can formulate the phase space theory of relativistic free particles using the Hamiltonian $H=\sqrt{p^2+m^2}$ and the Poisson bracket $[x,p]=1$. So there are no problems with Lorentz symmetry here.

There is a correspondence between these phase space theories and quantum theories. However, the corresponding quantum theory does not exist even on the level of pure mathematics because the position basis does not exist in the quantum theory.

So where does this correspondece break? And where does the position basis problem in the quantum theory manifest itself in this phase space theory?

Ryder Rude
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    What do you mean by "the position basis doesn't exist"? Are you asking about why we use quantum field theory rather than wavefunction based relativistic quantum mechanics? – By Symmetry Oct 17 '23 at 08:41
  • @BySymmetry i mean the $d^3p$ in $\psi (x,t)= \int d^3p \psi(p) e^{-i\omega_p t +px} $ is not Lorentz invariant, which makes $\psi(x)$ not Lorentz invariant – Ryder Rude Oct 17 '23 at 08:45
  • @BySymmetry The problem is that there is no, in a specific sense, position observable for a relativistic quantum theory. There are several posts here in PSE, but I cannot search for them right now. But a good starting point is the Malament theorem – Tobias Fünke Oct 17 '23 at 08:51
  • As far as I understand, the problem comes in when the position basis must also serve as single particle states. If we remove this requirement (which does not make sense to me anyway), then there does not seem to be any problem with a position basis. – flippiefanus Oct 17 '23 at 09:01
  • @flippiefanus there is no position observable at all – Tobias Fünke Oct 17 '23 at 09:19
  • see https://physics.stackexchange.com/q/762622/ – Valter Moretti Oct 17 '23 at 09:58
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    @Tobias Funke No, it is not true that there is no position observable at all: there is no position observable described in terms of PVMs, but there are many described in terms of POVMs... – Valter Moretti Oct 17 '23 at 10:01
  • @ValterMoretti It is interesting but where does the failure of the PVM position observable manifest itself in the corresponding phase space theory? I believe the failure is due to the Lorentz non-invariance of $d^3p$ in favor of $\frac{1}{2\omega _p} d^3p$? But if the phase space theory work out, why does canonical quantisation does not give a consistent theory with a position basis (in the PVM sense)? Instead, the correspondence between phase space and quantum theories breaks down here (but why) – Ryder Rude Oct 17 '23 at 10:41
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    The problem is the Hegerfeldt therem. There is a huge discussion in the literature. There is a summary of the various obstructions in this paper of mine https://arxiv.org/abs/2304.02133 https://link.springer.com/article/10.1007/s11005-023-01689-5 – Valter Moretti Oct 17 '23 at 11:30
  • @ValterMoretti Thank you. What I understand is that nothing goes wrong in, e.g., the geometric quantisation of the phase space theory with $H=\sqrt{p^2+m^2}$. I mean we do get a consistent quantum theory with a position basis, which is fine mathematically but the theory isnt experimentally correct because of superluminal communication. Is this correct? – Ryder Rude Oct 17 '23 at 15:14
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    Yes, in summary the problem is the one you wrote. – Valter Moretti Oct 17 '23 at 15:25
  • @ValterMoretti thanks for the links. However, it still seems to me that this superluminal communication problem has its root in the assumed 3-dimensional nature of these position eigenstates as candidates for single particle states. A coordinate basis for Minkowski space is four-dimensional and not suitable for single particle states. Why do these position eigenstates have to be 3-dimensional? – flippiefanus Oct 18 '23 at 03:18
  • Well, I do not understand well what you maean by three or four dimensional position eigenstates. This topic is extremely technical as, e.g., you may see from my paper. The story of this subject shows that common physical intuition is often the cause of wrong suppositions when dealing with these issues. It seems to me that you are referring to spacetime localization instead of spatial localization at a given time. That is a different subject and I am not very familiar with that. – Valter Moretti Oct 18 '23 at 05:45

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