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To my understanding: One-particle states in non-relativistic quantum mechanics can be seen as vectors from the space $L^2(\mathbb{R}^3)$ or any other general separable Hilbert space. Here, the time evolution is described by time evolution operators with parameter $t$. The moment we move to relativistic quantum mechanics (Klein-Gordon Eq., Dirac Eq, Pauli Eq), can the states now be seen as square integrable functions on Minkowski space, i.e. $L^2(Mink. Space)$? What is the meaning of time of here?

And the proof that the Schrödinger equation is not Lorentz invariant also implies that it is defined on $L^2(Mink. Space)$ instead of $L^2(\mathbb{R}^3)$ if we apply a Lorentz transformation at all, which needs a time axis.

Qmechanic
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Michiel
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Relativistic quantum theory is usually discussed in terms of quantum field theory. The relevant description involves fields of distribution valued Heisenberg picture observables. The states in quantum field theory are complex valued functionals of those field observables. For a brief introduction see

https://arxiv.org/abs/quant-ph/0112148

See also textbooks on QFT such as "Quantum field theory for the gifted amateur" by Lancaster and Blundell and "Quantum field theory in a nutshell" by Zee. For a deeper book see "The conceptual framework of quantum field theory" by Anthony Duncan.

QFT doesn't directly talk about space and time. Rather, QFT makes predictions about correlations among field observables and those field observables are functions of space-time points. In those theories space and time are not observables, they are just a background on which quantum fields evolve.

To get to a quantum theory of space and time they would have to be treated as observables. Some papers have been written on this topic, such as the Page and Wootters paper that discuss time observables in non-relativistic quantum theory:

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.27.2885

There are papers attempting to extend the Page-Wootters theory in various ways that may help with relativistic versions of the theory.

Kuypers has written a paper explaining that c-number time isn't required in the Heisenberg picture:

https://arxiv.org/abs/2108.02771

There is a paper by Kuypers and Rijavec on measurement of time in PW theory:

https://arxiv.org/abs/2406.14642

This paper by Singh has some material on treating space and time on an equal footing in a PW type theory:

https://arxiv.org/abs/2004.09139

alanf
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