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In Walter Greiner book about relativistic quantum mechanics, he writes the uncertainty relations for 4-position and 4-momentum in a neat way as:

$$[p^\mu, x^\nu] = i\hbar \eta^{\mu\nu}{\bf 1}$$

with Minkowski sign convention $(+,-,-,-)$. This gives manifestly all the uncertainty relations we want. But here, time is just a coordinate, so how do we interpret the commutation relation between time and energy?

Qmechanic
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Arthur
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    Related: https://physics.stackexchange.com/q/53802/2451 , https://physics.stackexchange.com/q/259334/2451 , https://physics.stackexchange.com/q/220697/2451 and links therein. – Qmechanic Oct 15 '20 at 09:53
  • @Qmechanic Thank you very much, that was super helpful ^_^ – Arthur Oct 15 '20 at 20:04
  • The takeaway that I took from my 2015 post, which Qmechanic linked above, is that in a covariant treatment of relativistic quantum mechanics, time can be an operator, and it is instead proper time which is just a parameter. The notation you cite from Greiner's book suggests that's what he's doing. Maybe I'll take a look at Greiner. – ziggurism Feb 07 '24 at 22:47

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