The relativistic uncertainty principle

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SUMMARY

The discussion centers on the relativistic uncertainty principle as presented in the textbook "Relativistic Quantum Theory" by Landau, Lifshitz, and Pitaevskii. The forum participants debate the validity of the uncertainty relation \(\triangle p \triangle t \sim \frac{\hbar}{c}\) and its implications for general relativity. It is concluded that the text primarily addresses special relativity, raising questions about the applicability of these principles in a general relativistic context. Participants suggest further reading of the original Russian edition for deeper insights.

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TL;DR
Statement: the product (scalar product in the Minkowski metric or in the local metric of a pseudo-Riemannian manifold) of the uncertainties of the relativistic coordinate and the 4-momentum has the order of h.
Could this statement be the first step towards quantum gravity? Or is it trivial or not true at all?
 
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Textbook of Landau, Lifshiz and Pitaevskii on relativistic quantum mechanics says
\triangle p \triangle t \ \sim \frac{\hbar}{c}
and for the rest frame of particle, e.g. electron
\triangle q \sim \frac{\hbar}{mc}
in the introduction.
 
Is this true in general relativity?
 
I do not think so. The text deals only with special relativity.
 
The first formula on the left has two factors, one of which is a number and the other is a vector. What do we get on the right? Or is there 4-momentum uncertainty taken modulo?
 
I am sorry to say I am not qualified to tell it to you. I recommend you to read their book "Relativistic Quantum Theory" 1968 that is original in Russian, if available.
 
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