The relativistic uncertainty principle

[bayakiv]

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?

 

[anuttarasammyak]

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.

 

[bayakiv]

Is this true in general relativity?

 

[anuttarasammyak]

I do not think so. The text deals only with special relativity.

 

[bayakiv]

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?

 

[anuttarasammyak]

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.