How is the 4-momentum 4-vector defined in relativistic QM?

[pmb_phy]

I've been wondering about relativistic quantum mechanics. Elsewhere I'm addressing some comments about this branch of physics but I have never studied it. Is the 4-momentum 4-vector defined in the same way in relativsitic QM or is there a difference? I'm wondering if the time component of 4-momentum is defined in the same way in relativistic QM as in classical relativity. Thanks.

Pete

 

[Hans de Vries]

I've been wondering about relativistic quantum mechanics. Elsewhere I'm addressing some comments about this branch of physics but I have never studied it. Is the 4-momentum 4-vector defined in the same way in relativsitic QM or is there a difference? I'm wondering if the time component of 4-momentum is defined in the same way in relativistic QM as in classical relativity. Thanks.

Pete

Yes, generally the metric is (+---), although Weinberg uses (-+++) as in (flat) GR.

In QFT the 4-momentum is typically associated with the phase change rates in the
time and space components corresponding to the plane wave eigenfunctions:

$$\psi(x)~=~e^{-iEt/\hbar + ipx/\hbar}$$Regards, Hans

 

[Fredrik]

It's defined as the $P^\mu$ that appears in the translation operator $e^{-iP^\mu a_\mu}$, where $a^\mu$ is the translation four-vector. This definition works in both relativistic and non-relativistic QM. (The best place to read about these things is chapter 2 of vol. 1 of Weinberg's QFT book).

In a relativistic quantum field theory, you can also construct the four-momentum operators expliclity from the Lagrangian, as the conserved quantities that Noether's theorem tells us must exist due to the invariance of the action under translations in space and time.