4-momentum in relativistic QM

  1. 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
     
  2. jcsd
  3. Hans de Vries

    Hans de Vries 1,135
    Science Advisor

    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:

    [tex]\psi(x)~=~e^{-iEt/\hbar + ipx/\hbar}[/tex]


    Regards, Hans
     
    Last edited: Jun 30, 2008
  4. Fredrik

    Fredrik 10,197
    Staff Emeritus
    Science Advisor
    Gold Member

    It's defined as the [itex]P^\mu[/itex] that appears in the translation operator [itex]e^{-iP^\mu a_\mu}[/itex], where [itex]a^\mu[/itex] 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.
     
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