1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Momentum operator of the quantized real Klein-Gordon field

  1. Aug 10, 2014 #1
    1. The problem statement, all variables and given/known data
    a+(k) creates particle with wave number vector k, a(k) annihilates the same; then the Klein-Gordon field operators are defined as ψ+(x) = ∑_k f(k) a(k) e^-ikx and ψ-(x) = ∑_k f(k) a+(k) e^ikx; the factor f contains constants and the ω(k). x is a Lorentz four vector, k is a 3-vector except in the exponential where it is also a four vector.

    2. Relevant equations
    The momentum operator is defined as P = ∫∫∫ d^3x 1/c^2 ∂/∂t ψ(x) ∇ψ(x) where ψ(x) = ψ+(x) + ψ-(x).
    Commutation relation is [a(k),a+(k')] = δ_kk'

    3. The attempt at a solution
    Upon straightforward calculation of the derivatives and insertion into the expression for the momentum operator I get the correct constant term for the infinite momentum, but not the number operator term for bosons. The result should be P = ∑_k hbar k ( a+(k) a(k) +1/2). I get the second but not the first term. The problem is that two terms appear in which I have a pair of creators and a pair of annihilators, respectively, but the mixed term vanishes after using the commutation relation. What am I missing? The problem is to be found on pages 40 and 41 of Mandl and Shaw, Quantum Field Theory. Thanks a lot for any kind of hint.
  2. jcsd
  3. Aug 10, 2014 #2


    User Avatar
    Science Advisor

    Could you post your exact calculation?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted