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Homework Statement
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.
Homework 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'
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.
