Sidney Coleman — Quantum Field Theory Lectures · Problem 3.2

September 30 - Causality, observables and quantum fields; Constructing the quantum field from Fock space operators axiomatically; Translational invariance; Lorentz invariance and Relativistic normalization; Field constructed satisfies Klein-Gordon equation — Sidney Coleman

Physics — QFT · klein-gordon-equation-free-scalar-field

Problem

The free scalar field of mass μ\mu is ϕ(x)=∫d3k(2π)3/22ωk⃗[ak⃗ e−ik⋅x+ak⃗† eik⋅x].\phi(x) = \int \frac{d^3k}{(2\pi)^{3/2}\sqrt{2\omega_{\vec{k}}}}\left[a_{\vec{k}}\,e^{-ik\cdot x} + a^\dagger_{\vec{k}}\,e^{ik\cdot x}\right]. Show, using k2=μ2k^2 = \mu^2, that this field satisfies the Klein-Gordon equation (□+μ2)ϕ(x)=0,□=∂μ∂μ.(\Box + \mu^2)\phi(x) = 0, \qquad \Box = \partial^\mu\partial_\mu.

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