Sidney Coleman — Quantum Field Theory Lectures · Problem 9.1

October 21 - Diagrammatic perturbation theory in Model 3; Vertex in model 1; Connected diagrams; Connected Wick diagrams; Model 1 solved; Model 2 begun — Sidney Coleman

Physics — QFT · model-1-coherent-state-alpha

Problem

In Model 1, HI=gρ(x)ϕ(x)\mathcal{H}_I = g\rho(x)\phi(x), the second connected Wick diagram gives O2=(−ig)2∫d4x1 d4x2 ϕ(x1)ϕ(x2)‾ ρ(x1)ρ(x2)=α+iβ.O_2 = (-ig)^2 \int d^4x_1\, d^4x_2\, \overline{\phi(x_1)\phi(x_2)}\, \rho(x_1)\rho(x_2) = \alpha + i\beta. Compute α\alpha directly. (You may check your answer using the unitarity argument: demanding UI(∞,−∞)U_I(\infty,-\infty) be unitary requires α=−∫d3k ∣f(k⃗)∣2\alpha = -\int d^3k\, |f(\vec{k})|^2, where f(k⃗)≡−ig(2π)3/22ωk⃗ ρ~(k⃗,ωk⃗)f(\vec{k}) \equiv \frac{-ig}{(2\pi)^{3/2}\sqrt{2\omega_{\vec{k}}}}\,\widetilde{\rho}(\vec{k},\omega_{\vec{k}}).)

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