Srednicki — Quantum Field Theory · Problem 13.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · lehmann-kallen-renormalization-factor

Problem

Consider an interacting scalar field theory in dd spacetime dimensions, L=−12Zφ∂μφ∂μφ−12Zmm2φ2−L1(φ),(13.18)\mathcal{L} = -\tfrac{1}{2}Z_\varphi\partial^\mu\varphi\partial_\mu\varphi - \tfrac{1}{2}Z_m m^2\varphi^2 - \mathcal{L}_1(\varphi) , \tag{13.18} where L1(φ)\mathcal{L}_1(\varphi) is a function of φ\varphi (and not its derivatives). The exact momentum-space propagator for φ\varphi can be expressed in Lehmann-K\"all\'en form by eq. (13.17). Find a formula for the renormalizing factor ZφZ_\varphi in terms of ρ(s)\rho(s). Hint: consider the commutator [φ(x),φ˙(y)][\varphi(x), \dot{\varphi}(y)].

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