Srednicki — Quantum Field Theory · Problem 5.1
Canonical Quantization of Scalar Fields — Mark Srednicki
Physics — QFT · lsz-reduction-formula
Problem
Work out the LSZ reduction formula for the complex scalar field that was introduced in problem 3.5. Note that we must specify the type ( or ) of each incoming and outgoing particle.Intuitive explanation
The LSZ formula connects scattering amplitudes to time-ordered correlation functions by "amputating" external legs—removing the free-field propagators attached to asymptotic states. For a complex scalar field, you have two particle types ( and ) that are distinct, so each external line carries a label specifying which type it is. The key insight is that incoming -particles and outgoing -particles (or vice versa) come from different annihilation/creation operators in the mode expansion, so you must carefully apply the right pole residues and operator orderings for each type. This labeling is what makes the complex field formula richer than the real scalar case.
Intuitive explanation
The LSZ formula connects scattering amplitudes to time-ordered correlation functions by "amputating" external legs—removing the free-field propagators attached to asymptotic states. For a complex scalar field, you have two particle types ( and ) that are distinct, so each external line carries a label specifying which type it is. The key insight is that incoming -particles and outgoing -particles (or vice versa) come from different annihilation/creation operators in the mode expansion, so you must carefully apply the right pole residues and operator orderings for each type. This labeling is what makes the complex field formula richer than the real scalar case.