 Källén–Lehmann spectral density

The Källén–Lehmann spectral density is the nonnegative invariant-mass measure in a scalar vacuum two-point function. With complete intermediate states it satisfies $2\pi \theta (p^0)\rho (p^2)=(2\pi {)}^4{\sum }_{\alpha }|⟨\alpha |\varphi (0)|0⟩{|}^2{\delta }^4(p-p_{\alpha })$, with continuous state measures included in the sum. A stable one-particle state of mass $m$, with covariant normalization and squared field overlap $Z$, contributes $Z\delta (\sigma -m^2)$. Other particles and multiparticle states contribute additional nonnegative weight. The isolated pole of the scalar propagator then has residue $Z$.