= Källén–Lehmann spectral density
{c}
{title2=$\rho(\sigma)$}
The <Källén–Lehmann spectral density> is the nonnegative invariant-mass measure in a scalar vacuum two-point function. With complete intermediate states it satisfies $2\pi\theta(p^0)\rho(p^2)=(2\pi)^4\sum_\alpha|\langle\alpha|\phi(0)|0\rangle|^2\delta^4(p-p_\alpha)$, 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 $Z$.
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