Källén–Lehmann spectral representation

The Källén–Lehmann spectral representation gives a general expression for the (time ordered) two-point function of an interacting quantum field theory as a sum of free propagators. It was discovered by Gunnar Källén and Harry Lehmann independently. This can be written as, using the mostly-minus metric signature,

${\displaystyle \Delta (p)=\int _{0}^{\infty }d\mu ^{2}\rho (\mu ^{2}){\frac {1}{p^{2}-\mu ^{2}+i\epsilon }},}$

where ${\displaystyle \rho (\mu ^{2})}$ is the spectral density function that should be positive definite. In a gauge theory, this latter condition cannot be granted but nevertheless a spectral representation can be provided. This belongs to non-perturbative techniques of quantum field theory.