 Källén–Lehmann spectral representation

For a Hermitian scalar field in a positive-norm Hilbert space with a Lorentz-invariant vacuum, the spectrum condition and completeness give the connected vacuum two-point function as a superposition of free mass-shell functions. Define the Källén–Lehmann spectral density by $\tilde{W}(p)=2\pi \theta (p^0)\rho (p^2)$, where $W(x)=⟨0|\varphi (x)\varphi (0)|0⟩$ and its vacuum expectation value has been subtracted if nonzero. Squared intermediate-state matrix elements make $\rho$ a nonnegative measure supported on nonnegative mass squared. Applying time ordering to each free mass shell gives the displayed Feynman propagator representation, using $i\Delta =⟨T\varphi \varphi ⟩$ and signature $(+---)$. Its positivity does not automatically extend to gauge-variant fields in an indefinite auxiliary space.

The Källén–Lehmann spectral representation is a fundamental concept in quantum field theory, particularly in the context of the study of quantum fields and their propagators. It provides a way to express correlation functions (or Green's functions) of quantum fields in terms of their spectral properties.