= Källén–Lehmann spectral representation
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{title2=$\Delta(p)=\int_0^\infty d\sigma\,\rho(\sigma)/(p^2-\sigma+i0)$}
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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 $\widetilde W(p)=2\pi\theta(p^0)\rho(p^2)$, where $W(x)=\langle0|\phi(x)\phi(0)|0\rangle$ 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 $i\Delta=\langle T\phi\phi\rangle$ and signature $(+---)$. Its positivity does not automatically extend to gauge-variant fields in an indefinite auxiliary space.
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