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.
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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 , where and its vacuum expectation value has been subtracted if nonzero. Squared intermediate-state matrix elements make a nonnegative measure supported on nonnegative mass squared. Applying time ordering to each free mass shell gives the displayed Feynman propagator representation, using and signature . Its positivity does not automatically extend to gauge-variant fields in an indefinite auxiliary space.