The Källén–Lehmann spectral representation, or simply Lehmann 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 in 1952, and independently by Harry Lehmann in 1954.
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.
[3] This belongs to non-perturbative techniques of quantum field theory.
In order to derive a spectral representation for the propagator of a field
It is immediate to realize that the spectral density function is real and positive.
So, one can write and we freely interchange the integration, this should be done carefully from a mathematical standpoint but here we ignore this, and write this expression as where From the CPT theorem we also know that an identical expression holds for
and so we arrive at the expression for the time-ordered product of fields where now a free particle propagator.
Now, as we have the exact propagator given by the time-ordered two-point function, we have obtained the spectral decomposition.