Källén–Lehmann spectral representation

Mathematical derivation

The following derivation employs the mostly-minus metric signature.

In order to derive a spectral representation for the propagator of a field \Phi(x), one considers a complete set of states {|n\rangle} so that, for the two-point function one can write

:\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle=\sum_n\langle 0|\Phi(x)|n\rangle\langle n|\Phi^\dagger(y)|0\rangle.

We can now use Poincaré invariance of the vacuum to write down

:\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle=\sum_n e^{-ip_n\cdot(x-y)}|\langle 0|\Phi(0)|n\rangle|^2.

Next we introduce the spectral density function

:\rho(p^2)\theta(p_0)(2\pi)^{-3}=\sum_n\delta^4(p-p_n)|\langle 0|\Phi(0)|n\rangle|^2.

Where we have used the fact that our two-point function, being a function of p_\mu, can only depend on p^2. Besides, all the intermediate states have p^2\ge 0 and p_00. It is immediate to realize that the spectral density function is real and positive. So, one can write

:\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle = \int\frac{d^4p}{(2\pi)^3}\int_0^\infty d\mu^2e^{-ip\cdot(x-y)}\rho(\mu^2)\theta(p_0)\delta(p^2-\mu^2)

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

:\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle = \int_0^\infty d\mu^2\rho(\mu^2)\Delta'(x-y;\mu^2)

where

:\Delta'(x-y;\mu^2)=\int\frac{d^4p}{(2\pi)^3}e^{-ip\cdot(x-y)}\theta(p_0)\delta(p^2-\mu^2).

From the CPT theorem we also know that an identical expression holds for \langle 0|\Phi^\dagger(x)\Phi(y)|0\rangle and so we arrive at the expression for the time-ordered product of fields

:\langle 0|T\Phi(x)\Phi^\dagger(y)|0\rangle = \int_0^\infty d\mu^2\rho(\mu^2)\Delta(x-y;\mu^2)

where now

:\Delta(p;\mu^2)=\frac{1}{p^2-\mu^2+i\epsilon}

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.

References

Bibliography

References

1. (1952). "On the Definition of the Renormalization Constants in Quantum Electrodynamics". Helvetica Physica Acta.
2. (1954). "Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder". Nuovo Cimento.