Källén–Lehmann spectral representation

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. This can be written as, using the mostly-minus metric signature,


 * $$\Delta(p)=\int_0^\infty d\mu^2\rho(\mu^2)\frac{1}{p^2-\mu^2+i\epsilon},$$

where $$\rho(\mu^2)$$ 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. This belongs to non-perturbative techniques of quantum field theory.

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 $$p_0>0$$. 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.