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,

$\mathrm{\Delta }(p)={\int }_0^{\mathrm{\infty }}d{\mu }^2\rho ({\mu }^2)\frac{1}{p^2-{\mu }^2+iϵ},$

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 $\mathrm{\Phi }(x)$, one considers a complete set of states $\{|n⟩\}$ so that, for the two-point function one can write

$⟨0|\mathrm{\Phi }(x){\mathrm{\Phi }}^{†}(y)|0⟩=\sum _n⟨0|\mathrm{\Phi }(x)|n⟩⟨n|{\mathrm{\Phi }}^{†}(y)|0⟩.$

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

$⟨0|\mathrm{\Phi }(x){\mathrm{\Phi }}^{†}(y)|0⟩=\sum _n{e}^{-i{p}_n\cdot (x-y)}|⟨0|\mathrm{\Phi }(0)|n⟩{|}^2.$

Next we introduce the spectral density function

$\rho (p^2)\theta (p_0)(2\pi {)}^{-3}=\sum _n{\delta }^4(p-p_n)|⟨0|\mathrm{\Phi }(0)|n⟩{|}^2$.

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

$⟨0|\mathrm{\Phi }(x){\mathrm{\Phi }}^{†}(y)|0⟩=\int \frac{d^{4}p}{(2\pi {)}^3}{\int }_0^{\mathrm{\infty }}d{\mu }^2{e}^{-ip\cdot (x-y)}\rho ({\mu }^2)\theta (p_0)\delta (p^2-{\mu }^2)$

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

$⟨0|\mathrm{\Phi }(x){\mathrm{\Phi }}^{†}(y)|0⟩={\int }_0^{\mathrm{\infty }}d{\mu }^2\rho ({\mu }^2){\mathrm{\Delta }}^{\prime }(x-y;{\mu }^2)$

where

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

From the CPT theorem we also know that an identical expression holds for $⟨0|{\mathrm{\Phi }}^{†}(x)\mathrm{\Phi }(y)|0⟩$ and so we arrive at the expression for the time-ordered product of fields

$⟨0|T\mathrm{\Phi }(x){\mathrm{\Phi }}^{†}(y)|0⟩={\int }_0^{\mathrm{\infty }}d{\mu }^2\rho ({\mu }^2)\mathrm{\Delta }(x-y;{\mu }^2)$

where now

$\mathrm{\Delta }(p;{\mu }^2)=\frac{1}{p^2-{\mu }^2+iϵ}$

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.

Bibliography

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