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Källén–Lehmann spectral representation gives a general expression for the two-point function of an interacting quantum field theory as a sum of free propagators. It was discovered by Gunnar Källén and Harry Lehmann independently. This can be written as

$\Delta (p)=\int _{0}^{\infty }d\mu ^{2}\rho (\mu ^{2}){\frac {1}{p^{2}-\mu ^{2}+i\epsilon }}$

being $\rho (\mu ^{2})$ 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

In order to derive a spectral representation for the propagator of a field $\Phi (x)$, one consider 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

$\langle 0|\Phi (x)\Phi ^{\dagger }(y)|0\rangle =\sum _{n}e^{-ip_{n}\cdot (x-y)}|\langle 0|\Phi (0)|n\rangle |^{2}.$

Let us introduce the spectral density function

$\rho (p^{2})\theta (p_{0})(2\pi )^{-3}=\sum _{n}\delta ^{4}(p-p_{n})|\langle 0|\Phi (0)|n\rangle |^{2}$.

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 $p^{2}\geq 0$ and $p_{0}>0$. It is immediate to realize that the spectral density function is real and positive. So, one can write

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

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

$\langle 0|\Phi (x)\Phi ^{\dagger }(y)|0\rangle =\int _{0}^{\infty }d\mu ^{2}\rho (\mu ^{2})\Delta '(x-y;\mu ^{2})$

being

$\Delta '(x-y;\mu ^{2})=\int {\frac {d^{4}p}{(2\pi )^{3}}}e^{-ip\cdot (x-y)}\theta (p_{0})\delta (p^{2}-\mu ^{2})$.

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

$\langle 0|T\Phi (x)\Phi ^{\dagger }(y)|0\rangle =\int _{0}^{\infty }d\mu ^{2}\rho (\mu ^{2})\Delta (x-y;\mu ^{2})$

being 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 chronologically ordered two-point function, we have obtained the spectral decomposition.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Bibliography

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• The Tangent Bundle

See also

• Quantum field theory
• Correlation functions
• Propagator