Källén–Lehmann spectral representation

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The 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.[1][2] 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.[3] This belongs to non-perturbative techniques of quantum field theory.

Mathematical derivation[edit]

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\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)

being

\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 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[edit]

  1. ^ Källén, Gunnar (1952). Helvetica Physica Acta 25: 417. doi:10.5169/seals-112316(pdf download available) 
  2. ^ Lehmann, Harry (1954). Nuovo Cimento 11: 342. doi:10.1007/bf02783624. 
  3. ^ Strocchi, Franco (1993). Selected Topics on the General Properties of Quantum Field Theory. Singapore: World Scientific. ISBN 981-02-1143-0. 

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