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 }},

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.^[3] 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

↑ Källén, Gunnar (1952). "On the Definition of the Renormalization Constants in Quantum Electrodynamics". Helvetica Physica Acta. 25: 417. doi:10.5169/seals-112316(pdf download available)
↑ Lehmann, Harry (1954). "Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder". Nuovo Cimento (in German). Società Italiana di Fisica. 11 (4): 342–357. doi:10.1007/bf02783624. ISSN 0029-6341.
↑ Strocchi, Franco (1993). Selected Topics on the General Properties of Quantum Field Theory. Singapore: World Scientific. ISBN 981-02-1143-0.

Bibliography

Weinberg, S. (1995). The Quantum Theory of Fields: Volume I Foundations. Cambridge University Press. ISBN 0-521-55001-7.
Peskin, Michael; Schoeder, Daniel (1995). An Introduction to Quantum Field Theory. Perseus Books Group. ISBN 0-201-50397-2.
Zinn-Justin, Jean (1996). Quantum Field Theory and Critical Phenomena (3rd ed.). Clarendon Press. ISBN 0-19-851882-X.

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