Källén–Lehmann spectral representation

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 $p^2\ge 0$ 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

1. 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)
2. 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.
3. 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.