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
where 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 , one consider a complete set of states so that, for the two-point function one can write
We can now use Poincaré invariance of the vacuum to write down
Let us introduce the spectral density function
- .
We have used the fact that our two-point function, being a function of , can only depend on . Besides, all the intermediate states have and . It is immediate to realize that the spectral density function is real and positive. So, one can write
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
being
- .
From CPT theorem we also know that holds an identical expression for and so we arrive at the expression for the chronologically ordered product of fields
being now
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.