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[edit]

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



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.


  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.