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

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[edit]

  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[edit]