Källén–Lehmann spectral representation

The Källén–Lehmann spectral representation, or simply Lehmann representation, gives a general expression for the (time ordered) two-point function of an interacting quantum field theory as a sum of free propagators. It was discovered by Gunnar Källén in 1952, and independently by Harry Lehmann in 1954.[1][2] This can be written as, using the mostly-minus metric signature,

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

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The following derivation employs the mostly-minus metric signature.

In order to derive a spectral representation for the propagator of a field  , one considers 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

 

Next we introduce the spectral density function

 .

Where 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

 

where

 .

From the CPT theorem we also know that an identical expression holds for   and so we arrive at the expression for the time-ordered product of fields

 

where now

 

a free particle propagator. Now, as we have the exact propagator given by the time-ordered two-point function, we have obtained the spectral decomposition.

References

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  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){{cite journal}}: CS1 maint: postscript (link)
  2. ^ Lehmann, Harry (1954). "Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder". Nuovo Cimento (in German). 11 (4): 342–357. Bibcode:1954NCim...11..342L. doi:10.1007/bf02783624. ISSN 0029-6341. S2CID 120848922.
  3. ^ Strocchi, Franco (1993). Selected Topics on the General Properties of Quantum Field Theory. Singapore: World Scientific. ISBN 978-981-02-1143-1.

Bibliography

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