Källén–Lehmann spectral representation explained

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,

\Delta(p) = \int ^{inf ty} _{0} d\mu ^{2\rho(\mu} ^{2 ) (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

The following derivation employs the mostly-minus metric signature.

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

\Phi(x) , one considers 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 ⋅ (x-y)}} |\langle 0 |\Phi( 0 )|n\rangle| ^{2.}

Next we introduce the spectral density function

\rho(p ^{-3} _{0 )( 2 \pi)} = \sum ^{4 (p - p} _{n)|\langle} 0 |\Phi( 0 )|n\rangle| ^{2} .

Where 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 (d ^{4p})/(( 2 \pi) ^{3}) \int ^{inf ty} _{0} d\mu ^{2e} ^{-ip ⋅ (x-y)} \rho(\mu ^{2- \mu} _{0 )\delta(p} ^{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 ^{inf ty} _{0} d\mu ^{2\rho(\mu} ^{2)\Delta'(x-y;\mu} ^{2)}

where

\Delta'(x - y;\mu ^{2 ) = \int (d ^{4p})/(( 2 \pi) ^{3})} e ^{-ip ⋅ (x-y)} \theta(p ^{2- \mu} _{0 )\delta(p} ^{2)} .

From the CPT theorem we also know that an identical expression holds for

\langle 0 |\Phi ^{\dagger(x)\Phi(y)|0\rangle} and so we arrive at the expression for the time-ordered product of fields\langle 0 |T\Phi(x)\Phi ^{\dagger(y)|0\rangle} = \int ^{inf ty} _{0} d\mu ^{2\rho(\mu} ^{2)\Delta(x-y;\mu} ^{2)}

where now

\Delta(p;\mu ^{2 ) = (1)/(p ^{2-\mu} ^{2+i\epsilon})}

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.

Bibliography

• Book: Weinberg, S. . Steven Weinberg . The Quantum Theory of Fields: Volume I Foundations . . 1995 . 978-0-521-55001-7 . registration .
• Book: Michael . Peskin . Michael Peskin . Daniel . Schroeder . An Introduction to Quantum Field Theory . . 1995 . 978-0-201-50397-5 . registration .
• Book: Zinn-Justin, Jean . jean Zinn-Justin. Quantum Field Theory and Critical Phenomena. . 1996 . 3rd . 978-0-19-851882-2.

Notes and References

1. Gunnar Källén . Källén . Gunnar . 1952 . On the Definition of the Renormalization Constants in Quantum Electrodynamics. Helvetica Physica Acta . 25 . 417 . 10.5169/seals-112316. (pdf download available).
2. Harry Lehmann . Lehmann . Harry . 1954 . Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder. Nuovo Cimento . 0029-6341. 11. 4. 342 - 357. 10.1007/bf02783624. de. 1954NCim...11..342L. 120848922 .
3. Book: Strocchi , Franco . Selected Topics on the General Properties of Quantum Field Theory . World Scientific . 1993 . Singapore . 978-981-02-1143-1.