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
The following derivation employs the mostly-minus metric signature.
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 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 chronologically ordered product of fields
where 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){{cite journal}}: CS1 maint: postscript (link)
^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.
^Strocchi, Franco (1993). Selected Topics on the General Properties of Quantum Field Theory. Singapore: World Scientific. ISBN978-981-02-1143-1.
källén, lehmann, spectral, representation, gives, general, expression, time, ordered, point, function, interacting, quantum, field, theory, free, propagators, discovered, gunnar, källén, harry, lehmann, independently, this, written, using, mostly, minus, metri. The Kallen Lehmann spectral 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 Kallen and Harry Lehmann independently 1 2 This can be written as using the mostly minus metric signature D p 0 d m 2 r m 2 1 p 2 m 2 i ϵ displaystyle Delta p int 0 infty d mu 2 rho mu 2 frac 1 p 2 mu 2 i epsilon where r m 2 displaystyle 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 EditThe following derivation employs the mostly minus metric signature In order to derive a spectral representation for the propagator of a field F x displaystyle Phi x one consider a complete set of states n displaystyle n rangle so that for the two point function one can write 0 F x F y 0 n 0 F x n n F y 0 displaystyle 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 Poincare invariance of the vacuum to write down 0 F x F y 0 n e i p n x y 0 F 0 n 2 displaystyle langle 0 Phi x Phi dagger y 0 rangle sum n e ip n cdot x y langle 0 Phi 0 n rangle 2 Let us introduce the spectral density function r p 2 8 p 0 2 p 3 n d 4 p p n 0 F 0 n 2 displaystyle rho p 2 theta p 0 2 pi 3 sum n delta 4 p p n langle 0 Phi 0 n rangle 2 We have used the fact that our two point function being a function of p m displaystyle p mu can only depend on p 2 displaystyle p 2 Besides all the intermediate states have p 2 0 displaystyle p 2 geq 0 and p 0 gt 0 displaystyle p 0 gt 0 It is immediate to realize that the spectral density function is real and positive So one can write 0 F x F y 0 d 4 p 2 p 3 0 d m 2 e i p x y r m 2 8 p 0 d p 2 m 2 displaystyle langle 0 Phi x Phi dagger y 0 rangle int frac d 4 p 2 pi 3 int 0 infty d mu 2 e ip cdot x y rho mu 2 theta p 0 delta p 2 mu 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 0 F x F y 0 0 d m 2 r m 2 D x y m 2 displaystyle langle 0 Phi x Phi dagger y 0 rangle int 0 infty d mu 2 rho mu 2 Delta x y mu 2 where D x y m 2 d 4 p 2 p 3 e i p x y 8 p 0 d p 2 m 2 displaystyle Delta x y mu 2 int frac d 4 p 2 pi 3 e ip cdot x y theta p 0 delta p 2 mu 2 From the CPT theorem we also know that an identical expression holds for 0 F x F y 0 displaystyle langle 0 Phi dagger x Phi y 0 rangle and so we arrive at the expression for the chronologically ordered product of fields 0 T F x F y 0 0 d m 2 r m 2 D x y m 2 displaystyle langle 0 T Phi x Phi dagger y 0 rangle int 0 infty d mu 2 rho mu 2 Delta x y mu 2 where now D p m 2 1 p 2 m 2 i ϵ displaystyle Delta p mu 2 frac 1 p 2 mu 2 i epsilon 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 Kallen 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 a href Template Cite journal html title Template Cite journal cite journal a CS1 maint postscript link Lehmann Harry 1954 Uber 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 Strocchi Franco 1993 Selected Topics on the General Properties of Quantum Field Theory Singapore World Scientific ISBN 978 981 02 1143 1 Bibliography EditWeinberg S 1995 The Quantum Theory of Fields Volume I Foundations Cambridge University Press ISBN 978 0 521 55001 7 Peskin Michael Schoeder Daniel 1995 An Introduction to Quantum Field Theory Perseus Books Group ISBN 978 0 201 50397 5 Zinn Justin Jean 1996 Quantum Field Theory and Critical Phenomena 3rd ed Clarendon Press ISBN 978 0 19 851882 2 Retrieved from https en wikipedia org w index php title Kallen Lehmann spectral representation amp oldid 1107574476, wikipedia, wiki, book, books, library,