Ensemble averaging

2013-12-05T08:34:20Z

159.226.234.17: /* Method */

{{quantum field theory}}

'''Källén–Lehmann spectral representation''' gives a general expression for the [[Correlation function (quantum field theory)|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.<ref name=kallen>{{cite journal |authorlink1=Gunnar Källén |last1=Källén |first1=Gunnar |last2= |first2= |year=1952 |title= |journal=Helvetica Physica Acta |publisher= |volume=25 |issue= |pages=417 |url= |doi=10.5169/seals-112316}}</ref><ref name=lehmann>{{cite journal |authorlink1=Harry Lehmann |last1=Lehmann |first1=Harry |last2= |first2= |year=1954 |title= |journal=Nuovo Cimento |publisher= |volume=11 |issue= |pages=342 |url= |doi= }}</ref> This can be written as

:<math>\Delta(p)=\int_0^\infty d\mu^2\rho(\mu^2)\frac{1}{p^2-\mu^2+i\epsilon}</math>

being <math>\rho(\mu^2)</math> 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.<ref name=strocchi>{{Cite book
| last = Strocchi
| first = Franco
| authorlink = Franco Strocchi
| coauthors =
| title = Selected Topics on the General Properties of Quantum Field Theory
| publisher = World Scientific
| year = 1993
| location = Singapore
| pages =
| url =
| doi =
| id =
| isbn = 981-02-1143-0}}</ref> This belongs to [[non-perturbative]] techniques of [[quantum field theory]].

==Mathematical derivation==

In order to derive a spectral representation for the propagator of a field <math>\Phi(x)</math>, one consider a complete set of states <math>\{|n\rangle\}</math> so that, for the [[Correlation function (quantum field theory)|two-point function]] one can write

:<math>\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle=\sum_n\langle 0|\Phi(x)|n\rangle\langle n|\Phi^\dagger(y)|0\rangle.</math>

We can now use [[Poincaré group|Poincaré invariance]] of the vacuum to write down

:<math>\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle=\sum_n e^{-ip_n\cdot(x-y)}|\langle 0|\Phi(0)|n\rangle|^2.</math>

Let us introduce the spectral density function

:<math>\rho(p^2)\theta(p_0)(2\pi)^{-3}=\sum_n\delta^4(p-p_n)|\langle 0|\Phi(0)|n\rangle|^2</math>.

We have used the fact that our two-point function, being a function of <math>p_\mu</math>, can only depend on <math>p^2</math>. Besides, all the intermediate states have <math>p^2\ge 0</math> and <math>p_0>0</math>. It is immediate to realize that the spectral density function is real and positive. So, one can write

:<math>\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle = \int\frac{d^4p}{(2\pi)^3}\int_0^\infty d\mu^2e^{-ip\cdot(x-y)}\rho(\mu^2)\theta(p_0)\delta(p^2-\mu^2)</math>

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

:<math>\langle 0|\Phi(x)\Phi^\dagger(y)|0\rangle = \int_0^\infty d\mu^2\rho(\mu^2)\Delta'(x-y;\mu^2)</math>

being

:<math>\Delta'(x-y;\mu^2)=\int\frac{d^4p}{(2\pi)^3}e^{-ip\cdot(x-y)}\theta(p_0)\delta(p^2-\mu^2)</math>.

From [[CPT theorem]] we also know that holds an identical expression for <math>\langle 0|\Phi^\dagger(x)\Phi(y)|0\rangle</math> and so we arrive at the expression for the chronologically ordered product of fields

:<math>\langle 0|T\Phi(x)\Phi^\dagger(y)|0\rangle = \int_0^\infty d\mu^2\rho(\mu^2)\Delta(x-y;\mu^2)</math>

being now

:<math>\Delta(p;\mu^2)=\frac{1}{p^2-\mu^2+i\epsilon}</math>

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==
{{reflist}}

==Bibliography==

*{{cite book |last=Weinberg |first=S. |authorlink=Steven Weinberg |title=The Quantum Theory of Fields: Volume I Foundations |edition= |publisher=[[Cambridge University Press]] |year=1995 |isbn=0-521-55001-7}}
*{{cite book |first1=Michael |last1=Peskin |authorlink1=Michael Peskin|first2=Daniel |last2=Schoeder |authorlink2=Daniel Schroeder|title=An Introduction to Quantum Field Theory |publisher=[[Perseus Books Group]] |year=1995 |isbn=0-201-50397-2 }}
*{{cite book |last=Zinn-Justin |first=Jean |authorlink=jean Zinn-Justin|title=Quantum Field Theory and Critical Phenomena|publisher=[[Clarendon Press]] |year=1996 |edition=3rd |isbn=0-19-851882-X}}

==External links==
*[http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/Källén-Lehmann_spectral_representation The Tangent Bundle]

==See also==

*[[Quantum field theory]]
*[[Correlation function (quantum field theory)|Correlation functions]]
*[[Propagator]]

{{DEFAULTSORT:Kallen-Lehmann Spectral Representation}}
[[Category:Quantum field theory]]
[[Category:Particle physics]]

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Ensemble averaging