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{{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>
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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]]

Latest revision as of 17:02, 9 October 2014


The feeling of freedom and self-control that you feel as you drive through the mountain trails can help you forget all the stress and worries of daily life. For more serious cyclists, next to bicycles themselves, proper shoes are the most important piece of equipment. Understanding your strengths, and the strengths of your team or business partner can be your best competitive edge this year. Music sounds better with a background like the Continental Divide, and you simply may find out once and for all, as there are music vacations and concerts happening just about each weekend in the summertime. However, due to its high price tag and light weight, it is seldom used for jumping and downhill races.

Adjust the tyre pressure too so it is at its optimum level. By just taking a few minutes to make sure your bike is well-tuned before you set out, you can keep from spending hours on issues later. Each rider has lanes, but some others only have a single lane. Now please continue on additionally so you can get added info for this topic. If you cherished this article and you simply would like to obtain more info regarding Size guaide mountain bike sizing. kindly visit the web-site. Now the most successful business gurus are looking to the interns for lessons.

The "derailleur" attached to the gear shifter moves the chain from one ring to another. Mulberry Gap Mountain Bike Get-A-Way is located 12 miles from downtown Ellijay in the Chattahoochee National Forest with some of the best single track mountain bike trails including the Georgia Pinhoti trail system. The whole set-up, including your bag, should weigh no more than 20kgs (44lbs). Once you have answered the above questions and you see all these features present in a bike, you are on your way to choosing the road bike that is right for you. These are stable forks whose weights are directly in proportion to their durability.

I would wake up at 3 o'clock in the morning to make calls to America. Or get the type motor that senses when you are pedaling harder and switches on automatically. This article is to help you when buying a new bike. With the tips given, I am sure that you and your new bike will work together for that ultimate mountain biking adventure. Within 2 weeks of breathing exercises, the warm up period disappeared.

If you can afford it, a full suspension mountain bike is always worth the purchase. If you can, try and drop off a gutter or curb, standing to allow your knees to absorb the impact. The price may generally differ depending on the condition of the bike and on how new the model really is. A twenty-inch wheeled bike are good for longer traveling time. It's about the journey and all that it entails, learning about yourself, making friends, being helped, and of course, helping others along the way".

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