en>Rich Farmbrough: clean up- spelling "et al." and gen fixes using AWB

2009-05-19T23:06:24Z

clean up- spelling "et al." and gen fixes using AWB

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{{Quantum field theory}}
In [[physics]], the '''Callan–Symanzik equation''' is a [[differential equation]] describing the evolution of the [[Correlation function (quantum field theory)|n-point correlation functions]] under variation of the energy scale at which the theory is defined and involves the beta-function of the theory and the anomalous dimensions. This equation has the following structure

:<math>\left[M\frac{\partial }{\partial M}+\beta(g)\frac{\partial }{\partial g}+n\gamma\right] G^{(n)}(x_1,x_2,\ldots,x_n;M,g)=0</math>

being <math>\beta(g)</math> the [[Beta function (physics)|beta function]] and <math>\gamma</math> the scaling of the fields.

In [[quantum electrodynamics]] this equation takes the form

:<math>\left[M\frac{\partial }{\partial M}+\beta(e)\frac{\partial }{\partial e}+n\gamma_2 +m\gamma_3\right]G^{(n,m)}(x_1,x_2,\ldots,x_n;M,e)=0</math>

being n and m the number of [[electron]]s and [[photon]]s respectively.

It was discovered independently by [[Curtis Callan]]<ref>C. G. Callan, Jr., ''Broken Scale Invariance in Scalar Field Theory'', Phys. Rev. D '''2''', 1541–1547 (1970). [http://prola.aps.org/abstract/PRD/v2/i8/p1541_1 APS]</ref> and [[Kurt Symanzik]]<ref>K. Symanzik, ''Small Distance Behaviour in Field Theory and Power Counting'', Commun. math. Phys. '''18''', 227 (1970). [http://www.springerlink.com/content/t261885775652350/ SpringerLink]</ref><ref>K. Symanzik, ''Small-Distance-Behaviour Analysis and Wilson Expansions'', Commun. math. Phys. '''23''', 49 (1971). [http://www.springerlink.com/content/v5847848m81028t8/ SpringerLink]</ref> in 1970. Later it was used to understand [[asymptotic freedom]].

This equation arises in the framework of [[renormalization group]]. It is possible to treat the equation using [[Perturbation theory (quantum mechanics)|perturbation theory]].

==See also==
*[[Exact renormalization group equation]]
*[[Beta function (physics)|Beta function]]

==Notes==
<references/>

==References==
* Jean Zinn-Justin, ''Quantum Field Theory and Critical Phenomena '', [[Oxford University Press]] 2003, ISBN 0-19-850923-5
* John Clements Collins, ''Renormalization'', [[Cambridge University Press]] 1986, ISBN 0-521-31177-2

{{DEFAULTSORT:Callan-Symanzik equation}}
[[Category:Quantum field theory]]
[[Category:Renormalization group]]
[[Category:Equations]]

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en>Rich Farmbrough: clean up- spelling "et al." and gen fixes using AWB