Feynman parametrization - Revision history

85.250.123.224: Added the symmetric form of the parametrization, used in some QED papers from the 60's.

2014-11-08T13:23:32Z

Added the symmetric form of the parametrization, used in some QED papers from the 60's.

← Older revision		Revision as of 15:23, 8 November 2014
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	'''~~Schwinger parametrization~~'~~'' is~~ a ~~technique for evaluating~~ [~~[loop integral]]s which arise from [[Feynman diagram]]s with one or more loops~~.

	~~Using~~ the ~~well-known observation that~~

	~~:<math>\frac{1}{A^n}=\frac{1}{(n-1)!}\int^\infty_0 du \, u^{n-1}e^{-uA},</math>~~

	~~[[Julian Schwinger]~~] ~~noticed that one may simplify the integral:~~

	~~:<math>\int \frac{dp}{A(p~~)~~^n}=\frac{1}{\Gamma(n)}\int dp \int^\infty_0 du \, u^{n-1}e^{-uA(p)}=\frac{1}{\Gamma(n)}\int^\infty_0 du \, u^{n-1} \int dp \, e^{-uA(p)},</math>~~

	~~for Re(n)>0~~.

	~~Another version of Schwinger parametrization is:~~

	:<~~math~~>~~\frac{1}{A}=-i\int^\infty_0 du \, e^{iuA},~~<~~/math~~>

	~~and it is easy to generalize this identity to n denominators~~.

	~~See also [[Feynman parametrization]]~~.

	[[~~Category~~:~~Quantum field theory]]~~


	~~{{applied~~-~~math~~-~~stub}}~~
	~~{{quantum~~-~~stub}}~~

en>David Eppstein: {{applied-math-stub}}

2013-10-03T21:29:01Z

← Older revision		Revision as of 23:29, 3 October 2013
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			'''Schwinger parametrization''' is a technique for evaluating [[loop integral]]s which arise from [[Feynman diagram]]s with one or more loops.

			Using the well-known observation that

			:<math>\frac{1}{A^n}=\frac{1}{(n-1)!}\int^\infty_0 du \, u^{n-1}e^{-uA},</math>

			[[Julian Schwinger]] noticed that one may simplify the integral:

			:<math>\int \frac{dp}{A(p)^n}=\frac{1}{\Gamma(n)}\int dp \int^\infty_0 du \, u^{n-1}e^{-uA(p)}=\frac{1}{\Gamma(n)}\int^\infty_0 du \, u^{n-1} \int dp \, e^{-uA(p)},</math>

			for Re(n)>0.

			Another version of Schwinger parametrization is:

			:<math>\frac{1}{A}=-i\int^\infty_0 du \, e^{iuA},</math>

			and it is easy to generalize this identity to n denominators.

			See also [[Feynman parametrization]].

			[[Category:Quantum field theory]]


			{{applied-math-stub}}
			{{quantum-stub}}

195.168.225.55: /* Derivation */ Fix of wrong order of A,B in derivation

2012-07-15T15:13:02Z

Derivation: Fix of wrong order of A,B in derivation

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