en>Phleg1: dab

2014-10-21T19:29:41Z

dab

en>Lagrange613: pronoun

2013-12-23T15:44:38Z

pronoun

← Older revision		Revision as of 17:44, 23 December 2013
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	~~The name~~ of the ~~author is Numbers but it~~'s ~~not~~ the ~~most masucline title out there~~. ~~One~~ of the ~~extremely very best issues in~~ the ~~globe for me~~ is ~~to do aerobics~~ and ~~now I~~'~~m attempting~~ to ~~earn cash with~~ it. ~~North Dakota~~ is ~~our beginning place~~. ~~He utilized to be unemployed but~~ now he is ~~a computer operator but his promotion by no means arrives.~~<br><br>~~Look at my blog~~ [http://www.1a-~~pornotube~~.~~com~~/~~blog~~/~~84958 1a~~-~~pornotube~~.~~com~~]		{{Quantum field theory}}
			In [[theoretical physics]], '''background field method''' is a useful procedure to calculate the [[effective action]] of a [[quantum field theory]] by expanding a quantum field around a classical "background" value ''B'':
			: <math> \phi(x) = B(x) + \eta (x)</math>.
			After this is done, the Green's functions are evaluated as a function of the background. This approach has the advantage that the [[gauge invariance]] is manifestly preserved if the approach is applied to [[gauge theory]].

			==Method==
			We typically want to calculate expressions like
			:<math> Z[J] = \int \mathcal D \phi e^{i \int d^d x (\mathcal L [\phi(x)] + J(x) \phi(x))} </math>
			where ''J''(''x'') is a source, <math>\mathcal L(x)</math> is the [[Lagrangian density]] of the system, ''d'' is the number of dimensions and φ(''x'') is a field.

			In the background field method, one starts by splitting this field into a classical background field ''B''(''x'') and a field η(''x'') containing additional quantum fluctuations:
			:<math> \phi(x) = B(x) + \eta(x) \,.</math>
			Typically, ''B''(''x'') will be a solution of the classical equations of motion
			:<math> \left.\frac{\delta S}{\delta \phi}\right\|_{\phi = B} = 0 </math>
			where ''S'' is the action, i.e. the space integral of the Lagrangian density. Fields obeying these equations typically yield the greatest contribution in a path integral, so it is natural to expand around them. Switching on a source ''J''(''x'') will change the equations into δ''S''/δφ\|<sub>φ = ''B''</sub> + ''J'' = 0.

			Then the action is expanded around the background ''B''(''x''):
			:<math> \int d^d x (\mathcal L [\phi(x)] + J(x) \phi(x)) = \int d^d x (\mathcal L [B(x)] + J(x) B(x)) + \int d^d x \left(\frac{\delta\mathcal L}{\delta \phi(x)} [B] + J(x)\right) \eta(x) + \frac12 \int d^d x d^d y \frac{\delta^2\mathcal L}{\delta \phi(x) \delta\phi(y)} [B] \eta(x) \eta(y) + \cdots </math>
			The second term in this expansion is zero by the equations of motion. The first term does not depend on any fluctuating fields, so that it can be brought out of the path integral. The result is
			:<math> Z[J] = e^{i \int d^d x (\mathcal L [B(x)] + J(x) B(x))} \int \mathcal D \eta e^{\frac i2 \int d^d x d^d y \frac{\delta^2\mathcal L}{\delta \phi(x) \delta\phi(y)} [B] \eta(x) \eta(y) + \cdots}. </math>
			The path integral which now remains is (neglecting the corrections in the dots) of Gaussian form and can be integrated exactly:
			:<math> Z[J] = C e^{i \int d^d x (\mathcal L [B(x)] + J(x) B(x))} \left(\det \frac{\delta^2\mathcal L}{\delta \phi(x) \delta\phi(y)} [B]\right)^{-1/2} + \cdots </math>
			where "det" signifies a [[functional determinant]] and ''C'' is a constant. The power of minus one half will naturally be plus one for [[Grassmann number\|Grassmann fields]].

			The above derivation gives the Gaussian approximation to the functional integral. Corrections to this can be computed, producing a diagrammatic expansion.

			==See also==
			[[BF theory]]

			==References==
			* {{cite book \| author=Peskin, Michael E.; Schroeder, Daniel V. \| title=Introduction to Quantum Field Theory \| publisher=Perseus Publishing \| year=1994 \| isbn=0-201-50397-2}}
			* [[Hagen Kleinert\|Kleinert, Hagen]], ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, [http://www.worldscibooks.com/physics/7305.html World Scientific (Singapore, 2009)] (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=11 online]. See [http://users.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic3.pdf Chapter 3], p. 321)
			* [[L. F. Abbott]], ''Introduction to the Background Field Method'', Acta Physica Polonica, Vol. B13, 1982, No 1-2. (also available [http://neurotheory.columbia.edu/~larry/AbbottActPhysPol82.pdf online].)

			[[Category:Quantum field theory]]

en>Chthonicdaemon at 10:16, 31 May 2012

2012-05-31T10:16:18Z

New page

The name of the author is Numbers but it's not the most masucline title out there. One of the extremely very best issues in the globe for me is to do aerobics and now I'm attempting to earn cash with it. North Dakota is our beginning place. He utilized to be unemployed but now he is a computer operator but his promotion by no means arrives.<br><br>Look at my blog [http://www.1a-pornotube.com/blog/84958 1a-pornotube.com]

Rosenbrock function - Revision history

en>Phleg1: dab

en>Lagrange613: pronoun

en>Chthonicdaemon at 10:16, 31 May 2012