en>Anita5192: Undid revision 632359490 by 173.69.149.160 (talk)Reverted good faith edits. WP:MOSMATH uses italics for differentials

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Undid revision 632359490 by 173.69.149.160 (talk)Reverted good faith edits. WP:MOSMATH uses italics for differentials

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	~~{{DISPLAYTITLE~~:1/~~''N'' expansion}}~~		I woke up last week and [http://Search.About.com/?q=realised+- realised -] I have been solitary for a little while at the moment and following much [http://www.reddit.com/r/howto/search?q=intimidation intimidation] from pals I now locate myself opted for web dating. They assured me that there are plenty of entertaining, pleasant and regular [http://www.netpaw.org luke bryan tour date] people to meet up, therefore here goes the pitch!<br>I strive to stay as physically fit as possible staying at the gymnasium several-times a week. I enjoy my athletics and try to play or see as many a potential. Being wintertime I am going to often at Hawthorn suits. Notice: If you considered purchasing a hobby I do not mind, I've noticed the carnage of fumbling matches at stocktake revenue.<br>My buddies and fam are wonderful and spending some time with them at tavern gigabytes or dinners is consistently a necessity. As I come to realize that you could never have a significant dialogue with the noise I have never been into night clubs. I likewise have 2 unquestionably cheeky and very cunning puppies that are invariably excited to meet up fresh people.<br><br>
	~~{\| class~~=~~wikitable align~~=~~right width=320~~
	~~\|colspan=2\|~~[~~[File~~:~~Three Gluon Vertex in t'Hooft notation~~.~~png\|320px]~~]<BR>~~How~~ a ~~three gluon vertex would appear in t'Hooft's double index notation~~. ~~This makes the analogy~~ to a ~~string theory that will appear~~ at ~~large N apparent~~.
	\|-
	~~!colspan=2\|Examples~~
	~~\|- valign=top align=center~~
	\|1<BR>~~[[Image:1 over N1~~.~~png\|157px]]~~
	\|2<BR>~~[[Image:1 over N2.png\|150px]]~~
	\|}

	~~In [[quantum field theory]] and [[statistical mechanics]], the '''1/''N'' expansion''' (also known as the ''large~~-~~N'' expansion) is a particular~~ [[perturbation theory\|perturbative]] analysis of quantum field theories with an [[internal symmetry]] [[group theory\|group]] such as [[special orthogonal group\|SO(N)]] or [[special unitary group\|SU(N)]]. It consists in deriving an expansion for the properties of the theory in powers of <math>1/N</~~math>, which is treated as a small parameter~~.		my web blog - luke bryan concert tour luke bryan today show concert [[http://lukebryantickets.citizenswebcasting.com lukebryantickets.citizenswebcasting.com]] 2014 ([http://lukebryantickets.asiapak.net look at this website])

	~~This technique is used in [[Quantum chromodynamics\|QCD]] (even though <math>N</math> is only 3 there) with the [[gauge group]] SU(3)~~. ~~Another application in [[particle physics]] is to the study of [[AdS/CFT]] dualities~~.

	~~It is also extensively used in [[condensed matter physics]] where it can be used to provide a rigorous basis for [[mean field theory]]~~.

	~~== Example ==~~
	~~Starting with a simple example - the [[orthogonal group\|O(N)~~]] ~~[[Quartic interaction\|φ<sup>4</sup>]] - the scalar field φ takes on values in the [[real number\|real]] [[vector representation]] of O~~(~~N). Using the~~ [[index notation]] for the N "[[Flavour (particle physics)\|flavor]]s" with the [[Einstein summation convention]] and because O(N) is orthogonal, no distinction will be made between covariant and contravariant indices. The [[Lagrangian density]] is given by

	:~~<math>\mathcal{L}={1\over 2}\partial^\mu \phi_a \partial_\mu \phi_a-{m^2\over 2}\phi_a \phi_a-{\lambda\over 8N}(\phi_a \phi_a)^2</math>~~

	~~where <math>a</math> runs from 1 to N. Note that N has been absorbed into the [[coupling constant\|coupling strength]] λ. This is crucial here.~~

	~~Introducing an [[auxiliary field]] F;~~

	~~:<math>\mathcal{L}={1\over 2}\partial^\mu \phi_a \partial_\mu \phi_a -{m^2\over 2}\phi_a \phi_a +{1\over 2}F^2-{\sqrt{\lambda /N}\over 2}F \phi_a \phi_a</math>~~

	~~In the [[Feynman diagram]]s, the graph breaks up into disjoint [[cycle (graph theory)\|cycles]], each made up of φ edges of the same flavor and the cycles are connected by F edges.~~

	~~Each 4-point vertex contributes λ/N and hence, 1/N. Each flavor cycle contributes N because there are N such flavors to sum over. Note that not all momentum flow cycles are flavor cycles.~~

	~~At least perturbatively, the dominant contribution to the 2k-point [[connected correlation function]] is of the order (1~~/~~N)<sup>k-1<~~/~~sup> and the other terms are higher powers of 1/N. Performing a 1/N expansion gets more and more accurate in the large N limit~~. ~~The vacuum energy density is proportional to N, but can be ignored due to non-compliance with general relativity assumptions~~.

	~~Due to~~ this structure, a different graphical notation to denote the Feynman diagrams can be used. Each flavor cycle can be represented by a vertex. The flavor paths connecting two external vertices are represented by a single vertex. The two external vertices along the same flavor path are naturally paired and can be replaced by a single vertex and an edge (not an F edge) connecting it to the flavor path. The F edges are edges connecting two flavor cycles/paths to each other (or a flavor cycle/path to itself). The interactions along a flavor cycle/path have a definite cyclic order and represent a special kind of graph where the order of the edges incident to a vertex matters, but only up to a cyclic permutation, and since this is a theory of real scalars, also an order reversal (but if we have SU(N) instead of SU(2), order reversals aren't valid). Each F edge is assigned a momentum (the momentum transfer) and there is an internal momentum integral associated with each flavor cycle.

	~~== QCD ==~~

	~~{{Main\|Quantum chromodynamics}}~~

	~~QCD is a SU(3) [[gauge theory]] involving [[gluon]]s and [[quark]]s. The [[Weyl fermion\|left-handed quark~~]~~]s belong to a triplet representation, the right-handed to an antitriplet representation (after charge-conjugating them~~) ~~and the gluons to a [[real number\|real]] [[adjoint representation]]. A quark edge is assigned a color and orientation and a gluon edge is assigned a color pair.~~

	~~In the large N limit, we only consider the dominant term. See [[AdS/CFT]].~~

	~~==References==~~
	*{{cite journal
	~~\|author=[[Gerardus 't Hooft\|G. 't Hooft]]~~
	~~\|title=A planar diagram theory for strong interactions~~
	~~\|journal=Nuclear Physics B~~
	~~\|volume=72~~
	~~\|issue=3~~
	~~\|page=461~~
	~~\|doi=10.1016/0550-3213(74)90154-0~~
	~~\|url=http://igitur-archive.library.uu.nl/phys/2005-0622-152933/UUindex.html~~
	~~\|year=1974\|bibcode = 1974NuPhB..72..461T }}~~

	~~{{DEFAULTSORT:1 N Expansion}}~~
	~~[[Category:Quantum chromodynamics]]~~
	~~[[Category:String theory]]~~
	~~[[Category:Statistical mechanics]]~~

5.102.253.17: Fixed disambiguation ambiguity

2014-01-17T21:40:05Z

Fixed disambiguation ambiguity

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Weak derivative - Revision history

en>Anita5192: Undid revision 632359490 by 173.69.149.160 (talk)Reverted good faith edits. WP:MOSMATH uses italics for differentials

5.102.253.17: Fixed disambiguation ambiguity

en>Straightontillmorning: /* Examples */ disambig characteristic function