Hankel transform - Revision history

en>Eric Kvaalen: Added a section on an alternative definition and its domain of definition. Moved a couple sentences.

2014-07-28T19:44:54Z

Added a section on an alternative definition and its domain of definition. Moved a couple sentences.

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en>AnomieBOT: Dating maintenance tags: {{Citation needed}}

2014-02-20T20:12:53Z

Dating maintenance tags: {{Citation needed}}

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	~~<!---{{Unreferenced\|date=December 2009}}~~
	~~{{Context\|see discussion\|date=October 2009}}-->~~
	~~{{Quantum field theory}}~~
	~~In [[physics]] a '''free field''' is a [[Field (physics)\|field]] '''without [[fundamental interaction\|interaction]]s''', which is described by the terms of motion and mass.~~

	~~== Description ==~~

	~~In [[classical physics]], a '''free field'''~~ is a ~~field whose [[equations of motion]] are given by [[linear]] [[partial differential equation]]s. Such linear PDE's have a unique solution for a given~~ initial ~~condition~~.		Luke is a superstar from the producing along with the job expansion initial 2nd to his 3rd restaurant recording, & , will be the confirmation. He broken on [http://okkyunglee.com justin moore tour] the scene in 2014 together with his unique mixture of [http://www.ffpjp24.org photos of luke bryan] [http://www.netpaw.org upcoming luke bryan concerts] downward-house convenience, movie celebrity great looks and lyrics, is defined t within a major way. The brand new recording Top on the land chart and #2 around the take maps, creating it the 2nd highest first appearance during that time of 2013 to get a country artist. <br><br>The boy of your , knows patience and willpower are key elements with regards to an effective career- . His very first recording, Stay Me, generated the very best hits “All My Pals Say” and “Country Person,” when his effort, Doin’ Thing, located the performer-3 right [http://lukebryantickets.hamedanshahr.com luke bryan live in concert] No. 3 men and women: More Phoning Is really a Very good Factor.”<br><br>From the tumble of 2006, Concerts: Bryan And that have an amazing listing of , including Metropolitan. “It’s much like you’re acquiring a endorsement to travel to the next level, states those performers that have been a part of the Concert toursaround in to a larger sized level of designers.” It packaged as one of the most successful organized tours in their ten-year background.<br><br>Feel free to surf to my web blog ... luke bryan in concert ([http://www.ladyhawkshockey.org sneak a peek at this web-site])

	In [~~[quantum field theory~~]~~], an [[operator valued distribution]] is a '''free field''' if it satisfies some linear partial differential equations such that~~ the ~~corresponding case~~ of ~~the same linear PDEs for a classical field (i~~.e. ~~not an operator) would be the [[Euler-Lagrange equation]~~] ~~for some~~ [~~[quadratic polynomial\|quadratic]] [[Lagrangian]]~~. ~~We can differentiate distributions by defining their derivatives via differentiated [[test function]]s~~. ~~See [[Schwartz distribution~~]~~] for more details. Since we are dealing not with ordinary distributions but operator valued distributions~~, it is ~~understood these PDEs aren'~~t ~~constraints on states but instead~~ a ~~description of the relations among the smeared fields~~. ~~Beside~~ the ~~PDEs,~~ the ~~operators also satisfy another relation~~, the ~~commutation/anticommutation relations~~.

	~~== Canonical Commutation Relation ==~~
	~~Basically~~, ~~[[commutator]] (for [[boson]]s)/[[anticommutator]] (for [[fermions]]) of two smeared fields is i times the [[Peierls bracket]] of the field~~ with ~~itself (which is really a distribution~~, ~~not a function) for~~ the ~~PDEs smeared over both test functions. This has~~ the ~~form of a~~ [~~[CCR~~/~~CAR algebra]]~~.

	~~CCR/CAR algebras with infinitely many degrees of freedom have many inequivalent irreducible unitary representations~~. ~~If the theory is defined over [[Minkowski space]], we may choose the unitary irrep containing a [[vacuum state]~~] ~~although that isn't always necessary~~.

	~~==Example==~~
	~~Let φ be an operator valued distribution~~ and ~~the (Klein-Gordon) PDE be~~

	:<~~math~~>~~\partial^\mu \partial_\mu \phi+m^2 \phi^2=0~~<~~/math~~>.

	~~This is a bosonic field. Let's call~~ the ~~distribution given by the Peierls bracket Δ.~~

	~~Then~~,

	:~~<math>\{\phi(x)~~,~~\phi(y)\}=\Delta(x;y)</math>~~

	~~where here~~, ~~φ is~~ a ~~classical field and {,} is~~ the ~~Peierls bracket~~.

	~~Then, the [[canonical commutation relation]] is~~

	:<~~math~~>~~[\phi[f],\phi[g]]=i\Delta[f,g] \,~~<~~/math~~>.

	~~Note that Δ is a distribution over two arguments, and so, can be smeared as well~~.

	~~Equivalently, we could have insisted that~~

	~~:<math>\mathcal{T}\{[(~~(~~\partial^\mu \partial_\mu+m^2)\phi)~~[~~f],\phi[g]]\}=-i\int d^dx f(x)g(x)<~~/~~math>~~

	~~where <math>\mathcal{T}<~~/~~math> is the [[time ordering]] operator and that if the supports of f and g are spacelike separated,~~

	~~:<math>[\phi[f],\phi[g]]=0</math>.~~

	~~== See also ==~~
	*[[Normal order]]
	*[[Wick's theorem]]

	~~== References ==~~
	* Michael E. ~~Peskin and Daniel V~~. ~~Schroeder, ''An Introduction to Quantum Field Theory'', Addison-Wesley, Reading, 1995. p19~~-~~p29~~

	~~{{DEFAULTSORT:Free Field}}~~
	~~[[Category:Quantum field theory]~~]

en>Anrnusna: /* References */journal name, replaced: Proc. Lond Math. Soc. → Proceedings of the London Mathematical Society, J. Opt. → J. Opt. Soc. Am. (6) using AWB

2013-10-17T12:00:29Z

References: journal name, replaced: Proc. Lond Math. Soc. → Proceedings of the London Mathematical Society, J. Opt. → J. Opt. Soc. Am. (6) using AWB

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en>Bibcode Bot: Adding 0 arxiv eprint(s), 3 bibcode(s) and 1 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2012-08-30T00:44:51Z

Adding 0 arxiv eprint(s), 3 bibcode(s) and 1 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

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