Nanoparticle tracking analysis - Revision history

87.14.187.173: /* Comparison to dynamic light scattering */

2014-12-18T23:43:35Z

Comparison to dynamic light scattering

← Older revision		Revision as of 01:43, 19 December 2014
Line 1:		Line 1:
	~~{{Unreferenced\|date=December 2009}}~~		The writer is known by the title of Numbers Lint. His spouse doesn't like it the way he does but what he really likes doing is to do aerobics and he's been doing it for quite a while. Managing people is his occupation. California is our beginning place.<br><br>Also visit my blog over the counter std test - [http://www.neweracinema.com/tube/user/D69E use this link],
	~~In [[theoretical physics]], a '''Fierz identity'''~~ is ~~an identity that allows one to rewrite ''[[bilinear]]s{{dn\|date=December 2013}} of~~ the ~~product''~~ of ~~two [[spinor]]s as a [[linear combination]] of ''products of the bilinears~~'~~' of~~ the ~~individual spinors. It~~ is ~~named after Swiss physicist [[Markus Fierz]].~~

	~~There is a version of the Fierz identities for [[Dirac spinor]]s~~ and ~~there is another version for [[Weyl spinor]]~~s~~. And there are versions~~ for ~~other dimensions besides 3+1 dimensions.~~

	~~Spinor bilinears can be thought of as elements of~~ a ~~[[Clifford Algebra]]~~. ~~Then the Fierz identity~~ is ~~the concrete realization of the [[Clifford Algebra#Relation to the exterior algebra\|relation to the exterior algebra]]~~.
	~~The identities for a generic scalar written as the contraction of two Dirac bilinears of the same type can be written with coefficients according the following table~~.

	~~{\| class="wikitable"~~
	\|-
	~~! Product~~
	~~! S~~
	~~! V~~
	~~! T~~
	~~! A~~
	~~! P~~
	\|-
	~~\| S × S =~~
	~~\| 1/4~~
	~~\| 1/4~~
	~~\| -1/4~~
	~~\| -1/4~~
	~~\| 1/4~~
	\|
	\|-
	~~\| V × V =~~
	~~\| 1~~
	~~\| -1/2~~
	~~\| 0~~
	~~\| -1/2~~
	~~\| -1~~
	\|-
	~~\| T × T =~~
	~~\| -3/2~~
	~~\| 0~~
	~~\| -1/2~~
	~~\| 0~~
	~~\| -3/2~~
	\|-
	~~\| A × A =~~
	~~\| -1~~
	~~\| -1/2~~
	~~\| 0~~
	~~\| -1/2~~
	~~\| 1~~
	\|-
	~~\| P × P =~~
	~~\| 1/4~~
	~~\| -1/4~~
	~~\| -1/4~~
	~~\| 1/4~~
	~~\| 1/4~~
	\|-
	\|}

	~~For example the V × V product can be expanded as,~~
	:<~~math~~>
	~~\left(\chi^\dagger\gamma^0\gamma^\mu\psi\right)\left(\psi^\dagger\gamma^0\gamma_\mu \chi\right)=~~
	~~\left(\chi^\dagger\gamma^0\chi\right)\left(\psi^\dagger\gamma^0\psi\right)-~~
	~~\frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\psi\right)-~~
	~~\frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\gamma_5\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\gamma_5\psi\right)~~
	~~-\left(\chi^\dagger\gamma^0\gamma^5\chi\right)\left(\psi^\dagger\gamma^0\gamma_5\psi\right).~~<~~/math~~>

	~~Simplifications arise when~~ the ~~considered spinors are chiral or~~ [~~[Majorana spinor]]s as some term in the expansion can be vanishing.~~

	~~==References==~~
	~~A derivation of identities for rewriting any scalar contraction of Dirac bilinears can be found in 29~~.3.~~4 of~~
	~~{{cite book\|author=L. B. Okun\|title=Leptons and Quarks\|publisher=North-Holland\|year=1980\|isbn=978-0-444-86924-1}}~~

	~~See also appendix B.1.2 in {{cite book\|author=T. Ortin\|title=Gravity and Strings\|publisher=Cambridge University Press\|year=2004\|isbn=978-0-521-82475-0}}~~

	~~{{DEFAULTSORT:Fierz Identity}}~~
	~~[[Category:Quantum field theory~~]]
	~~[[Category:Spinors]]~~
	~~[[Category:Mathematical identities]]~~


	~~{{Phys-stub}}~~

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2013-03-15T08:31:15Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3870163

← Older revision		Revision as of 10:31, 15 March 2013
Line 1:		Line 1:
	~~Let me first start by~~ [~~http://Bossip.com/634680/celebrities-accused-~~of~~-carrying-stds3920/ introducing~~] ~~myself~~. ~~My title std testing at home~~ is ~~Boyd Butts even though it~~ is ~~not~~ the ~~name on my birth certification~~. ~~Doing~~ [~~http:~~//~~Www.Huffingtonpost.ca~~/~~colette~~-~~kenney~~/~~dating~~-~~with~~-~~herpes_b_1609226.html ceramics] is what her family and her appreciate. For years~~ ~~[http:~~//~~livestudentjobs.co.uk~~/~~groups~~/~~solid~~-~~advice~~-~~for~~-~~handling~~-a-~~candida~~/ ~~livestudentjobs.co.uk] he's been operating as a meter reader and it's something~~ ~~at home std test he~~ ~~[http:~~//~~Www.Adosphere.com~~/~~poyocum home std test kit] truly enjoy. Years ago we moved to North Dakota and I love each day residing~~ right ~~here~~.<br>~~<br>Stop by my webpage ::~~ [~~http://gcjcteam~~.~~org/index~~.~~php?mid~~=~~etc_video&document_srl~~=~~655020&sort_index~~=~~regdate&order_type~~=~~desc gcjcteam~~.~~org~~]		{{Unreferenced\|date=December 2009}}
			In [[theoretical physics]], a '''Fierz identity''' is an identity that allows one to rewrite ''[[bilinear]]s{{dn\|date=December 2013}} of the product'' of two [[spinor]]s as a [[linear combination]] of ''products of the bilinears'' of the individual spinors. It is named after Swiss physicist [[Markus Fierz]].

			There is a version of the Fierz identities for [[Dirac spinor]]s and there is another version for [[Weyl spinor]]s. And there are versions for other dimensions besides 3+1 dimensions.

			Spinor bilinears can be thought of as elements of a [[Clifford Algebra]]. Then the Fierz identity is the concrete realization of the [[Clifford Algebra#Relation to the exterior algebra\|relation to the exterior algebra]].
			The identities for a generic scalar written as the contraction of two Dirac bilinears of the same type can be written with coefficients according the following table.

			{\| class="wikitable"
			\|-
			! Product
			! S
			! V
			! T
			! A
			! P
			\|-
			\| S × S =
			\| 1/4
			\| 1/4
			\| -1/4
			\| -1/4
			\| 1/4
			\|
			\|-
			\| V × V =
			\| 1
			\| -1/2
			\| 0
			\| -1/2
			\| -1
			\|-
			\| T × T =
			\| -3/2
			\| 0
			\| -1/2
			\| 0
			\| -3/2
			\|-
			\| A × A =
			\| -1
			\| -1/2
			\| 0
			\| -1/2
			\| 1
			\|-
			\| P × P =
			\| 1/4
			\| -1/4
			\| -1/4
			\| 1/4
			\| 1/4
			\|-
			\|}

			For example the V × V product can be expanded as,
			:<math>
			\left(\chi^\dagger\gamma^0\gamma^\mu\psi\right)\left(\psi^\dagger\gamma^0\gamma_\mu \chi\right)=
			\left(\chi^\dagger\gamma^0\chi\right)\left(\psi^\dagger\gamma^0\psi\right)-
			\frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\psi\right)-
			\frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\gamma_5\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\gamma_5\psi\right)
			-\left(\chi^\dagger\gamma^0\gamma^5\chi\right)\left(\psi^\dagger\gamma^0\gamma_5\psi\right).</math>

			Simplifications arise when the considered spinors are chiral or [[Majorana spinor]]s as some term in the expansion can be vanishing.

			==References==
			A derivation of identities for rewriting any scalar contraction of Dirac bilinears can be found in 29.3.4 of
			{{cite book\|author=L. B. Okun\|title=Leptons and Quarks\|publisher=North-Holland\|year=1980\|isbn=978-0-444-86924-1}}

			See also appendix B.1.2 in {{cite book\|author=T. Ortin\|title=Gravity and Strings\|publisher=Cambridge University Press\|year=2004\|isbn=978-0-521-82475-0}}

			{{DEFAULTSORT:Fierz Identity}}
			[[Category:Quantum field theory]]
			[[Category:Spinors]]
			[[Category:Mathematical identities]]


			{{Phys-stub}}

82.68.24.217: /* Comparison to dynamic light scattering */

2012-05-22T11:19:49Z

Comparison to dynamic light scattering

New page

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