Scale invariance - Revision history

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2015-01-07T23:54:50Z

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	In [[quantum field theory]], the '''Dirac adjoint''' defines the [[dual vector space\|dual]] operation of a [[Dirac spinor]]. The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors. Since the usual [[Hermitian adjoint]] lacks the Lorentz symmetry of the system, the Dirac adjoint must be used instead.

	~~Possibly to avoid confusion with the usual [[Hermitian adjoint]], some textbooks do not provide a name for the Dirac adjoint but simply call it "<var>ψ</var>-bar".~~

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	~~Let <var>ψ</var> be~~ a ~~[[Dirac spinor]]~~. ~~Then its Dirac adjoint is defined as~~

	~~:<math>\bar\psi \equiv \psi^\dagger \gamma^0</math>~~

	~~where <var>ψ</var><sup>†</sup> denotes~~ the ~~[[Hermitian adjoint]]~~ of ~~the spinor~~ <~~var~~>ψ<~~/var~~> and ~~<var>γ<~~/~~var><sup>0<~~/~~sup> is the time-like [[gamma matrices\|gamma matrix]]~~.

	~~== Spinors under Lorentz transformations ==~~

	~~The [[Lorentz group]]~~ of ~~[[special relativity]] is not [[compact space\|compact]~~], ~~therefore [[Representations of the Lorentz group\|representation]]~~s ~~of [[Lorentz transformation]]s in~~ the ~~Dirac spinor space are~~ not ~~[[unitary operator\|unitary]]~~. ~~That~~ is~~, in general,~~

	:<~~math~~>~~\lambda^\dagger \ne \lambda^{-1}~~<~~/math>~~

	~~where <var>λ</var~~> is the ~~corresponding Lorentz transformation that maps spinors:~~

	~~:<math>\psi \mapsto \lambda \psi</math>.~~

	~~The Hermitian adjoint~~ of ~~spinors transforms according~~ to

	~~:<math>\psi^\dagger \mapsto \psi^\dagger \lambda^\dagger</math>~~.

	~~Therefore~~, ~~using only~~ the ~~Hermitian adjoint, one finds~~ that ~~{{nowrap\|~~<~~var~~>ψ<~~/var~~><sup>†</sup> <var>ψ</var>}} is not a [[Lorentz scalar]] and {{nowrap\|<var>ψ</var><sup>†</sup> <var>γ</var><sup><var>μ</var></sup> <var>ψ</var>}} is not even [[self-adjoint operator\|Hermitian]].

	~~Using the definition~~, ~~one finds that~~ the ~~Dirac adjoint~~ of ~~spinors transforms according~~ to

	~~:<math>\bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0</math>~~.

	~~Using~~ the ~~identity {{nowrap\|<var>γ</var><sup>0</sup> <var>λ</var><sup>†</sup> <var>γ</var><sup>0</sup> {{=}} <var>λ</var><sup>-~~1~~</sup>}}~~, the ~~transformation reduces~~ to

	~~:<math>\bar\psi \mapsto \bar\psi \lambda^{-~~1~~}</math>,~~

	~~which possesses the required Lorentz symmetry for {{nowrap\|{{overline\|<var>ψ~~<~~/var~~>}} <~~var~~>~~ψ</var>}} and {{nowrap\|{{overline\|<var>ψ</var>}} <var>γ</var><sup><var>μ</var></sup> <var>ψ</var>}}.~~

	~~== Usage ==~~

	~~Using~~ the ~~Dirac adjoint,~~ the ~~[[probability current\|probability four~~-~~current]] <var>~~'~~''J'''</var> for a spin-1/2 particle field can be written as~~

	~~:<math>J^\mu = c \bar\psi \gamma^\mu \psi</math>~~

	~~where <var>c</var> is the speed~~ of ~~light~~ and ~~the components~~ of ~~<var>'''J''~~'<~~/var~~> ~~represent the probability density~~ <~~var>ρ</var> and the probability 3-current <var~~>'''~~j'''</var>:~~

	~~:<math>\boldsymbol J =~~ (~~c \rho~~, ~~\boldsymbol j~~)</~~math>~~.

	~~Taking {{nowrap\|<var>μ<~~/~~var> {{=}} 0}} and using the relation~~ for ~~[[gamma matrices]]~~

	:<~~math~~>~~\left(\gamma^0\right)^2 = I~~<~~/math~~>,

	~~the probability density becomes~~

	~~:<math>\rho = \psi^\dagger \psi</math>~~.

	~~== See also ==~~

	*[[Dirac equation]]
	*[[Rarita-Schwinger equation]]

	~~== References ==~~

	*B. Bransden and C. ~~Joachain (2000). ''Quantum Mechanics''~~, ~~2e, Pearson~~. ~~ISBN 0-582-35691-1.~~
	*M. Peskin and ~~D. Schroeder (1995). ''An Introduction~~ to ~~Quantum Field Theory'', Westview Press. ISBN 0-201-50397-2.~~
	*A. Zee (2003). ''Quantum Field Theory in a Nutshell'', Princeton University Press. ~~ISBN 0-691-01019-6.~~

	~~[[Category:Quantum field theory]]~~
	~~[[Category:Spinors]]~~
	~~[[Category:Mathematical notation]]~~

en>Mark viking: /* See also */ Removed scale space, as it is already linked to above

2014-01-15T21:30:29Z

See also: Removed scale space, as it is already linked to above

← Older revision		Revision as of 23:30, 15 January 2014
Line 1:		Line 1:
			In [[quantum field theory]], the '''Dirac adjoint''' defines the [[dual vector space\|dual]] operation of a [[Dirac spinor]]. The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors. Since the usual [[Hermitian adjoint]] lacks the Lorentz symmetry of the system, the Dirac adjoint must be used instead.

			Possibly to avoid confusion with the usual [[Hermitian adjoint]], some textbooks do not provide a name for the Dirac adjoint but simply call it "<var>ψ</var>-bar".

	~~Chances are they call me Gabrielle. I am~~ a ~~particular cashier and I'm trying pretty good financially~~. ~~As an important girl what I sincerely like~~ is ~~going on to~~ [~~http:~~//~~Www~~.~~Alexa.com/search?q~~=~~karaoke&r~~=~~topsites_index&p~~=~~bigtop karaoke~~] ~~but I~~ in ~~no way made a dime going without running shoes~~. ~~My husband and I select to reside~~ in ~~Guam but I will attain~~ to ~~move in~~ a ~~suitable year or two~~. ~~See what precisely new on my web blog here~~: ~~http~~://~~prometeu~~.~~net~~<br><br>~~Also visit my webpage;~~ [~~http~~://~~prometeu~~.~~net clash of clans hack ifunbox~~]		== Definition ==

			Let <var>ψ</var> be a [[Dirac spinor]]. Then its Dirac adjoint is defined as

			:<math>\bar\psi \equiv \psi^\dagger \gamma^0</math>

			where <var>ψ</var><sup>†</sup> denotes the [[Hermitian adjoint]] of the spinor <var>ψ</var> and <var>γ</var><sup>0</sup> is the time-like [[gamma matrices\|gamma matrix]].

			== Spinors under Lorentz transformations ==

			The [[Lorentz group]] of [[special relativity]] is not [[compact space\|compact]], therefore [[Representations of the Lorentz group\|representation]]s of [[Lorentz transformation]]s in the Dirac spinor space are not [[unitary operator\|unitary]]. That is, in general,

			:<math>\lambda^\dagger \ne \lambda^{-1}</math>

			where <var>λ</var> is the corresponding Lorentz transformation that maps spinors:

			:<math>\psi \mapsto \lambda \psi</math>.

			The Hermitian adjoint of spinors transforms according to

			:<math>\psi^\dagger \mapsto \psi^\dagger \lambda^\dagger</math>.

			Therefore, using only the Hermitian adjoint, one finds that {{nowrap\|<var>ψ</var><sup>†</sup> <var>ψ</var>}} is not a [[Lorentz scalar]] and {{nowrap\|<var>ψ</var><sup>†</sup> <var>γ</var><sup><var>μ</var></sup> <var>ψ</var>}} is not even [[self-adjoint operator\|Hermitian]].

			Using the definition, one finds that the Dirac adjoint of spinors transforms according to

			:<math>\bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0</math>.

			Using the identity {{nowrap\|<var>γ</var><sup>0</sup> <var>λ</var><sup>†</sup> <var>γ</var><sup>0</sup> {{=}} <var>λ</var><sup>-1</sup>}}, the transformation reduces to

			:<math>\bar\psi \mapsto \bar\psi \lambda^{-1}</math>,

			which possesses the required Lorentz symmetry for {{nowrap\|{{overline\|<var>ψ</var>}} <var>ψ</var>}} and {{nowrap\|{{overline\|<var>ψ</var>}} <var>γ</var><sup><var>μ</var></sup> <var>ψ</var>}}.

			== Usage ==

			Using the Dirac adjoint, the [[probability current\|probability four-current]] <var>'''J'''</var> for a spin-1/2 particle field can be written as

			:<math>J^\mu = c \bar\psi \gamma^\mu \psi</math>

			where <var>c</var> is the speed of light and the components of <var>'''J'''</var> represent the probability density <var>ρ</var> and the probability 3-current <var>'''j'''</var>:

			:<math>\boldsymbol J = (c \rho, \boldsymbol j)</math>.

			Taking {{nowrap\|<var>μ</var> {{=}} 0}} and using the relation for [[gamma matrices]]

			:<math>\left(\gamma^0\right)^2 = I</math>,

			the probability density becomes

			:<math>\rho = \psi^\dagger \psi</math>.

			== See also ==

			*[[Dirac equation]]
			*[[Rarita-Schwinger equation]]

			== References ==

			*B. Bransden and C. Joachain (2000). ''Quantum Mechanics'', 2e, Pearson. ISBN 0-582-35691-1.
			*M. Peskin and D. Schroeder (1995). ''An Introduction to Quantum Field Theory'', Westview Press. ISBN 0-201-50397-2.
			*A. Zee (2003). ''Quantum Field Theory in a Nutshell'', Princeton University Press. ISBN 0-691-01019-6.

			[[Category:Quantum field theory]]
			[[Category:Spinors]]
			[[Category:Mathematical notation]]

en>Accelerometer: Undid revision 509445381 by 98.227.140.142 (talk)

2012-08-27T16:38:35Z

Undid revision 509445381 by 98.227.140.142 (talk)

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