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Revision as of 16:34, 3 July 2019 editIncnis Mrsi (talk | contribs)Extended confirmed users, Pending changes reviewers, Rollbackers11,646 edits →Lorentz transformations: new section← Previous edit Revision as of 22:16, 3 July 2019 edit undoXxanthippe (talk | contribs)Extended confirmed users, Pending changes reviewers20,432 edits Reverted good faith edits by Incnis Mrsi (talk): No OR please (TW)Tag: UndoNext edit →
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:<math> \rho = \psi^\dagger\psi\,</math> . :<math> \rho = \psi^\dagger\psi\,</math> .

== Lorentz transformations ==

A spinor (of any kind) technically can’t be respected by Lorentz transformations&nbsp;– respective reps of the ] are ], namely, are defined up to ±1. Spinors must be transformed under the suitable ], such as {{math|SL(2, ℂ) ≅ Spin<sup>+</sup>(1,3)}} for this case. Note that {{math|O<sup>+</sup>(1,3) ≅ ](2, ℂ)}} and is doubly covered by {{math|SL(2, ℂ)}}. ] (]) 16:34, 3 July 2019 (UTC)

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Unless there are objections, I shall replace the Usage section with the below. The probability density is incorrect in the current version. Some formatting may need to be improved. Xxanthippe 05:56, 25 February 2007 (UTC)

Usage

Using the Dirac adjoint, the conserved probability four-current density for a spin-1/2 particle field

j μ = ( c ρ , j ) {\displaystyle j^{\mu }=(c\rho ,j)\,}

where ρ {\displaystyle \rho \,} is the probability density and j the probability current 3-density can be written as

j μ = c ψ ¯ γ μ ψ {\displaystyle j^{\mu }=c{\bar {\psi }}\gamma ^{\mu }\psi }

where c is the speed of light. Taking μ = 0 {\displaystyle \mu =0} and using the relation for Gamma matrices

( γ 0 ) 2 = I {\displaystyle \left(\gamma ^{0}\right)^{2}=I\,}

the probability density becomes

ρ = ψ † ψ {\displaystyle \rho =\psi ^{\dagger }\psi \,} .
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