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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

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

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

${\displaystyle j^{\mu }=c{\bar {\psi }}\gamma ^{\mu }\psi }$

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

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

the probability density becomes

${\displaystyle \rho =\psi ^{\dagger }\psi \,}$ .

Lorentz transformations

A spinor (of any kind) technically can’t be respected by Lorentz transformations – respective reps of the Lorentz group are projective, namely, are defined up to ±1. Spinors must be transformed under the suitable Spin group, such as SL(2, ℂ) ≅ Spin(1,3) for this case. Note that O(1,3) ≅ PSL(2, ℂ) and is doubly covered by SL(2, ℂ). Incnis Mrsi (talk) 16:34, 3 July 2019 (UTC)

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