- Dirac adjoint
In quantum field theory, the Dirac adjoint of a Dirac spinor is defined to be the dual spinor , where is the time-like gamma matrix. Possibly to avoid confusion with the usual Hermitian adjoint , some textbooks do not give a name to the Dirac adjoint, simply calling it "psi-bar".
The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors. For example, is not a Lorentz scalar, and is not even Hermitian. One source of trouble is that if λ is the spinor representation of a Lorentz transformation, so that
Using the Dirac adjoint, the conserved probability four-current density for a spin-1/2 particle field
where is the probability density and j the probability current 3-density can be written as
where c is the speed of light. Taking μ = 0 and using the relation for Gamma matrices
the probability density becomes
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