In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor.
The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors, replacing the usual role of the Hermitian adjoint.
Possibly to avoid confusion with the usual Hermitian adjoint, some textbooks do not provide a name for the Dirac adjoint but simply call it "ψ-bar".
Then its Dirac adjoint is defined as where
denotes the Hermitian adjoint of the spinor
is the time-like gamma matrix.
The Lorentz group of special relativity is not compact, therefore spinor representations of Lorentz transformations are generally not unitary.
is a projective representation of some Lorentz transformation, then, in general, The Hermitian adjoint of a spinor transforms according to Therefore,
Dirac adjoints, in contrast, transform according to Using the identity
transforms as a Lorentz scalar and
Using the Dirac adjoint, the probability four-current J for a spin-1/2 particle field can be written as where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j: Taking μ = 0 and using the relation for gamma matrices the probability density becomes