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. 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 "ψ-bar".
Let ψ be a Dirac spinor. Then its Dirac adjoint is defined as
Spinors under Lorentz transformations
where λ is the corresponding Lorentz transformation that maps spinors:
The Hermitian adjoint of spinors transforms according to
Using the definition, one finds that the Dirac adjoint of spinors transforms according to
Using the identity γ0 λ† γ0 = λ-1, the transformation reduces to
which possesses the required Lorentz symmetry for ψ ψ and ψ γμ ψ.
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