Dirac adjoint


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".

Definition

Let ψ be a Dirac spinor. Then its Dirac adjoint is defined as
where ψ† denotes the Hermitian adjoint of the spinor ψ and γ0 is the time-like gamma matrix.

Spinors under Lorentz transformations

The Lorentz group of special relativity is not compact, therefore spinor representations of Lorentz transformations are generally not unitary. That is, if λ is a projective representation of some Lorentz transformation,
then, in general,
The Hermitian adjoint of a spinor transforms according to
Therefore, is not a Lorentz scalar and is not even Hermitian.
Dirac adjoints, in contrast, transform according to
Using the identity, the transformation reduces to
Thus, transforms as a Lorentz scalar and as a four-vector.

Usage

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 and using the relation for gamma matrices
the probability density becomes