Dirac adjoint

Definition

Let \psi be a Dirac spinor. Then its Dirac adjoint is defined as

:\bar\psi \equiv \psi^\dagger \gamma^0

where \psi^\dagger denotes the Hermitian adjoint of the spinor \psi, and \gamma^0 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 \lambda is a projective representation of some Lorentz transformation,

:\psi \mapsto \lambda \psi,

then, in general,

:\lambda^\dagger \ne \lambda^{-1}.

The Hermitian adjoint of a spinor transforms according to

:\psi^\dagger \mapsto \psi^\dagger \lambda^\dagger.

Therefore, \psi^\dagger\psi is not a Lorentz scalar and \psi^\dagger\gamma^\mu\psi is not even Hermitian.

Dirac adjoints, in contrast, transform according to

:\bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0.

Using the identity \gamma^0 \lambda^\dagger \gamma^0 = \lambda^{-1}, the transformation reduces to

:\bar\psi \mapsto \bar\psi \lambda^{-1},

Thus, \bar\psi\psi transforms as a Lorentz scalar and \bar\psi\gamma^\mu\psi 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

:J^\mu = c \bar\psi \gamma^\mu \psi

where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j:

:\boldsymbol J = (c \rho, \boldsymbol j).

Taking and using the relation for gamma matrices

:\left(\gamma^0\right)^2 = I,

the probability density becomes

:\rho = \psi^\dagger \psi.

References