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 "-bar".

Definition

be a Dirac spinor. Then its Dirac adjoint is defined as

denotes the Hermitian adjoint is the time-like gamma matrix.

Spinors under Lorentz transformations

The Lorentz group of special relativity is not compact, therefore spinor representationss 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

is not a Lorentz scalar is not even Hermitian.

Dirac adjoints, in contrast, transform according to

, the transformation reduces to

as a four-vector.

Usage

Using the Dirac adjoint, the probability four-current for a spin-1/2 particle field can be written as

where is the speed of light and the components of represent the probability density and the probability 3-current :

Taking and using the relation for gamma matrices

the probability density becomes

See also

• Dirac equation

• Rarita–Schwinger equation

References