Dirac adjoint

Template:Short description 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 "Template:Math-bar".

Definition

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

$\overline{\psi }\equiv {\psi }^{†}{\gamma }^0$

where ${\psi }^{†}$ 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,

${\lambda }^{†}\ne {\lambda }^{-1}.$

The Hermitian adjoint of a spinor transforms according to

${\psi }^{†}\mapsto {\psi }^{†}{\lambda }^{†}.$

Therefore, ${\psi }^{†}\psi$ is not a Lorentz scalar and ${\psi }^{†}{\gamma }^{\mu }\psi$ is not even Hermitian.

Dirac adjoints, in contrast, transform according to

$\overline{\psi }\mapsto {\left(\lambda \psi \right)}^{†}{\gamma }^0.$

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

$\overline{\psi }\mapsto \overline{\psi }{\lambda }^{-1},$

Thus, $\overline{\psi }\psi$ transforms as a Lorentz scalar and $\overline{\psi }{\gamma }^{\mu }\psi$ as a four-vector.

Usage

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

$J^{\mu }=c\overline{\psi }{\gamma }^{\mu }\psi$

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

$𝑱=(c\rho ,𝒋).$

Taking Template:Nowrap and using the relation for gamma matrices

${\left({\gamma }^0\right)}^2=I,$

the probability density becomes

$\rho ={\psi }^{†}\psi .$

See also

• Dirac equation
• Rarita–Schwinger equation

References

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