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 $\psi$ 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 }\neq \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 μ = 0 and using the relation for gamma matrices

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

,

the probability density becomes

\rho =\psi ^{\dagger }\psi

.

References

B. Bransden and C. Joachain (2000). Quantum Mechanics, 2e, Pearson. ISBN 0-582-35691-1.
M. Peskin and D. Schroeder (1995). An Introduction to Quantum Field Theory, Westview Press. ISBN 0-201-50397-2.
A. Zee (2003). Quantum Field Theory in a Nutshell, Princeton University Press. ISBN 0-691-01019-6.