Dirac adjoint

Dual to the Dirac spinor

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 ${\displaystyle \psi }$ be a Dirac spinor. Then its Dirac adjoint is defined as

${\displaystyle {\bar {\psi }}\equiv \psi ^{\dagger }\gamma ^{0}}$

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

${\displaystyle \psi \mapsto \lambda \psi }$,

then, in general,

${\displaystyle \lambda ^{\dagger }\neq \lambda ^{-1}}$.

The Hermitian adjoint of a spinor transforms according to

${\displaystyle \psi ^{\dagger }\mapsto \psi ^{\dagger }\lambda ^{\dagger }}$.

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

Dirac adjoints, in contrast, transform according to

${\displaystyle {\bar {\psi }}\mapsto \left(\lambda \psi \right)^{\dagger }\gamma ^{0}}$.

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

${\displaystyle {\bar {\psi }}\mapsto {\bar {\psi }}\lambda ^{-1}}$,

Thus, ${\displaystyle {\bar {\psi }}\psi }$ transforms as a Lorentz scalar and ${\displaystyle {\bar {\psi }}\gamma ^{\mu }\psi }$ 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

${\displaystyle J^{\mu }=c{\bar {\psi }}\gamma ^{\mu }\psi }$

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

${\displaystyle {\boldsymbol {J}}=(c\rho ,{\boldsymbol {j}})}$.

Taking μ = 0 and using the relation for gamma matrices

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

the probability density becomes

${\displaystyle \rho =\psi ^{\dagger }\psi }$.

See also

• Dirac equation
• Rarita–Schwinger equation

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.