Dirac adjoint

In quantum field theory, the Dirac adjoint $\bar\psi$ of a Dirac spinor $\ \psi$ is defined to be the dual spinor $\ \psi^{\dagger} \gamma^0$, where $\ \gamma^0$ is the time-like gamma matrix. Possibly to avoid confusion with the usual Hermitian adjoint $\psi^\dagger$, some textbooks do not give a name to the Dirac adjoint, simply calling it "psi-bar".

Motivation

The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors. For example, $\psi^\dagger\psi$ is not a Lorentz scalar, and $\psi^\dagger\gamma^\mu\psi$ is not even Hermitian. One source of trouble is that if λ is the spinor representation of a Lorentz transformation, so that

$\psi\to\lambda\psi,$

then

$\psi^\dagger\to\psi^\dagger\lambda^\dagger.$

Since the Lorentz group of special relativity is not compact, λ will not be unitary, so $\lambda^\dagger\neq\lambda^{-1}$. Using $\bar\psi$ fixes this problem, in that it transforms as

$\bar\psi\to\bar\psi\lambda^{-1}.$

Usage

Using the Dirac adjoint, the conserved probability four-current density for a spin-1/2 particle field

$j^\mu = (c\rho, j)\,$

where $\rho\,$ is the probability density and j the probability current 3-density can be written as

$j^\mu = c\bar\psi\gamma^\mu\psi$

where c is the speed of light. Taking μ = 0 and using the relation for Gamma matrices

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

the probability density becomes

$\rho = \psi^\dagger\psi\,$ .

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.